Research Article Local Morrey and Campanato Spaces on...

Research ArticleLocal Morrey and Campanato Spaces onQuasimetric Measure Spaces

Krzysztof Stempak1 and Xiangxing Tao2

1 Instytut Matematyki i Informatyki Politechnika Wrocławska Wybrzeze Wyspianskiego 27 50-370 Wrocław Poland2Department of Mathematics Zhejiang University of Science and Technology Hangzhou Zhejiang 310023 China

Correspondence should be addressed to Xiangxing Tao xxtaozusteducn

Received 17 February 2014 Accepted 15 April 2014 Published 25 May 2014

Academic Editor Dachun Yang

Copyright copy 2014 K Stempak and X Tao This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We define and investigate generalized local Morrey spaces and generalized local Campanato spaces within a context of a generalquasimetric measure spaceThe locality is manifested here by a restriction to a subfamily of involved ballsThe structural propertiesof these spaces and the maximal operators associated to them are studied In numerous remarks we relate the developed theorymostly in the ldquoglobalrdquo case to the cases existing in the literature We also suggest a coherent theory of generalized Morrey andCampanato spaces on open proper subsets of R119899

1 Introduction

Aquasimetric on a nonempty set119883 is amapping119889 119883times119883 rarr[0infin) which satisfies the following conditions

(i) for every 119909 119910 isin 119883 119889(119909 119910) = 0 if and only if 119909 = 119910(ii) for every 119909 119910 isin 119883 119889(119909 119910) = 119889(119910 119909)(iii) there is a constant119870 ge 1 such that for every 119909 119910 119911 isin

119883

119889 (119909 119911) le 119870 (119889 (119909 119910) + 119889 (119910 119911)) (1)

The pair (119883 119889) is then called a quasimetric space if 119870 = 1then 119889 is a metric and (119883 119889) is a metric space

Given 119903 gt 0 and 119909 isin 119883 let

119861 (119909 119903) = 119910 isin 119883 119889 (119909 119910) lt 119903 (2)

be the ldquoquasimetricrdquo ball related to 119889 of radius 119903 and withcenter 119909 If (119883 119889) is a quasimetric space then T

119889 the

topology in 119883 induced by 119889 is canonically defined bydeclaring 119866 sub 119883 to be open that is 119866 isin T

119889 if and only

if for every 119909 isin 119866 there exists 119903 gt 0 such that 119861(119909 119903) sub 119866 (atthis point one easily checks directly that the topology axiomsare satisfied for such a definition note however that the balls

themselvesmay not be open sets)Observe that this definitionis consistent with the definition of metric topology in casewhen 119889 is a genuine metric Moreover the topology T

119889is

metrizable see for instance [1] for referencesTwo quasimetrics 119889 and 1198891015840 on119883 are said to be equivalent

if 119888minus1

1198891015840

(119909 119910) le 119889(119909 119910) le 1198881198891015840

(119909 119910) with some 119888 ge 1 beingindependent of 119909 119910 isin 119883 It is clear that for equivalentquasimetrics induced topologies coincide Moreover for any119886 gt 0 119889119886 is a quasimetric as well and T

119889= T

119889119886 A

quasimetric 119889 is called a 119902-metric for 0 lt 119902 le 1 providedthat

119889 (119909 119911) le (119889(119909 119910)119902

+ 119889(119910 119911)119902

)1119902 (3)

holds uniformly in 119909 119910 119911 isin 119883 It is easily checked that a 119902-metric enjoys the open ball property that is every ball relatedto 119889 is an open set in (119883T

119889) It is also known (see [1]) that

given 119889 for 119902 determined by the equality (2119870)119902 = 2 119889119902

defined by

119889119902(119909 119910)

= inf

119899

sum

119895=1

119889(119909119895minus1 119909

119895)119902

119909 = 1199090 119909

1 119909

119899= 119910 119899 ge 1

(4)

Hindawi Publishing CorporationJournal of Function SpacesVolume 2014 Article ID 172486 15 pageshttpdxdoiorg1011552014172486

2 Journal of Function Spaces

is a metric on 119883 which is equivalent to 119889119902 more precisely119889

119902le 119889

119902

le 4119889119902 Consequently 119889

(119902)= (119889

119902)1119902 is a 119902-metric

equivalent with 119889 more precisely 119889(119902)

le 119889 le 41119902

119889(119902)

Thus every quasimetric admits an equivalent 119902-metric thatpossesses the open ball property

In what follows if (119883 119889) is a given quasimetric spacethen 119883 is considered as a topological space equipped withthe (metrizable) topology T

119889 It may happen that a ball in

119883 is not a Borel set (ie it does not belong to the Borel 120590-algebra generated byT

119889) see for instance [1] as an example

To avoid such pathological cases the assumption that allballs are Borel sets must be made Then if 119883 is additionallyequipped with a Borel measure 120583 which is finite on boundedsets and nontrivial in the sense that 120583(119883) gt 0 we say that(119883 119889 120583) is a quasimetric measure space (we do not assumethat 120583(119861) gt 0 for every ball 119861) In this paper we additionallyassume (similar to the assumption (13) made in [2]) that

all balls in 119883 are open (5)

taking into account what was mentioned above this assump-tion does not narrow the generality of our considerations

Let (119883 119889 120583) be a quasimetric measure space Define thefunction 120588

0 119883 rarr [0infin) by setting

1205880(119909) = inf 120576 120583 (119861 (119909 120576)) gt 0 119909 isin 119883 (6)

Observe that if 1205880(119909) gt 0 for some 119909 isin 119883 then

120583(119861(119909 1205880(119909))) = 0 this is a consequence of the continuity

property from below of the measure 120583 The property ldquo120583(119861) gt0 for every ball 119861rdquo is equivalent with the statement that 120588

0equiv

0Given a function 120588 119883 rarr (0infin] such that 120588

0(119909) lt 120588(119909)

for every 119909 isin 119883 let B120588(119909) = B

120588119889(119909) denote the family of

balls (related to 119889) centered at 119909 and with radius 119903 satisfying1205880(119909) lt 119903 lt 120588(119909) (clearly balls with different radii but which

coincide are identified as sets) Then we set

B120588=B

120588119889= ⋃

119909isin119883

B120588(119909) (7)

Thus B120588denotes the family of all 120588-local balls in 119883 with

positive measure In case the lower estimate on the radius1205880(119909) lt 119903 is disregarded we shall write B

120588for the resulting

family of ballsBy a 120588-local integrability of a real or complex-valued

function on 119883 we mean its integrability with respect to thefamily of balls fromB

120588 thus 119891 isin 1198711

loc120588(119883) = 1198711

loc120588(119883 119889 120583)

provided that int119861

|119891|119889120583 lt infin for every ball 119861 isin B120588(and

thus also for every 119861 isin B120588) Note that this notion of local

integrability does not refer to compactness Similarly for 1 le119901 lt infin we define 119871119901

loc120588(119883) = 119891 |119891|119901

isin 1198711

loc120588(119883)If 120588(119909) =infin for some 119909 isin 119883 then we will refer to 120588 as

a locality function and to objects associated to 120588 as ldquolocalrdquoobjects If 120588 equiv infin identically then we shall skip the infinsubscript writing B 1198711

loc(119883) 119872119901120601(119883) L

119901120601(119883) and so on

(thus B denotes the family of all balls in 119883) and refer to thissetting as to the global one Notice that the proofs of all resultsstated in the paper contain 120588 = infin as a special case

Parallel to the main theory we shall also develop analternative theory in the framework of closed balls 119861(119909 119903) =119910 isin 119883 119889(119909 119910) le 119903 Note that in the metric case 119861(119909 119903)is indeed a closed set and in general if all balls are assumedto be Borel sets then 119861(119909 119903) is Borel too The definitions ofMorrey and Campanato spaces based on closed balls (in factbeing closed cubes) in the framework of (R119899

119889(infin)

120583)occur inthe literature compare for instance [3] Clearly taking closedballs makes no difference with respect to the theory based onopen balls when 120583 has the property that 120583(120597119861) = 0 for everyball 119861 where 120597119861 = 119861 119861 this happens for instance when119889120583(119909) = 119908(119909)119889119909 where 119908 ge 0 and 119889119909 denotes Lebesguemeasure onR119899 In general however the two alternative waysmay give different outcomes Relevant comments indicatingcoincidences or differences of both theories will be given inseveral places

The general notion of local maximal operators was intro-duced in [4] and some objects associated to them mostly theBMO spaces were investigated there in the setting ofmeasuremetric spaces The present paper enhances investigationdone in [4] in several directions First the broader contextof quasimetric measure spaces is considered Second thecondition 120583(119861) gt 0 for every ball 119861 is not assumedThird several variants of generalized maximal operators areadmitted into our investigation All this makes the developedtheory more flexible in possible applications

Throughout the paper we use a standard notation Whilewriting estimates we use the notation 119878 ≲ 119879 to indicate that119878 le 119862119879 with a positive constant 119862 independent of significantquantities We shall write 119878 ≃ 119879 when simultaneously 119878 ≲ 119879and 119879 ≲ 119878 for instance 119889 ≃ 1198891015840 means the equivalence ofquasimetrics 119889 and 1198891015840 and so forth By 119871119901

(119883) = 119871119901

(119883 120583)1 le 119901 lt infin we shall denote the usual Lebesgue 119871119901 spaceon the measure space (119883 120583) Whenever we refer to a ball weunderstand that its center and radius have been chosen (ingeneral these need not be uniquely determined by 119861 as a set)Thenwriting 120591119861 for a given ball119861 = 119861(119909 119903) and 120591 gt 0 meansthat 120591119861 = 119861(119909 120591119903) For a function 119891 isin 1198711

loc120588(119883) its averagein a ball 119861 = 119861(119909 119903) isinB

120588will be denoted by

⟨119891⟩119861=

1

120583 (119861)int119861

119891119889120583 (8)

and similarly for any other Borel set 119860 0 lt 120583(119860) lt infinand any 119891 whenever the integral makes sense When thesituation is specified to the Euclidean setting of R119899 we shallconsider either the metric 119889(2) induced by the norm sdot

2or

119889(infin) induced by sdot

infin

2 Generalized Local Maximal Operators

By defining and investigating generalized local Morrey andCampanato spaces on quasimetric measure spaces we adaptthe general approach to these spaces presented by Nakai [2](and follow the notation used there) and extend the conceptof locality introduced in [4] Also we find it more convenientto work with relevant maximal operators when investigating

Journal of Function Spaces 3

the aforementioned spaces An interesting concept of local-ization of Morrey and Campanato spaces on metric measurespaces recently appeared in [5] this concept is howeverdifferent from our concept On the other hand the concept oflocality forMorrey andCampanato spaces onmetricmeasurespaces that appeared in the recent paper [6] is consistent withthe one we develop see Remark 15 for further details

Let 120601 be a positive function defined on B120588 In practice

120601 will be usually defined on B the family of all balls in119883 Then a tempting alternative way of thinking about 120601 isto treat it as a function 120601 119883 times R

+rarr R

+and then to

define 120601(119861) = 120601(119909 119903) for 119861 = 119861(119909 119903) There is however apitfall connected with the fact that in general the mapping119883 times R

+ni (119909 119903) 997891rarr 119861(119909 119903) isin B is not injective Hence we

assume that 120601 possesses the following property

120601 (1199091 119903

1) = 120601 (119909

2 119903

2) whenever 119861 (119909

1 119903

1) = 119861 (119909

2 119903

2)

(9)

(Thus for instance when119883 is bounded ie diam(119883) = 119877119883lt

infin the function 120601 must obey the following rule for every119909

1 119909

2isin 119883 and 119903

1 119903

2gt 119877

119883 120601(119909

1 119903

1) = 120601(119909

2 119903

2))

Clearly working with a general 120601 cannot lead to fully sat-isfactory results Therefore in what follows we shall imposesome additionalmild (andnatural) assumptions on120601 in orderto develop the theory Frequently in such assumptions 120601and 120583 will be interrelated Of particular interest will be thefunctions

120601119898120572(119861) = 120583 (119861)

120572

120601119903120572(119861) = 119903(119861)

120572

(10)

where 120572 isin R and 119903(119861) denotes the radius of 119861 (the 119898 and 119903stand for measure and radius resp) It is necessary to pointout here that for the second function in fact we considera selector 119861 997891rarr 119903(119861) assigning to any 119861 one of its possibleradii (clearly this subtlety does not occur when for instance119883 = R119899) We shall frequently test the constructed theory onthese two functions Finally let usmention that itmay happenthat for a constant 119899 gt 0 (playing the role of the dimension)we have

120583 (119861) ≃ 119903(119861)119899

(11)

uniformly in 119861 isinB120588 Then

120601119898120572(119861) ≃ 120601

119903120572(119861)

119899

119861 isinB120588 (12)

Let the system (119883 119889 120583 120588 120601) be given In what followsby an admissible function on 119883 we mean either a Borelmeasurable complex-valued function (when the complexcase is considered) or a Borelmeasurable functionwith valuesin the extended real number system R = R cup plusmninfin (whenthe real case is investigated) Given 1 le 119901 lt infin we define thegeneralized local fractional maximal operator 119872

119901120601120588acting

on any admissible 119891 by

119872119901120601120588119891 (119909) = sup

119909isin119861isinB120588

1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(13)

where the supremum is taken over all the balls from B120588

which contain 119909 and its centered version by

119872119888

119901120601120588119891 (119909) = sup

1205880(119909)lt119903lt120588(119909)

1

120601 (119861 (119909 119903))

times (1

120583 (119861 (119909 119903))int119861(119909119903)

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(14)

On the other hand we define the generalized local sharpfractional maximal operator119872

119901120601120588for any admissible 119891 by

119872119901120601120588119891 (119909) = sup

119909isin119861isinB120588

inf119911isinC

1

120601 (119861)

times (1

120583 (119861)int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(15)

and similarly for its centered version119872119888119901120601120588

(If spaces of realfunctions are considered then the infimum is taken over 119911 isinR the analogous agreement applies in similar places)

An alternative way of defining the local sharp maximaloperator is

119901120601120588119891 (119909) = sup

119909isin119861isinB120588

1

120601 (119861)

times (1

120583 (119861)int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(16)

but this makes sense only for119891 isin 1198711

loc120588(119883) Similar commentapplies to the analogous definition of 119888

119901120601120588 Clearly

119872119901120601120588119891 (119909) ≃

119901120601120588119891 (119909) 119872

119888119901120601120588119891 (119909) ≃

119888119901120601120588119891 (119909)

(17)

uniformly in 119891 isin 1198711

loc120588(119883) and 119909 isin 119883 Observe an advantageof using119872

119901120601120588instead of

119901120601120588

119901120601120588119891 is defined for 119891 isin

1198711

loc120588(119883) only while 119872119901120601120588119891 makes sense for much wider

class of admissible functionsFor 120601 = 1 that is when 120601 equiv 1 the maximal operators

11987211120588 119872

11120588 and

11120588 and their centered counterparts

were defined and investigated in [4] (in the setting of ametricmeasure space in addition satisfying 120583(119861) gt 0 for every ball119861)

Another property to be immediately noted is

1198721199011120601120588119891 (119909) le 119872

1199012120601120588119891 (119909) 119909 isin 119883 (18)

that holds for 1 le 1199011lt 119901

2lt infin by an application ofHolderrsquos

inequality similar relation is valid for119872119901120601120588

and 119901120601120588

andfor the centered versions of the three operators


Finally in case of considering maximal operators basedon closed balls we shall use the notations119872

119901120601120588119872

119901120601120588 and

so forth To be precise the definition of119872119901120601120588

is

119872119901120601120588119891 (119909) = sup

119909isin119861isinB120588

1

120601 (119861)

(1

120583 (119861)

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(19)

where B120588denotes the family of all closed balls 119861 = 119861(119909 119903)

such that 1205880(119909) lt 119903 lt 120588(119909) and similarly for other maximal

operators considered above Note that if 1205880(119909) = inf 120576

120583(119861(119909 120576)) gt 0 then 1205880(119909) = 120588

0(119909) this is a consequence

of continuity property of the measure 120583

Remark 1 It may be worth mentioning that the following(local) variant of the Hardy-Littlewood maximal operator

120581120588119891 (119909) = sup

119909isin119861isinB120588

1

120583 (120581119861)int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 119909 isin 119883 (20)

and its centered version 119888

120581120588 where 120581 gt 1 is given both

fall within the scheme presented here 120581120588

coincides with119872

1120601120581120588 where 120601

120581(119861) = 120583(120581119861)120583(119861) See [7 p 493] where

120588 equiv infin and 120581 = 5 [8 p 126] where 120588 equiv infin 120581 gt 1 and 120581

is considered in the setting of R119899 and closed cubes and [9p 469] where 120588 equiv infin 120581 = 3 and 119888

3is considered in the

setting of R119899 and open (Euclidean) balls This variant is animportant substitute of the usual Hardy-Littlewood maximaloperator (the limiting case of 120581 = 1) and is used frequentlyin the nondoubling case Analogous comment concerns yetanother variant of the Hardy-Littlewood maximal operator

119901120581120588119891 (119909) = (

120581120588(10038161003816100381610038161198911003816100381610038161003816

119901

) (119909))1119901

119909 isin 119883 (21)

1 le 119901 lt infin (see [9 p 470] where its centered version isconsidered for 120588 equiv infin and 120581 = 3) Also the local fractionalmaximal operator

119872(120572)

119891 (119909) = sup119909isin119861isinB

120588

1

120583(119861)1minus120572120591

int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 119909 isin 119883 (22)

where 120583 is a Borel measure on119883 satisfying the upper growthcondition

120583 (119861 (119909 119903)) ≲ 119903120591

(23)

for some 0 lt 120591 le 119899 with 119899 playing the role of a dimensionuniformly in 119903 gt 0 and 119909 isin 119883 (if119883 = R119899 120583 is Lebesgue mea-sure and 120591 = 119899 then119872(120572) is the classical fractional maximaloperator) is covered by the presented general approach since119872

(120572) coincides with1198721120601119898minus120572120591

Finally a mixture of both

119872(120572)

120581119901119891 (119909) = sup

119909isin119861isinB120588

(1

120583(120581119861)1minus119901120572120591

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(24)

considered in [10] in the setting of (R119899

119889(infin)

) coincides with119872

119901120601120581

where

120601120581(119861) = 120601

120581119901120572(119861) = (

120583(120581119861)1minus119901120572120591

120583 (119861))

1119901

(25)

An interesting discussion of mapping properties of(global) fractional maximal operators in Sobolev and Cam-panato spaces in measure metric spaces equipped with adoubling measure 120583 in addition satisfying the lower boundcondition 120583(119861(119909 119903)) ≳ 119903120591 is done by Heikkinen et al in [11]Investigation of local fractional maximal operators (from thepoint of viewof their smoothing properties) defined in propersubdomains of the Euclidean spaces was given by Heikkinenet al in [12] See also comments at the end of Section 31

The following lemma enhances [4 Lemmas 21 and 31]By treating the centered case we have to impose someassumptions on 120588

0 120588 and 120601 Namely we assume that 120588

0is an

upper semicontinuous function (usc for short) 120588 is a lowersemicontinuous function (lsc for short) and 120601 satisfies

forall119903 gt 0 the mapping 119883 ni 119910 997891997888rarr 120601 (119910 119903) is usc (26)

It may be easily checked that in case 119889 is a genuine metric 1205880

is usc and 120601119898120572

120572 isin R satisfies (26)

Lemma2 For any admissible119891 and 1 le 119901 lt infin the functions119872

119901120601120588119891119872119888

119901120601120588119891119872

119901120601120588119891 and119872119888

119901120601120588119891 are lsc hence Borel

measurable and the same is true for 119901120601120588119891 and 119888

119901120601120588119891

when 119891 isin 1198711

loc120588(119883)

Proof In the noncentered case no assumption on 1205880 120588 and 120601

is required Indeed fix 119891 consider the level set 119865120582= 119865

120582(119891) =

119909 isin 119883 119872119901120601120588119891(119909) gt 120582 and take a point 119909

0from this set

This means that there exists a ball 119861 isin B120588such that 119909

0isin 119861

and

1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

gt 120582 (27)

But the same ball 119861 considered for any 119910 isin 119861 also gives119872

119901120601120588119891(119910) gt 120582 hence 119861 sub 119865

120582 which shows that the level

set is open Exactly the same argument works for the level set119909 isin 119883

119901120601120588119891(119909) gt 120582 except for the fact that now in

(27) 119891 is replaced by 119891 minus ⟨119891⟩119861 Finally consider the level set

119865120582= 119909 isin 119883 119872

119901120601120588119891(119909) gt 120582 and take a point 119909

0from this

setThere exists a ball 119861 isinB120588and 120576 gt 0 such that 119909

0isin 119861 and

for every 119911 isin C we have (1120601(119861))(⟨|119891 minus 119911|119901⟩119861)1119901

gt 120582+120576 Butthe same ball 119861 is good enough for any 119910 isin 119861 in the sensethat119872

119901120601120588119891(119910) gt 120582 and hence 119861 sub 119865

120582 which shows that

the level set is openIn the centered case we use the assumptions imposed on

1205880 120588 and 120601 For119872119888

119901120601120588 we write the level set 119865119888

120582= 119865

119888

120582(119891) =

119909 isin 119883 119872119888

119901120601120588119891(119909) gt 120582 as a union of open sets

119865119888

120582= ⋃

119903gt0

119910 isin 119883 1

120601 (119861 (119910 119903))(⟨10038161003816100381610038161198911003816100381610038161003816

119901

⟩119861(119910119903)

)

1119901

gt 120582

cap 119910 isin 119883 120588 (119910) gt 119903 gt 1205880(119910)

(28)


Each intersection on the right hand side is an open setIndeed

119910 isin 119883 120588 (119910) gt 119903 gt 1205880(119910) = 119910 isin 119883 120588 (119910) gt 119903

cap 119910 isin 119883 1205880(119910) lt 119903

(29)

is open since by assumption 120588 is lsc and 1205880is usc On the

other hand for every fixed 119903 gt 0 the function

119883 ni 119910 997891997888rarr1

120601 (119861 (119910 119903))(⟨10038161003816100381610038161198911003816100381610038161003816

119901

⟩119861(119910119903)

)

1119901

(30)

is lsc as well To show this note that the limit of anincreasing sequence of lsc functions is a lsc function andhence it suffices to consider 119891 = 120594

119860 120583(119860) lt infin But then

119883 ni 119910 997891997888rarr1

120601 (119861 (119910 119903))

120583(119860 cap 119861 (119910 119903))1119901

120583 (119861 (119910 119903))

(31)

is lsc as a product of three lsc functions 119883 ni 119910 997891rarr

120583(119860 cap 119861(119910 119903))1119901 is lsc by continuity of 120583 from above 119883 ni

119910 997891rarr (1120583(119861(119910 119903))) is lsc by continuity of 120583 from belowand finally 119883 ni 119910 997891rarr 120601(119861(119910 119903))

minus1 is lsc as well by theassumption (26) imposed on 120601

Exactly the same argument works for the level set 119865119888120582=

119909 isin 119883 119888119901120601120588119891(119909) gt 120582 except for the fact that now in

relevant places 119891 has to be replaced by 119891 minus ⟨119891⟩119861 Finally for

the level set 119865119888120582= 119909 isin 119883 119872

119888119901120601120588119891(119909) gt 120582 an argument

similar to that given above combinedwith that used for119872119901120601120588

does the job

To relate maximal operators based on closed balls withthese based on open balls we must assume something moreon the function 120601 Namely we assume that 120601 is defined on theunion B

120588cupB

120588(rather than on B

120588only) and consider the

following continuity condition for every1199100isin 119883 and 120588

0(119910

0) lt

1199030lt 120588(119910

0)

120601 (119861 (1199100 119903

0)) = lim

119903rarr119903+

0

120601 (119861 (1199100 119903))

120601 (119861 (1199100 119903

0)) = lim

119903rarr119903minus

0

120601 (119861 (1199100 119903))

(32)

Note that 120601119898120572

120572 isin R satisfies (32) due to the continuityproperty of measure in particular 120601 equiv 1 satisfies (32)

We then have the following

Lemma 3 Assume that (32) holds Then for 1 le 119901 lt infin wehave

119872119901120601120588119891 (119909) = 119872

119901120601120588119891 (119909) 119891 isin 119871

119901

119897119900119888120588(119883) 119909 isin 119883 (33)

119901120601120588119891 (119909) =

119901120601120588119891 (119909) 119891 isin 119871

119901

119897119900119888120588(119883) 119909 isin 119883 (34)

and the analogous identities for their centered counterpartsConsequently for any 119891 isin 119871119901

119897119900119888120588(119883) the functions 119872

119901120601120588119891

119872119888

119901120601120588119891

119901120601120588119891 and

119888119901120601120588119891 are lsc and hence Borel

measurable

Proof For every 1199100isin 119883 and 119903

0gt 0 we have

119861 (1199100 119903

0) = ⋂

119903gt1199030

119861 (1199100 119903) 119861 (119910

0 119903

0) = ⋃

0lt119903lt1199030

119861 (1199100 119903)

(35)

To prove ge in (33) it is sufficient to check that for any 1198610=

119861(1199100 119903

0) isinB

120588 such that 119909

0isin 119861

0 the following holds

1

120601 (1198610)(

1

120583 (1198610)int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le sup119909isin119861isinB

120588

1

120601 (119861)

(1

120583 (119861)

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(36)

Let 119903119899rarr 119903

minus

0and 119903

119899gt 120588

0(119910

0) Then using the second

part of (32) continuity of 120583 from below and the monotoneconvergence theorem gives

1

120601 (119861 (1199100 119903

119899))

(1

120583 (119861 (1199100 119903

119899))

int119861(1199100119903119899)

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

997888rarr1

120601 (1198610)(

1

120583 (1198610)int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(37)

Similarly to prove le in (33) it suffices to check that for any119861

0= 119861(119910

0 119903

0) isinB

120588 such that 119909

0isin 119861


1

120601 (1198610)

(1

120583 (1198610)

int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901


120588

1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(38)

Let 119903119899rarr 119903

+

0and 119903

119899lt 120588(119910

0)Then using the first part of (32)

continuity of 120583 from above and the dominated convergencetheorem gives

1

120601 (119861 (1199100 119903

119899))(

1

120583 (119861 (1199100 119903

119899))int119861(1199100119903119899)

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

997888rarr1

120601 (1198610)

(1

120583 (1198610)

int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(39)

The proof of (34) follows the line of the proof of (33) withthe additional information that

⟨119891⟩119861(11991001199030)= lim

119899rarrinfin

⟨119891⟩119861(1199100119903119899)

⟨119891⟩119861(11991001199030)= lim

119899rarrinfin

⟨119891⟩119861(1199100119903119899)

(40)

(note that 119871119901

loc120588(119883) sub 1198711

loc120588(119883)) Finally the proofs of thecentered versions go analogously

Given (119883 119889) let 120581119870be the ldquodilation constantrdquo appearing

in the version of the basic covering theorem for a quasimetric


space with a constant 119870 in the quasitriangle inequality see[13Theorem 12] It is easily seen that 120581

119870= 119870(3119870+2) suffices

(so that if119889 is ametric then119870 = 1 and120581119870= 5) (119883 119889) is called

geometrically doubling provided that there exists119873 isin N suchthat every ball with radius 119903 can be covered by at most119873 ballsof radii (12)119903 In the case when (119883 119889 120583) is such that 120588

0equiv 0

we say (cf [4 p 243]) that 120583 satisfies the 120588-local 120591-condition120591 gt 1 provided that

120583 (120591119861) ≲ 120583 (119861) 119861 isinB120588 (41)

In what follows when the 120588-local 120591-condition is invoked wetacitly assume that 120588

0equiv 0

The following lemma enhances [4 Proposition 22]

Proposition 4 Suppose that 120601 and 120583 satisfy one of the follow-ing two assumptions

(i) 1 ≲ 120601 and 120583 satisfies the 120588-local 120581119870-condition

(ii) 120583(120581119870119861)120583(119861) ≲ 120601(119861) uniformly in 119861 isinB

120588 and (119883 119889)

is geometrically doubling

Then1198721120601120588

maps1198711

(119883 120583) into1198711infin

(119883 120583) boundedly and con-sequently119872

1120601120588is bounded on 119871119901

(119883 120583) for any 1 lt 119901 lt infin

Proof The assumption 1 ≲ 120601 simply guarantees that119872

1120601120588≲ 119872

11120588 while the condition 120583(120581119870119861)120583(119861) ≲ 120601(119861)

implies 1198721120601120588

≲ 1120581119870120588 To verify the weak type (1 1)

of both maximal operators in the latter replacement notethat for 119872

11120588 this is simply the conclusion of a versionfor quasimetric spaces of [4 Proposition 22] while for

1120581119870120588 the result is essentially included in [7 Proposition

35] (120581119870replaces 5 and the argument presented in the proof

easily adapts to the local setting) Thus each of the operators119872

11120588 and 1120581119870120588

is bounded on 119871119901

(119883 120583) by applyingMarcinkiewicz interpolation theorem and hence the claimfor119872

1120601120588follows

Remark 5 It is probablyworth pointing out that in the settingof R119899 closed cubes and an arbitrary Borel measure 120583 on R119899

which is finite on bounded sets the maximal operator 120581is

of weak type (1 1) with respect to 120583 and thus is bounded on119871

119901

(120583) for any 120581 gt 1 (since 120581120588le

120581 the same is true

for 120581120588) The details are given in [8 p 127] The same is

valid for open (Euclidean) balls see [14Theorem 16] In [14]Sawano also proved that for an arbitrary separable locallycompact metric space equipped with a Borel measure whichis finite on bounded sets (every such a measure is Radon)for every 120581 ge 2 the associated centered maximal operator

119888

120581is of weak type (1 1) with respect to 120583 and the result

is sharp with respect to 120581 See also Terasawa [15] where thesame result except for the sharpness is proved without theassumption on separability of a metric space but with anadditional assumption on the involved measure

3 Local Morrey and Campanato Spaces

The generalized local Morrey and Campanato spaces in thesetting of the given system (119883 119889 120583 120588 120601)

119871119901120601120588

(119883) = 119871119901120601120588

(119883 119889 120583)

L119901120601120588

(119883) =L119901120601120588

(119883 119889 120583)

(42)

1 le 119901 lt infin are defined by the requirements10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

= sup119909isin119883

119872119901120601120588119891 (119909) lt infin (43)

10038171003817100381710038171198911003817100381710038171003817L119901120601120588

= sup119909isin119883

119872119901120601120588119891 (119909) lt infin (44)

respectively Note that the identities

sup119909isin119883

119872119888

119901120601120588119891 (119909) = sup

119909isin119883

119872119901120601120588119891 (119909)

sup119909isin119883

119872119888119901120601120588119891 (119909) = sup

119909isin119883

119872119901120601120588119891 (119909)

(45)

hold for any admissible 119891 Therefore using the centeredversions of the operators119872

119901120601120588and119872

119901120601120588in (43) and (44)

does not affect the spaces and the norms Also

sup119909isin119883

119888119901120601120588119891 (119909) = sup

119909isin119883

119901120601120588119891 (119909) (46)

and hence using either 119888119901120601120588

or 119901120601120588

in place of 119872119901120601120588

in (44) does not affect the spaces and due to (17) thenorms remain equivalent It is also worth noting that in thedefinitions of the spaces 119871

119901120601120588(119883) and L

119901120601120588(119883) a priori

we do not require 119891 to belong to 119871119901

loc120588(119883) but a posterioriindeed 119871

119901120601120588(119883) sub 119871

119901

loc120588(119883) andL119901120601120588(119883) sub 119871

119901

loc120588(119883)Other properties to be observed are the inequality

119872119901120601120588119891 (119909) le 119872

119901120601120588119891 (119909) 119909 isin 119883 (47)

which holds for any admissible 119891 and gives10038171003817100381710038171198911003817100381710038171003817L119901120601120588

le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119871119901120601120588

(119883) subL119901120601120588

(119883) (48)

and the continuous embeddings

1198711199011120601120588(119883) sub 119871

1199012120601120588(119883) L

1199011120601120588(119883) subL

1199012120601120588(119883)

(49)

for 1 le 1199011lt 119901

2lt infin that follow from (18) and its version for

119872119901120601120588

When 120588

0equiv 0 and 119889 is a metric for 119901 = 1 and 120601 =

1 the space L11120588(119883) coincides with the local BMO space

BMO120588(119883) = BMO

120588(119883 119889 120583) defined and investigated in [4]

in the setting of a metric measure space satisfying 120583(119861) gt 0for every ball 119861

Since sdot L119901120601120588

is merely a seminorm a genuine norm isgenerated by considering the quotient spaceL

119901120601120588(119883)119882

0120588

where the subspace1198820120588

is

1198820120588= 119891 isinL

119901120601120588(119883)

10038171003817100381710038171198911003817100381710038171003817L119901120601120588

= 0 (50)


Unlikely to the case of 120588 equiv infin1198820120588

may be bigger than thespace of constant functions As it was explained in [4 p 249]119882

0120588coincides with the space of functions which are constant

120583-ae on each of 120588-components of 119883 where 120588-componentsare obtained by means of the equivalence relation sim

120588and

119909sim120588119910 provided that there exist balls 119861

1 119861

119898 sub B

120588such

that 119909 isin 1198611 119910 isin 119861

119898 and 119861

119894cap 119861

119894+1= 0 119894 = 1 119898 minus 1

In what follows we shall abuse slightly the language (infact we already did it) using in several places the term norminstead of (the proper term) seminorm

The definition of the generalized local Morrey and Cam-panato spaces based on closed balls requires using in (43)and (44) the operators 119872

119901120601120588and 119872

119901120601120588 respectively The

resulting spaces are then denoted by 119871119901120601120588(119883) andL

119901120601120588(119883)

respectively Lemma 3 immediately leads to Corollary 6

Corollary 6 Assume that (32) holds Then for 1 le 119901 lt infinwe have

119871119901120601120588

(119883) = 119871119901120601120588

(119883) L119901120601120588

(119883) =L119901120601120588

(119883) (51)

with identity of the corresponding norms in the first case andequivalence of norms in the second case

Remark 7 Consider the global case that is 120588 equiv infin Inthe setting of R119899 equipped with the Euclidean distanceand Lebesgue measure the classical Morrey and Campanatospaces 119871

119901120582andL

119901120582(in the notation from [16]) correspond

to the choice of 120601 = 120601119898120572

(up to a multiplicative constant)where 1 le 119901 lt infin 120572 = (120582119899 minus 1)119901 and 0 le 120582 le 119899 + 119901 andare explicitely given by

119871119901120582= 119871

119901120582(R

119899

)

= 119891 10038171003817100381710038171198911003817100381710038171003817(119901120582)

= sup119909isinR119899119903gt0

(1

119903120582int119861(119909119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

L119901120582=L

119901120582(R

119899

)

= 119891 10038171003817100381710038171198911003817100381710038171003817119901120582

= sup119909isinR119899119903gt0

inf119911isinC(1

119903120582int119861(119909119903)

1003816100381610038161003816119891 (119910) minus 1199111003816100381610038161003816

119901

119889119910)

1119901

lt infin

(52)

If 120582 = 0 then clearly L1199010C cong 119871

1199010= 119871

119901

(R119899

) It is alsoknown (see [16] for references) that for 0 lt 120582 lt 119899L

119901120582C cong

119871119901120582

for 120582 = 119899 L119901119899= BMO(R119899

) and 119871119901119899= 119871

infin

(R119899

) andfor 119899 lt 120582 le 119899+119901L

119901120582= Lip

120572(R119899

) with 120572 = (120582 minus 119899)119901 HereC = C(R119899

) denotes the space of all constant functions onR119899

Recall that a quasimetric measure space (119883 119889 120583) is calleda space of homogeneous type provided that120583 is doubling thatis it satisfies

120583 (119861 (119909 2119903)) ≲ 120583 (119861 (119909 119903)) (53)

uniformly in 119909 isin 119883 and 119903 gt 0 clearly the doubling conditionimplies that 120588

0equiv 0

In the framework of a space of homogeneous type(119883 119889 120583) a systematic treatment of generalized CampanatoMorrey and Holder spaces was presented by Nakai [2] Werefer to this paper for a discussion (among other things) ofthe relations between these spaces In the nondoubling casethat is in the setting of 119883 = R119899 and a Borel measure 120583 thatsatisfies the growth condition (23) a theory of Morrey spaceswas developed by Sawano and Tanaka [3] and Sawano [17]for details see Remarks 13 and 14

Remark 8 Of course it may happen that 119871119901120601120588(119883 119889 120583) is

trivial in the sense that it contains only the null functionThetriviality of 119871

119901120601120588(119883 119889 120583) is equivalent with the statement

that for every nonnull function 119891 isin 119871119901

loc120588(119883) there exists119909

0isin 119883 such that119872

119901120601120588119891(119909

0) = infin For instance if 119883 = R119899

with Lebesgue measure 120588 equiv infin 1 le 119901 lt infin and 120601 = 120601119898120572

with 120572 lt minus1119901 then 119872

119901120601119891(119909

0) = infin for every nonnull

119891 isin 119871119901

loc(R119899

) and every 1199090isin R119899 (so that 119871

119901120582(R119899

) = 0 forevery 120582 lt 0) Similarly it may happen that L

119901120601120588(119883 119889 120583) is

trivial in the sense that it consists of functions from1198820120588

onlyThis time the triviality of L

119901120601120588(119883 119889 120583) is equivalent with

the statement that for every function 119891 isin 119871119901

loc120588(119883) 1198820120588

there exists 1199090isin 119883 such that119872

119901120601120588119891(119909

0) = infin For instance

if 119883 = R119899 with Lebesgue measure 120588 equiv infin 1 le 119901 lt infinand 120601 = 120601

119898120572 with 120572 gt minus1 then 119872

119901120601119891(119909

0) = infin for

every 119891 isin 119871119901

loc(R119899

) C(R119899

) and every 1199090isin R119899 (so that

L119901120582(R119899

) = C(R119899

) for every 120582 gt 119899 + 119901 in particular (48)then implies that 119871

119901120582(R119899

) = 0 for 120582 gt 119899 + 119901)See also [18] for further remarks on triviality of 119871

119901120601(R119899

)

(the global case R119899 equipped with the Euclidean metric 119889(2)

and Lebesgue measure) In the same place [18] the followinginteresting observation is made Let 120601 (0infin) rarr (0infin)

be a function 120601(119861) = 120601(119903) for 119861 = 119861(119909 119903) 119909 isin R119899 and let1 le 119901 lt infin be given If 120601(119903) = inf

0lt119905le119903120601(119905) gt 0 for every

119903 gt 0 then 120601 is decreasing and 119871119901120601(R119899

) = 119871119901120601(R119899

) withequivalency of norms Similarly if inf

119903le119905ltinfin120601(119905)119905

119899119901

gt 0 forevery 119903 gt 0 then for 120601(119903) = 119903minus119899119901inf

119903le119905ltinfin120601(119905)119905

119899119901 120601(119903)119903119899119901is increasing and 119871

119901120601(R119899

) = 119871119901

120601(R119899

) with equivalency ofnorms

In the Euclidean setting ofR119899 with Lebesguemeasure thedefinition of the classical Morrey and Campanato spaces byusing either the Euclidean balls or the Euclidean cubes (withsides parallel to the axes) gives the same outcome Choosingballs or cubes means using either the metric 119889(2) or 119889(infin) Inthe general setting we consider two equivalent quasimetricson119883 and possibly different 120588 and 120601 functions

The result that follows compares generalized localMorreyand Campanato spaces for the given system (119883 119889 120583 120588 120601)


with these of (119883 1198891015840

120583 1205881015840

1206011015840

) under convenient and in somesense natural assumptions

Proposition 9 Let 1 le 119901 lt infin and the system (119883 119889 120583 120588 120601)be given and suppose that the triple (1198891015840

1205881015840

1206011015840

) is different from(119889 120588 120601) Assume also that there exists 119899

0isin N such that for

any ball 119861 isin B120588119889 there exists a covering 1198611015840

1 119861

1015840

1198990

of 119861consisting of balls from B

12058810158401198891015840 such that

1206011015840

(1198611015840

119895) 120583(119861

1015840

119895)1119901

≲ 120601 (119861) 120583(119861)1119901

119895 = 1 1198990 (54)

Then10038171003817100381710038171198911003817100381710038171003817119871119901120601120588119889

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601101584012058810158401198891015840

119891 isin 119871119901120601101584012058810158401198891015840 (119883) (55)

and consequently 119871119901120601101584012058810158401198891015840(119883) sub 119871

119901120601120588119889(119883) Similarly if for

any ball 119861 isin B120588119889 there exists a ball 1198611015840

isin B12058810158401198891015840 such that

119861 sub 1198611015840 and

1206011015840

(1198611015840

) 120583(1198611015840

)1119901

≲ 120601 (119861) 120583(119861)1119901

(56)

then10038171003817100381710038171198911003817100381710038171003817L119901120601120588119889

≲10038171003817100381710038171198911003817100381710038171003817L119901120601101584012058810158401198891015840

119891 isinL119901120601101584012058810158401198891015840 (119883) (57)

and henceL119901120601101584012058810158401198891015840(119883) subL

119901120601120588119889(119883)

Proof To prove the first claim take 119891 isin 119871119901120601101584012058810158401198891015840(119883) and 119861 isin

B120588119889 and consider a covering of 119861 119861 sub cup119899

0

119895=1119861

1015840

119895 consisting of

balls from B12058810158401198891015840 and satisfying (54) We have

1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le

1198990

sum

119895=1

1

120601 (119861) 120583(119861)1119901

(int1198611015840

119895

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲ sum

119895

1

1206011015840 (1198611015840

119895) 120583(119861

1015840

119895)1119901

(int1198611015840

119895

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le 1198990

10038171003817100381710038171198911003817100381710038171003817119871119901120601101584012058810158401198891015840

(58)

where in the second sum summation goes only over these 119895rsquosfor which 120583(1198611015840

119895) gt 0 Taking the supremum over the relevant

balls 119861 on the left hand side shows the required estimateand hence the inclusion To prove the second claim take119891 isin L

119901120601101584012058810158401198891015840(119883) and 119861 isin B

120588119889 and consider 1198611015840 119861 sub 1198611015840

satisfying (56) Then

inf119911isinC

1

120601 (119861)(1

120583 (119861)int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

≲ inf119911isinC

1

1206011015840 (1198611015840) 120583(1198611015840)1119901

(int1198611015840

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817L119901120601101584012058810158401198891015840

(59)

Taking again the supremum over the relevant balls 119861 on theleft hand side shows the second required estimate and hencethe second inclusion

Corollary 10 Under the assumptions of Proposition 9 andthe analogous assumptions but with the roles of (119889 120588 120601) and(119889

1015840

1205881015840

1206011015840

) switched we have

119871119901120601120588119889

(119883) = 119871119901120601101584012058810158401198891015840 (119883)

L119901120601120588119889

(119883) =L119901120601101584012058810158401198891015840 (119883)

(60)

with equivalency of the corresponding norms

Remark 11 In the case when in the system (119883 119889 120583 120588 120601)only 120601 is replaced by 1206011015840 it may happen that 1206011015840

(119861) ≲ 120601(119861)

uniformly in 119861 isin B120588 Then the conclusion of Proposition 9

is obvious but at the same moment this is the simplest caseof the assumptionmade in Proposition 9 with 119899

0= 1 and the

covering of 119861 consisting of 119861

The following example generalizes the situation of equiv-alency of theories based on the Euclidean balls or cubesmentioned above

Example 12 Let (119883 119889 120583) be a space of homogeneous typeAssume that 1198891015840 is a quasimetric equivalent with 119889 and 120588 =120588

1015840

equiv infin Given 120572 isin R let 120601 = 120601(119889)

119898120572and 1206011015840

= 120601(1198891015840

)

119898120572 Then for

1 le 119901 lt infin and 120572 ge minus1119901 we have

119872119901120601119889119891 (119909) ≃ 119872

11990112060110158401198891015840119891 (119909)

119872119901120601119889119891 (119909) ≃ 119872

11990112060110158401198891015840119891 (119909)

(61)

uniformly in 119891 and 119909 isin 119883 and consequently

119871119901120601119889

(119883) = 11987111990112060110158401198891015840 (119883) L

119901120601119889(119883) =L

11990112060110158401198891015840 (119883)

(62)

with equivalency of the corresponding norms Indeedassuming that 119888minus1

1198891015840

le 119889 le 1198881198891015840 for a 119888 gt 1 we have (in what

follows 1198611015840 means a ball related to 1198891015840)

119861 (119909 119903) sub 1198611015840

(119909 119888119903) sub 119861 (119909 1198882

119903) (63)

and hence we take 1198611015840

(119909 119888119903) as a covering of 119861(119909 119903) isin B119889

The doubling property of 120583 then implies

120583 (1198611015840

(119909 119888119903)) le 120583 (119861 (119909 1198882

119903)) le 119862120583 (119861 (119909 119903)) (64)

and therefore (54) follows with 120601 and 1206011015840 declared as aboveThe ldquodualrdquo estimate follows analogously

Remark 13 Sawano and Tanaka [3] defined and investigatedMorrey spaces in the setting of (R119899

119889(infin)

120583) where 120583 is aBorel measure onR119899 finite on bounded sets (recall that everysuch measure is automatically a Radon measure) which maybe nondoubling


For a parameter 119896 gt 1 and 1 le 119901 le 119902 lt infin the MorreyspaceM119902

119901(119896 120583) (in the notation of [3] but with the roles of 119901

and 119902 switched) is the space of functions on R119899 satisfying

sup120583(119876)gt0

120583(119896119876)1119902minus1119901

(int119876

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (65)

where the supremum is taken over all (closed) cubes with theproperty 120583(119876) gt 0 The space M119902

119901(119896 120583) coincides with our

space 119871119901120601(R119899

119889(infin)

120583) (ie 120588 equiv infin) where

120601 (119876) = 120583(119876)minus1119901

120583(119896119876)1119901minus1119902

(66)

It was proved in [3 Proposition 11] (the growth condition(23) did not intervene there) thatM119902

119901(119896 120583) does not depend

on the choice of 119896 gt 1 This corresponds to the situationof 119889 = 1198891015840

= 119889(infin) 120588 = 1205881015840

equiv infin 120601 as above and 1206011015840

(119876) =

120583(119876)minus1119901

120583(1198961015840

119876)1119901minus1119902 1198961015840 gt 1 in Corollary 10 since as it can

be easily observed for 1 lt 119896 lt 1198961015840 lt infin say we have 120601 le 1206011015840

and on the other hand the assumption of Proposition 9 issatisfied due to simple geometrical properties of cubes in R119899

(see [3 p 1536] for details)

Remark 14 Sawano [17] defined and investigated the so-called generalized Morrey spaces in the same setting of(R119899

119889(infin)

120583) (with closed cubes) For a parameter 119896 gt 1

and a nondecreasing function 120595 (0infin) rarr (0infin) thespaceL119901120595

(119896 120583) was defined as the space of functions onR119899

satisfying

sup120583(119876)gt0

(1

120595 (120583 (119896119876))int119876

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (67)

The spaceL119901120595

(119896 120583) coincides with our space 119871119901120601(R119899

119889(infin)

120583) where

120601 (119876) = (120595 (120583 (119896119876))

120583 (119876))

1119901

(68)

(Note that for 120595(119905) = 1199051minus119901119902 1 le 119901 le 119902 lt infin we alsohaveL119901120595

(119896 120583) =M119902

119901(119896 120583)) It was proved in [1 Proposition

12] (again the growth assumption did not intervene there)that L119901120595

(119896 120583) is independent of 119896 gt 1 with equivalencyof norms This result may also be seen as a consequence ofCorollary 10 Indeed by taking 1206011015840

(119876) = (120595(120583(1198961015840

119876))120583(119876))1119901

in this corollary for 1 lt 119896 lt 1198961015840 lt infin we have 120601 le 1206011015840 Onthe other hand the assumption of Proposition 9 is satisfied bythe argument already mentioned in Remark 13 (geometricalproperties of cubes in R119899)

Remark 15 Recently Liu et al [6] defined and investigatedthe local Morrey spaces in the setting of a locally doublingmetric measure space (119883 119889 120583) The latter means that themeasure 120583 possesses the doubling and the reverse doublingproperties only on a class of admissible balls This class B

119886

is defined with an aid of an admissible function 119898 119883 rarr

(0infin) and a parameter 119886 isin (0infin) and agrees with our

class B120588for the locality function 120588(119909) = 119886119898(119909) (in [6] an

assumption of geometrical nature is imposed on 119898) ThentheMorrey-type spaceM119901119902

B119886

(119883) 1 le 119902 le 119901 lt infin was definedas the space of functions on119883 satisfying

sup119861isinB119886

120583(119861)1119901

(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (69)

The investigations in the general setting were next specifiedin [6] to the important example of the Gauss measure space(R119899

119889(2)

120574119899) where 120574

119899denotes the Gauss measure 119889120574

119899(119909) =

120587minus1198992 exp(minus1199092

2)119889119909 The importance of this example lies

in the fact that the measure space (R119899

120574119899) is the natural

environment for theOrnstein-Uhlenbeck operator minus(12)Δ+119909 sdot nabla In the context of (R119899

119889(2)

120574119899) the Campanato-type

space E119901119902

B119886

(120574119899) was also defined as the space of functions on

119883 satisfying

100381710038171003817100381711989110038171003817100381710038171198711(120574

119899)+ sup

119861isinB119886

120583(119861)1119901

(1

120583 (119861)int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816

119901

119889120583)

1119901

lt infin

(70)

(the additional summand 1198911198711(120574119899)was added due to the

specific character of the involved measure space)

Remark 16 In [19 Theorems 4 and 5] an example ofBorel measure 120583 in R2 was provided (120583 being absolutelycontinuous with respect to Lebesgue measure) such thatBMO(2)

(R2

120583) (= L11(R

2

119889(2)

120583)) and BMO(infin)

(R2

120583) (=L

11(R2

119889(infin)

120583)) differ

In the final example of this section we analyse a specificcase that shows that in general thingsmay occur unexpected

Example 17 Take 119883 = N 119889 to be the 0 minus 1 metric on N and120583 to be the measure on N such that 120583(119899) = 119886

119899 where 119886

119899gt 0

and sum119886119899= 1 (so that 120583(N) = 1) Note that 120583 is nondoubling

it is not even locally doubling and if 119861 is a ball then either119861 = 119899 for some 119899 isin N or 119861 = N and hence 0 lt 120583(119861) lt infinfor every ball 119861 Then 119871119901

loc(N) = ℓ119901

(N 120583) for 1 le 119901 lt infin and1205880equiv 0 For simplicity we now treat the case 120601 = 1 onlyConsider first 120588 equiv infin Then for any 119891 = (119891(119895))

119895isinN isin

ℓ119901

(N 120583)1198721199011119891 and119872119888

1199011119891 are constant functions

1198721199011119891 (119899) = 119872

1198881199011119891 (119899) = (sum

119895

1003816100381610038161003816119891 (119895) minus ⟨119891⟩N

1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(71)

where ⟨119891⟩N = sum119895119891(119895)119886

119895 HenceL

1199011(N) = ℓ119901

(N 120583) 119891L1199011

≃ 119891 minus ⟨119891⟩Nℓ119901(N120583) 119882

0= C and L

1199011(N)C is identified


with ℓ119901(N 120583) whereC = C(N) denotes the space of constantsequences Similarly for any 119891 isin ℓ119901(N 120583)

1198721199011119891 (119899) = 119872

119888

1199011119891 (119899)

= max

1003816100381610038161003816119891 (119899)1003816100381610038161003816 (sum

119895

1003816100381610038161003816119891 (119895)1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(72)

and hence 1198711199011(N) = ℓ

infin

(N) cap ℓ119901(N 120583) with identity ofnorms

Consider now the case of 120588 equiv 1ThenB120588consists of balls

of the form 119861 = 119899 119899 isin N 119871119901

loc120588(N) = 119904(N) where 119904(N)denotes the space of all sequences on N and119872

1199011120588119891(119899) = 0for any119891 isin 119904(N) and 119899 isin N and henceL

1199011120588(N) = 119904(N) and119891L

1199011120588= 0 for 119891 isin 119904(N) In addition every 120588-component

is of the form 119899 119899 isin N and hence1198820120588= 119904(N) Similarly

for any 119891 isin 119904(N) we have1198721199011120588119891(119899) = |119891(119899)| 119899 isin N and

hence 1198711199011120588(N) = ℓ

infin

(N) and sdotL1199011120588= sdot

ℓinfin

31 Morrey and Campanato Spaces on Open Proper Subsetsof R119899 In this subsection we suggest a coherent theory ofgeneralized Morrey and Campanato spaces on open propersubsets of R119899

As it wasmentioned in [4 p 259] indecisions accompanychoosing a suitable definition of BMO(Ω) for a general openproper subsetΩ ofR119899 equipped with Lebesgue measure Theldquorightrdquo way seems to be the following BMO(Ω) consists ofthose functions 119891 isin 1198711

loc(Ω) such that

10038171003817100381710038171198911003817100381710038171003817BMO(Ω)

= sup119861subΩ

1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816 lt infin (73)

where the supremum is taken over all closed balls (orclosed cubes if one prefers then the character 119861 should bereplaced by119876) entirely contained inΩ see [20]Throughoutthis section |119860| stands for the Lebesgue measure of 119860 ameasurable subset ofΩ Note that such a definition has a localflavor the locality function entering the scene is

120588Ω(119909) = 119889 (119909 120597Ω) 119909 isin Ω (74)

where the distance from 119909 isin Ω to 120597Ω is given by

119889 (119909 120597Ω) = inf 119889 (119909 119910) 119910 isin 120597Ω (75)

and 119889 = 119889(2) or 119889 = 119889(infin)Similar indecisions accompany the process of choosing

a suitable definition of Morrey and Campanato spaces for ageneral open proper subset Ω sub R119899 The spaces 119871119901120582

(Ω)1 le 119901 lt infin and 0 le 120582 le 119899 determined by

10038171003817100381710038171198911003817100381710038171003817119871119901120582(Ω)

= sup119909isinΩ

sup0lt119903lt119903

0

(1

119903120582int119861(119909119903)capΩ

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

(76)

where 1199030= diamΩ (see also [21]) were originally introduced

by Morrey [22] (with a restriction to open and bounded

subsets) For a definition of L119901120582

(Ω) (nowadays called afterCampanato the Campanato space) also with a restriction toopen and bounded subsets see [23]

An alternative way of defining generalized Morrey andCampanato spaces on open proper (not necessarily bounded)subset Ω sub R119899 is by using our general approach with thelocality function 120588

Ωgiven above To fix the attention let us

assume for a moment that 119889 = 119889(infin) Thus for a given

function 120601 B120588(Ω) rarr (0infin) we define 119871

119901120601(Ω) =

119871119901120601120588Ω

(Ω) andL119901120601(Ω) =L

119901120601120588Ω

(Ω) Explicitely thismeansthat for 120588 = 120588

Ω

119871119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817119871119901120601

(Ω)

= sup119861isinB120588(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

L119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817L119901120601

(Ω)

= sup119861isinB120588(Ω)

inf119911isinC

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910) minus 1199111003816100381610038161003816

119901

119889119910)

1119901

lt infin

(77)

by the definition of 120588 B120588(Ω) is the family of all closed balls

entirely contained inΩGiven a parameter 0 lt 119896 le 1 we now define the locality

function 120588119896= 120588

119896Ωas

120588119896(119909) = 119896 sdot 119889 (119909 120597Ω) 119909 isin Ω (78)

so that 1205881= 120588

Ω Then for a function 120601 as above we define

119871119901120601119896(Ω) = 119871

119901120601120588119896

(Ω) and L119901120601119896(Ω) = L

119901120601120588119896

(Ω) Thestructure of the above definition of 119871

119901120601119896(Ω) and L

119901120601119896(Ω)

reveals that if Ω is not connected then the defined spacesare isometrically isomorphic to the direct sums of thecorresponding spaces built on the connected componentsof Ω with ℓinfin norm for the direct sum of the given spacesIndeed if for instance119891 isin 119871

119901120601119896(Ω)Ω = ⋃

119895isin119869Ω

119895(119869 is finite

or countable) where eachΩ119895is a connected component ofΩ

and 119891119895denotes the restriction of 119891 to Ω

119895 then

10038171003817100381710038171198911003817100381710038171003817119871119901120601119896

(Ω)= sup

119909isinΩ

sup119909isin119861isinB

119896(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901


= sup119895isin119869

sup119909isinΩ119895


119896(Ω119895)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

= sup119895isin119869

10038171003817100381710038171003817119891

119895

10038171003817100381710038171003817119871119901120601119895119896(Ω119895)

(79)

Thus without loss of generality we can assume (and we dothis) thatΩ is connected

The analogous definitions (and comments associated tothem) obey 119889 = 119889(2) To distinguish between the two casescorresponding to the choice of 119889(2) or 119889(infin) when necessarywe shall write 120588(2)

119896and 120588(infin)

119896 119871B

119901120601119896(Ω) and 119871Q

119901120601119896(Ω) and so

forthAlso the family of balls related to120588(2)


BΩ

119896 while the family of cubes related to 120588(infin)

119896will be denoted

by QΩ

119896

In what follows rather than considering a general 120601 welimit ourselves to the specific case of 120601 = 120601

119898120572 Clearly

120601119898120572

satisfies (32) and hence distinguishing between openor closed balls (or open or closed cubes) is not necessaryWe write 119871Q

119901120572119896(Ω) in place of 119871Q

119901120601119898120572

119896(Ω) and similarly in

other occurences Our goal is to prove that the definitions onMorrey and Campanato spaces do not depend on choosingballs or cubes this is contained inTheorem 20

The following propositions partially contain [24 Theo-rems 35 and 39] as special cases

Proposition 18 Let 1 le 119901 lt infin and 120572 ge minus1119901 begiven The spaces 119871Q

119901120572119896(Ω) are independent of the choice

of the scale parameter 119896 isin (0 1) with equivalence of thecorresponding norms The analogous statement is valid for thespacesLQ

119901120572119896(Ω)

Proof Let 0 lt 119896 lt 119898 lt 1 We shall prove the inequalities10038171003817100381710038171198911003817100381710038171003817119871Q119901120572119898

(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817119871Q119901120572119896

(Ω) 119891 isin 119871

Q119901120572119896

(Ω) (80)10038171003817100381710038171198911003817100381710038171003817LQ119901120572119898

(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) 119891 isinL

Q119901120572119896

(Ω) (81)

which give the inclusions 119871Q119901120572119896(Ω) sub 119871

Q119901120572119898

(Ω) andLQ

119901120572119896(Ω) sub LQ

119901120572119898(Ω) The inequalities opposite to (80)

and (81) (with 119862 = 1) are obvious and thus the oppositeinclusions follow

Consider first the case of (80) There exists 119873 = 119873(119898

119896 119899) such that bisecting any cube 119876 isin QΩ

119898119873 times results in

obtaining a family 119876119895 of 2119899119873 congruent subcubes of119876 each

of them in QΩ

119896 Thus

1

|119876|120572(1

|119876|int119876

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le1

|119876|120572sum

119895

(1

|119876|int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le sum

119895

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572(1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le 21198991198731003817100381710038171003817119891

1003817100381710038171003817119871Q119901120572119896

(Ω)

(82)

and the result follows

Considering (81) we shall apply the procedure similarto that used in the proof of [24 Theorem 35] Take 119891 isin

LQ119901120572119896(Ω) and 119876 isin QΩ

119898 QΩ

119896 Then

inf119911isinC

1

|119876|120572(1

|119876|int119876

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

=1

|119876|120572+1119901

10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

(83)

where 119871119901

(119876)C is the quotient space and sdot 119871119901(119876)C is the

quotient norm Since the dual to 119871119901

(119876)C is identified with119871

1199011015840

0(119876) where 1199011015840 is the exponent conjugate to 119901 1119901+11199011015840

=

1 and 1198711199011015840

0(119876) denotes the subspace of 119871119901

(119876) consisting offunctions 119891 with int

119876

119891 = 0 therefore

10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

= sup10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

120595 isin 1198711199011015840

0(119876)

100381710038171003817100381712059510038171003817100381710038171199011015840le 1

(84)

According to [24 Lemma 31] there exist constants 119862 =

119862(119896119898 1199011015840

119899) and119873 = 119873(119896119898 119899) such that for every119876 isin QΩ

119898

and every function 120595 isin 1198711199011015840

0(119876) there exist subcubes 119876

119895isin

QΩ

119896 1 le 119895 le 119873 of 119876 and functions 120595

119895isin 119871

1199011015840

0(119876) 1 le

119895 le 119873 such that supp(120595119895) sub 119876

119895and 120595

1198951199011015840le 119862120595

1199011015840 for

119895 = 1 119873 and 120595 = sum119873

119895=1120595

119895 Take 120595 isin 119871119901

1015840

0(119876) satisfying

1205951199011015840 le 1 where 119876 is the cube chosen earlier and select

subcubes 119876119895119873

1and functions 120595

119895119873

1with properties as above

Then

1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816100381610038161003816

int119876

119891(

119873

sum

119895=1

120595119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

119891120595119895

100381610038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

(119891 minus 119891119876119895

)120595119895

100381610038161003816100381610038161003816100381610038161003816

le 119862

119873

sum

119895=1

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572+1119901

(int119876119895

100381610038161003816100381610038161003816119891 minus 119891

119876119895

100381610038161003816100381610038161003816

119901

)

1119901

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω)

(85)

Hence

1

|119876|120572+1119901

100381710038171003817100381711989110038171003817100381710038171198711(119876)C

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) (86)

and consequently 119891LQ119901120572119898

(Ω)le 119862119873119891LQ

119901120572119896(Ω)

Proposition 19 Let 1 le 119901 lt infin and 120572 ge minus1119901 be givenThe spaces 119871B

119901120572119896(Ω) are independent of the choice of the scale

parameter 119896 isin (0 1) with equivalence of the correspond-ing norms The analogous statement is valid for the spacesLB

119901120572119896(Ω)


Proof The present proof mimics the one of Proposition 18since essentially it suffices to replace the character Q by Band assuming 0 lt 119896 lt 119898 lt 1 are given to use the followinggeometrical properties of Euclidean balls The first one saysthat there exists119873 = 119873(119896119898 119899) such that every ball 119861 isin BΩ

119898

may be covered by a family 119861119895119873

1of balls each of them inBΩ

119896

and with radii smaller than that of 119861 The second one (moresophisticated) is contained in [24 Lemma 38] and says thatthere exist constants 119862 = 119862(119896119898 1199011015840

119899) and 119873 = 119873(119896119898 119899)such that for every 119861 isin BΩ

119898and every function 120595 isin 119871119901

1015840

0(119861)

there exist balls 119861119895isin BΩ

119896 1 le 119895 le 119873 with radii smaller

than that of 119861 and functions 120595119895isin 119871

1199011015840

0(119861) 1 le 119895 le 119873 such

that supp(120595119895) sub 119861

119895and 120595

1198951199011015840le 119862120595

1199011015840 for 119895 = 1 119873

and 120595 = sum119873

119895=1120595

119895 (The fact that radii of 119861

119895are smaller than

the radius of119861 is not directly indicated in the statement of [24Lemma 38] but it is implicitly contained in the constructionincluded in the proof of that lemma)

For the sake of completeness we include an outline of theproof of the first aforementioned property We shall use thefollowing simple geometrical fact given 0 le 1198771015840

lt 119877 and 120576 gt1 minus 119877

1015840

119877 there exists 1198731015840

= 1198731015840

(1198771015840

119877 120576 119899) such that for anysphere 119878(119909

0 119877) = 119909 isin R119899

119909 minus 11990902= 119877 one can find

points 1199091 119909

1198731015840 on that sphere such that

119861 (1199090 2119877 minus 119877

1015840

) 119861 (1199090 119877

1015840

) sub cup1198731015840

119895=1119861 (119909

119895 120576119877) (87)

(if 1198771015840

= 0 then we set 119861(1199090 0) = 0)

Now take any 119861 = 119861(1199090 119903) isin BΩ

119898 In fact we shall prove

the aforementioned property for the ldquomaximalrdquo ball 119861(1199090 119903

0)

with 1199030= 119898119889

(2)

(1199090 120597Ω) Let 119877

119895= (119895119904)119903

0 119895 isin N where

119904 isin N is large enough (to be determined in the last stepof the argument) Using the above geometrical fact on eachsphere 119878

119895= 119878(119909

0 119877

119895) 119895 = 1 2 119904 minus 1 we choose finite

number of points such that the balls centered at these pointsandwith radii equal (2119904)119903

0covering the annulus119861(119909

0 119877

119895+1)

119861(1199090 119877

119895minus1) More precisely given 119895 = 1 2 119904 minus 1 we apply

the geometrical fact with 119877 = 119877119895= (119895119904)119903

0and 1198771015840

= 119877119895minus1=

((119895 minus 1)119904)1199030(so that 1198771015840

119877 = ((119895 minus 1)119895)) and 120576 = 2119895 (so that120576 gt 1 minus 119877

1015840

119877) It is clear that the union of all chosen ballscovers 119861(119909

0 119903

0) and there is 119873 = sum

119904minus1

119895=1119873

1015840

((119895 minus 1)119895 2119895 119899)

of them To verify that each of these balls is in BΩ

119896 take

119861(119909lowast

0 119903

lowast

0) with center lying on the sphere 119878(119909

0 119877

119904minus1) (this is

the worst case) Since for 120596 isin Ω119888 we have 119889(2)

(120596 1199090) le

119889(2)

(120596 119909lowast

0) + 119889

(2)

(119909lowast

0 119909

0) and 119889(2)

(119909lowast

0 119909

0) = ((119904 minus 1)119904)119903

0=

((119904 minus 1)119904)119898119889(2)

(1199090 120597Ω) it is clear that

119889(2)

(119909lowast

0 120597Ω) ge 119889

(2)

(1199090 120597Ω) (1 minus 119898

119904 minus 1

119904) (88)

Hence if 119904 is chosen to be the least positive integer with theproperty (2119904)(1(1 minus 119898)) lt 119896 then

119903lowast

0=2

1199041199030=2

119904119898119889

(2)

(1199090 120597Ω) lt

2

119904

1

1 minus 119898119889

(2)

(119909lowast

0 120597Ω)

le 119896119889(2)

(119909lowast

0 120597Ω)

(89)

and the required property follows (note that 119904 depends on 119896and 119898 and hence 119873 depends on 119896 119898 and 119899 as claimed)

The results of Propositions 18 and 19 allow us to define119871Q119901120572(Ω) = 119871

Q11990112057212

(Ω) and sdot 119871Q119901120572

(Ω)= sdot

119871Q11990112057212

(Ω)(the

choice of 119896 = 12 being ldquorandomrdquo) and similarly for119871B119901120572(Ω) LQ

119901120572(Ω) LB

119901120572(Ω) and the corresponding norms

The following theorem partially contains [24 Theorem 42]as a special case

Theorem 20 Let 1 le 119901 lt infin and 120572 ge minus1119901 be given Thenwe have

119871Q119901120572(Ω) = 119871

B119901120572(Ω) L

Q119901120572(Ω) =L

B119901120572(Ω) (90)

with equivalence of the corresponding norms

Proof We focus on proving the statement concerning theCampanato spaces the argument for the Morrey spaces isanalogous (and slightly simpler) Given a cube 119876 or a ball 119861by 119861

119876or 119876

119861 we will denote the ball circumscribed on 119876 or

the cube circumscribed on 119861 respectively By the inequality sdot

infinle sdot

2le radic119899 sdot

infin it is clear that for 0 lt 119896 le 1radic119899

119861119876isinBΩ

radic119899119896 if119876 isin QΩ

119896 and119876

119861isin QΩ

radic119899119896 if 119861 isinBΩ

119896 Moreover

|119861119876| = 119888

1|119876| and |119876

119861| = 119888

2|119861| where 119888

1and 119888

2 depend on the

dimension 119899 onlyFix 0 lt 119896 lt 1radic119899 and take 119891 isin LQ

119901120572radic119899119896(Ω) For any 119861

and 119876119861defined above

inf119911isinC

1

|119861|120572(1

|119861|int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

le 1198881015840

2inf119911isinC

1

1003816100381610038161003816119876119861

1003816100381610038161003816

120572(1

1003816100381610038161003816119876119861

1003816100381610038161003816

int119876119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

(91)

Consequently 119891LB119901120572119896

(Ω)le 119888

1015840

2119891LQ

119901120572radic119889119896

(Ω)which also shows

that LQ119901120572radic119899119896

(Ω) sub LB119901120572119896(Ω) The results of Propositions

18 and 19 now give LQ119901120572(Ω) sub LB

119901120572(Ω) and sdot LB

119901120572(Ω)

le

119862 sdot LQ119901120572

(Ω)The opposite inclusion and inequality are proved

in an analogous way

Clearly the concept of Morrey and Campanato spaceson open proper subsets of R119899 may be generalized to openproper subsets of a general quasimetric space 119883 See [4Section 5] where the concept of local maximal operators insuch framework was mentioned Finally we mention that thepresented concept of locality for open proper subdomains inthe Euclidean spaces is rather common See for instance therecent paper [25] where the regularity of the local Hardy-Littlewood operator was studied and the paper [12] wherenotions of local fractional operators were introduced andstudied in both cases in the setting of Ω sub R119899 with thelocality functionΩ ni 119909 997891rarr dist(119909R119899

Ω)


4 Boundedness of Operators onLocal Morrey Spaces

Boundedness of classical operators of harmonic analysis onMorrey spaces was investigated in a vast number of paperssee for instance [3 5 17 26ndash30] and references cited there

In this section we assume the system (119883 119889 120583 120588) to befixed We begin with a result on the boundedness of localHardy-Littlewood maximal operator between local Morreyspaces For the notational convenience let 120591

119870= 119870(2119870 + 1)

where 119870 is the constant from the quasitriangle inequality (if119889 is a metric then 119870 = 1 and 120591

119870= 3) Observe that the

assumptions we impose in Proposition 21 are satisfied forinstance when 120583(119861) ≃ 119903(119861)119899 for some 119899 gt 0 uniformly in119861 isin B

120588 120593(119903) = 119903120572 minus 119899119901 le 120572 le 0 and 120583 satisfies the

120588-local 120591119870-condition Recall also that when it comes to the

boundedness of 11987211120588 on 119871119901

(119883 120583) we have the conclusionof Proposition 4 to our disposal

Proposition 21 Let 1 le 119901 lt infin and 120593 (0infin) rarr (0infin) bea nonincreasing function such that 120601(119861) = 120593(119903(119861)) satisfies

120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ 120601 (119861) 120583(119861)1119901

(92)

uniformly in 119861 isin B120588 If119872

11120588 is bounded on 119871119901

(119883 120583) thenit is also bounded from 119871

119901120601120591119870120588(119883) to 119871

119901120601120588(119883)

Proof For the notational convention let119872120588= 119872

11120588 thatis

119872120588119891 (119909) = sup

119909isin119861isinB120588

1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 (93)

Take 119891 isin 119871119901120601120591119870120588(119883) fix a ball 119861 isin B

120588 and consider the

splitting119891 = 1198911+119891

2adjusted to119861 in the sense that119891

1= 119891120594

120591119870119861

Then for any 119909 isin 119861

119872120588119891

2(119909) le sup

1198611015840

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 (94)

where the supremum is taken over all balls 1198611015840

isinB120588such that

119909 isin 1198611015840 and 119903(1198611015840

) gt 119903(119861) If 119909 isin 119861 and 1198611015840 is one of such ballsthen the fact that 120593 is nonincreasing gives

1

120601 (119861)

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 le

1

120593 (119903 (119861))(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le1

120601 (1198611015840)(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

(95)

Consequently

1

120601 (119861)119872

120588119891

2(119909) le

10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119909 isin 119861 (96)

This estimate subadditivity of119872120588 and the assumption that

119872120588is bounded on 119871119901

(119883 120583) give

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891)

119901

119889120583)

1119901

le sum

119894=12

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891

119894)119901

119889120583)

1119901

le 119888119901

1

120601 (119861) 120583(119861)1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

le 1198881015840

119901

1

120601 (120591119870119861) 120583(120591

119870119861)

1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

le 11988810158401015840

119901

10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

(97)

This shows the required estimate 119872120588119891

119871119901120601120588

≲ 119891119871119901120601120591119870120588

Remark 22 Consider the global case 120588 equiv infin To rediscoverthe classical result of Chiarenza and Frasca [31 Theorem1] which is the boundedness of the usual Hardy-Littlewoodoperator on the space 119871

119901120582(R119899

) 1 lt 119901 lt infin and 0 lt 120582 lt 119899(see Remark 7) take 120593(119903) = 119903

(120582minus119899)119901 which is decreasingThe assumption (92) with 120583 being Lebesgue measure andthe metric being 119889(infin) is obviously satisfied (clearly the usualHardy-Littlewood operator is also bounded on 119871119901

(R119899

))Similarly if (119883 119889 120583) is a space of homogeneous type and

120593(119903) = 119903120572 120572 le 0 so that 120601 = 120601

119903120572 then condition (92)

is satisfied and hence 119872120583= 119872

11 the Hardy-Littlewoodoperator associated to 120583 maps boundedly 119871

119901120601119903120572

into itself

In the literature several variants of fractional integralsover quasimetric measure spaces are considered Here weshall consider a variant in the setting of a quasimetricmeasurespaces (119883 119889 120583) with 120583 satisfying the upper growth condition(23) with 120591 = 119899 For any appropriate function 119891 and 0 lt 120573 lt119899 we define the fractional integral operator 119868

120573by letting

119868120573119891 (119909) = int

119883

119891 (119910)

119889(119909 119910)119899minus120573

119889120583 (119910) 119909 isin 119883 (98)

For functions 120593 120595 (0infin) rarr (0infin) we shall consider thefollowing conditions (compare them with the assumptionsimposed in [32])

120593 (119903)

119903is nonincreasing

int

infin

119903

119905120573minus1

120593 (119905) 119889119905 ≲ 120595 (119903) 119903 gt 0

(99)

Proposition 23 Let 1 le 119901 119902 lt infin and 120593 120595 (0infin) rarr(0infin) be functions satisfying (99) In addition assume that120601(119861) = 120593(119903(119861)) and Ψ(119861) = 120595(119903(119861)) satisfy

120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ Ψ (119861) 120583(119861)1119902

(100)


uniformly in119861 isin B If 119868120573is bounded from119871119901

(119883 120583) to119871119902

(119883 120583)then for any 120588 it is also bounded from 119871

119901120601(119883) to 119871

119902Ψ120588(119883)

Proof Since sdot 119871119902Ψ120588

le sdot 119871119902Ψ

it is sufficient to consider thecase 120588 equiv infin The estimate to be proved is

1

Ψ (119861)(1

120583 (119861)int119861

1003816100381610038161003816100381611986812057311989110038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(101)

uniformly in 119861 isin B and 119891 isin 119871119901120601(119883) Take 119861 = 119861(119886 119903) and

119891 isin 119871119901120601(119883) and consider the decomposition 119891 = 119891

1+ 119891

2

1198911= 119891120594

120591119870119861 and 120591

119870= 2119870 It suffices to verify (101) with 119891

replaced by 119891119894 119894 = 1 2 on the left hand side of this estimate

For 1198911 using the assumption on the 119871119901

minus 119871119902 boundedness of

119868120573and (100) we write

1

Ψ (119861)(1

120583 (119861)int119861

10038161003816100381610038161003816119868120573119891

1

10038161003816100381610038161003816

119902

119889120583)

1119902

le1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171003817119868120573119891

1

10038171003817100381710038171003817119871119902(120583)

≲1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171198911

1003817100381710038171003817119871119901(120583)

≲1

120601 (120591119870119861)(

1

120583 (120591119870119861)int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(102)

For 1198912 note that for any 119909 isin 119861 we have 119861(119886 120591

119870119903)

119888

sub 119861(119909 119903)119888

(119860119888 denotes the completion of 119860 in119883) and therefore

10038161003816100381610038161003816119868120573119891

2(119909)10038161003816100381610038161003816le int

(120591119870119861)119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le int119861(119909119903)

119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

=

infin

sum

119895=0

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le

infin

sum

119895=0

(2119895

119903)120573minus119899

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲

infin

sum

119895=0

(2119895

119903)120573 1

120583 (119861 (119909 2119895+1119903))int119861(1199092119895+1

119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲ (

infin

sum

119895=0

(2119895

119903)120573

120601 (119861 (119909 2119895+1

119903)))10038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ int

infin

119903

119905120573minus1

120593 (119905) 11988911990510038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ 120595 (119903)10038171003817100381710038171198911003817100381710038171003817119871119901120601

(103)

With this pointwise estimate it follows that

1

Ψ (119861 (119886 119903))(

1

120583 (119861 (119886 119903))int119861(119886119903)

10038161003816100381610038161003816119868120573119891

2

10038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(104)

The proof is complete

Remark 24 Garcıa-Cuerva and Gatto proved that [33 Corol-lary 33] for a metric measure space (119883 119889 120583) satisfying (23)119868120573is bounded from 119871119901

(120583) to 119871119902

(120583) provided that 1 lt 119901 lt 119899120573and 1119902 = 1119901 minus 120573119899 The assumption that 119889 is a metric maybe relaxed see [1] and in fact we can assume (119883 119889 120583) to be aquasi-metricmeasure space satisfying (23) Let 120572 lt minus120573Then120593(119903) = 119903

120572 and 120595(119903) = 119903120572+120573 satisfy (99) In addition if weassume that 120583 satisfies 120583(119861(119909 119903)) ≃ 119903119899 uniformly in 119909 isin 119883and 119903 gt 0 then (100) holds with constraints on 119901 119902 120573 asabove Therefore with all these assumptions 119868

120573is bounded

from 119871119901120601(119883) to 119871

119902Ψ(119883)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research was initiated when Krzysztof Stempak visitedthe Department of Mathematics of Zhejiang University ofScience and Technology China in April 2013 He is thankfulfor the warm hospitality he received The authors would liketo thank the referees for their careful commentsThe researchof Krzysztof Stempak is supported by NCN of Poland underGrant 201309BST102057The research of Xiangxing Tao issupported by NNSF of China under Grants nos 11171306 and11071065

References

[1] K Stempak ldquoOn quasi-metric measure spacesrdquo preprint Inpress

[2] ENakai ldquoTheCampanatoMorrey andHolder spaces on spacesof homogeneous typerdquo Studia Mathematica vol 176 no 1 pp1ndash19 2006

[3] Y Sawano and H Tanaka ldquoMorrey spaces for non-doublingmeasuresrdquo Acta Mathematica Sinica English Series vol 21 no6 pp 1535ndash1544 2005

[4] C-C Lin K Stempak and Y-S Wang ldquoLocal maximal opera-tors onmeasuremetric spacesrdquo PublicacionsMatematiques vol57 no 1 pp 239ndash264 2013

[5] D Yang D Yang and Y Zhou ldquoLocalized Morrey-Campanatospaces on metric measure spaces and applications toSchrodinger operatorsrdquo Nagoya Mathematical Journal vol 198pp 77ndash119 2010

[6] L Liu Y Sawano and D Yang ldquoMorrey-type spaces on Gaussmeasure spaces and boundedness of singular integralsrdquo Journalof Geometric Analysis vol 24 no 2 pp 1007ndash1051 2014

[7] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010


[8] X Tolsa ldquoBMO1198671 and Calderon-Zygmund operators for nondoubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001

[9] F Nazarov S Treil and A Volberg ldquoWeak type estimatesand Cotlar inequalities for Calderon-Zygmund operators onnonhomogeneous spacesrdquo International Mathematics ResearchNotices no 9 pp 463ndash487 1998

[10] W Chen and E Sawyer ldquoA note on commutators of fractionalintegrals with 119877119861119872119874(120583) functionsrdquo Illinois Journal of Mathe-matics vol 46 no 4 pp 1287ndash1298 2002

[11] THeikkinen J Lehrback JNuutinen andHTuominen ldquoFrac-tional maximal functions in metric measure spacesrdquo Analysisand Geometry in Metric Spaces vol 1 pp 147ndash162 2012

[12] T Heikkinen J Kinnunen J Korvenpaa and H TuominenldquoRegularity of the local fractional maximal functionrdquo httparxivorgabs13104298

[13] J Heinonen Lectures on Analysis on Metric Spaces SpringerNew York NY USA 2001

[14] Y Sawano ldquoSharp estimates of the modified Hardy-Littlewoodmaximal operator on the nonhomogeneous space via coveringlemmasrdquo Hokkaido Mathematical Journal vol 34 no 2 pp435ndash458 2005

[15] Y Terasawa ldquoOuter measures and weak type (1 1) estimatesof Hardy-Littlewoodmaximal operatorsrdquo Journal of Inequalitiesand Applications vol 2006 Article ID 15063 13 pages 2006

[16] J Petree ldquoOn the theory of L119901120582

spacesrdquo Journal of FunctionalAnalysis vol 4 pp 71ndash87 1969

[17] Y Sawano ldquoGeneralized Morrey spaces for non-doubling mea-suresrdquoNonlinearDifferential Equations andApplications vol 15no 4-5 pp 413ndash425 2008

[18] E Nakai ldquoA characterization of pointwise multipliers on theMorrey spacesrdquo Scientiae Mathematicae vol 3 no 3 pp 445ndash454 2000

[19] J Mateu P Mattila A Nicolau and J Orobitg ldquoBMO fornondoubling measuresrdquo Duke Mathematical Journal vol 102no 3 pp 533ndash565 2000

[20] P W Jones ldquoExtension theorems for BMOrdquo Indiana UniversityMathematics Journal vol 29 no 1 pp 41ndash66 1980

[21] C T Zorko ldquoMorrey spacerdquo Proceedings of the AmericanMathematical Society vol 98 no 4 pp 586ndash592 1986

[22] C B Morrey Jr ldquoOn the solutions of quasi-linear ellipticpartial differential equationsrdquo Transactions of the AmericanMathematical Society vol 43 no 1 pp 126ndash166 1938

[23] S Campanato ldquoProprieta di una famiglia di spazi funzionalirdquoAnnali della Scuola Normale Superiore di Pisa vol 18 pp 137ndash160 1964

[24] C-C Lin and K Stempak ldquoAtomic 119867119901 spaces and their dualson open subsets of R119899rdquo Forum Mathematicum 2013

[25] H Luiro ldquoOn the regularity of the Hardy-Littlewood maximaloperator on subdomains of 119877119899rdquo Proceedings of the EdinburghMathematical Society vol 53 no 1 pp 211ndash237 2010

[26] Eridani H Gunawan E Nakai and Y Sawano ldquoCharacteriza-tions for the generalized fractional integral operators onMorreyspacesrdquoMathematical Inequalities amp Applications In press

[27] Y Sawano S Sugano and H Tanaka ldquoGeneralized fractionalintegral operators and fractional maximal operators in theframework of Morrey spacesrdquo Transactions of the AmericanMathematical Society vol 363 no 12 pp 6481ndash6503 2011

[28] W Yuan W Sickel and D Yang Morrey and CampanatoMeet Besov Lizorkin and Triebel vol 2005 of Lecture Notes inMathematics Springer Berlin Germany 2010

[29] I Sihwaningrum and Y Sawano ldquoWeak and strong typeestimates for fractional integral operators on Morrey spacesover metric measure spacesrdquo Eurasian Mathematical Journalvol 4 no 1 pp 76ndash81 2013

[30] Y Shi and X Tao ldquoSome multi-sublinear operators on general-izedMorrey spaces with non-doublingmeasuresrdquo Journal of theKorean Mathematical Society vol 49 no 5 pp 907ndash925 2012

[31] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni Serie VII vol 7 no 3-4 pp 273ndash279 1987

[32] E Nakai ldquoHardy-Littlewood maximal operator singular inte-gral operators and the Riesz potentials on generalized MorreyspacesrdquoMathematische Nachrichten vol 166 pp 95ndash103 1994

[33] J Garcıa-Cuerva and A E Gatto ldquoBoundedness propertiesof fractional integral operators associated to non-doublingmeasuresrdquo Studia Mathematica vol 162 no 3 pp 245ndash2612004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


is a metric on 119883 which is equivalent to 119889119902 more precisely119889

119902le 119889

119902

le 4119889119902 Consequently 119889

(119902)= (119889

119902)1119902 is a 119902-metric

equivalent with 119889 more precisely 119889(119902)

le 119889 le 41119902

119889(119902)

Thus every quasimetric admits an equivalent 119902-metric thatpossesses the open ball property

In what follows if (119883 119889) is a given quasimetric spacethen 119883 is considered as a topological space equipped withthe (metrizable) topology T

119889 It may happen that a ball in

119883 is not a Borel set (ie it does not belong to the Borel 120590-algebra generated byT

119889) see for instance [1] as an example

To avoid such pathological cases the assumption that allballs are Borel sets must be made Then if 119883 is additionallyequipped with a Borel measure 120583 which is finite on boundedsets and nontrivial in the sense that 120583(119883) gt 0 we say that(119883 119889 120583) is a quasimetric measure space (we do not assumethat 120583(119861) gt 0 for every ball 119861) In this paper we additionallyassume (similar to the assumption (13) made in [2]) that

all balls in 119883 are open (5)

taking into account what was mentioned above this assump-tion does not narrow the generality of our considerations

Let (119883 119889 120583) be a quasimetric measure space Define thefunction 120588

0 119883 rarr [0infin) by setting

1205880(119909) = inf 120576 120583 (119861 (119909 120576)) gt 0 119909 isin 119883 (6)

Observe that if 1205880(119909) gt 0 for some 119909 isin 119883 then

120583(119861(119909 1205880(119909))) = 0 this is a consequence of the continuity

property from below of the measure 120583 The property ldquo120583(119861) gt0 for every ball 119861rdquo is equivalent with the statement that 120588

0equiv

0Given a function 120588 119883 rarr (0infin] such that 120588

0(119909) lt 120588(119909)

for every 119909 isin 119883 let B120588(119909) = B

120588119889(119909) denote the family of

balls (related to 119889) centered at 119909 and with radius 119903 satisfying1205880(119909) lt 119903 lt 120588(119909) (clearly balls with different radii but which

coincide are identified as sets) Then we set

B120588=B

120588119889= ⋃

119909isin119883

B120588(119909) (7)

Thus B120588denotes the family of all 120588-local balls in 119883 with

positive measure In case the lower estimate on the radius1205880(119909) lt 119903 is disregarded we shall write B

120588for the resulting

family of ballsBy a 120588-local integrability of a real or complex-valued

function on 119883 we mean its integrability with respect to thefamily of balls fromB

120588 thus 119891 isin 1198711

loc120588(119883) = 1198711

loc120588(119883 119889 120583)

provided that int119861

|119891|119889120583 lt infin for every ball 119861 isin B120588(and

thus also for every 119861 isin B120588) Note that this notion of local

integrability does not refer to compactness Similarly for 1 le119901 lt infin we define 119871119901

loc120588(119883) = 119891 |119891|119901

isin 1198711

loc120588(119883)If 120588(119909) =infin for some 119909 isin 119883 then we will refer to 120588 as

a locality function and to objects associated to 120588 as ldquolocalrdquoobjects If 120588 equiv infin identically then we shall skip the infinsubscript writing B 1198711

loc(119883) 119872119901120601(119883) L

119901120601(119883) and so on

(thus B denotes the family of all balls in 119883) and refer to thissetting as to the global one Notice that the proofs of all resultsstated in the paper contain 120588 = infin as a special case

Parallel to the main theory we shall also develop analternative theory in the framework of closed balls 119861(119909 119903) =119910 isin 119883 119889(119909 119910) le 119903 Note that in the metric case 119861(119909 119903)is indeed a closed set and in general if all balls are assumedto be Borel sets then 119861(119909 119903) is Borel too The definitions ofMorrey and Campanato spaces based on closed balls (in factbeing closed cubes) in the framework of (R119899

119889(infin)

120583)occur inthe literature compare for instance [3] Clearly taking closedballs makes no difference with respect to the theory based onopen balls when 120583 has the property that 120583(120597119861) = 0 for everyball 119861 where 120597119861 = 119861 119861 this happens for instance when119889120583(119909) = 119908(119909)119889119909 where 119908 ge 0 and 119889119909 denotes Lebesguemeasure onR119899 In general however the two alternative waysmay give different outcomes Relevant comments indicatingcoincidences or differences of both theories will be given inseveral places

The general notion of local maximal operators was intro-duced in [4] and some objects associated to them mostly theBMO spaces were investigated there in the setting ofmeasuremetric spaces The present paper enhances investigationdone in [4] in several directions First the broader contextof quasimetric measure spaces is considered Second thecondition 120583(119861) gt 0 for every ball 119861 is not assumedThird several variants of generalized maximal operators areadmitted into our investigation All this makes the developedtheory more flexible in possible applications

Throughout the paper we use a standard notation Whilewriting estimates we use the notation 119878 ≲ 119879 to indicate that119878 le 119862119879 with a positive constant 119862 independent of significantquantities We shall write 119878 ≃ 119879 when simultaneously 119878 ≲ 119879and 119879 ≲ 119878 for instance 119889 ≃ 1198891015840 means the equivalence ofquasimetrics 119889 and 1198891015840 and so forth By 119871119901

(119883) = 119871119901

(119883 120583)1 le 119901 lt infin we shall denote the usual Lebesgue 119871119901 spaceon the measure space (119883 120583) Whenever we refer to a ball weunderstand that its center and radius have been chosen (ingeneral these need not be uniquely determined by 119861 as a set)Thenwriting 120591119861 for a given ball119861 = 119861(119909 119903) and 120591 gt 0 meansthat 120591119861 = 119861(119909 120591119903) For a function 119891 isin 1198711

loc120588(119883) its averagein a ball 119861 = 119861(119909 119903) isinB


⟨119891⟩119861=

1

120583 (119861)int119861

119891119889120583 (8)

and similarly for any other Borel set 119860 0 lt 120583(119860) lt infinand any 119891 whenever the integral makes sense When thesituation is specified to the Euclidean setting of R119899 we shallconsider either the metric 119889(2) induced by the norm sdot

2or

119889(infin) induced by sdot

infin

2 Generalized Local Maximal Operators

By defining and investigating generalized local Morrey andCampanato spaces on quasimetric measure spaces we adaptthe general approach to these spaces presented by Nakai [2](and follow the notation used there) and extend the conceptof locality introduced in [4] Also we find it more convenientto work with relevant maximal operators when investigating





+rarr R

+and then to




120601 (1199091 119903

1) = 120601 (119909

2 119903

2) whenever 119861 (119909

1 119903

1) = 119861 (119909

2 119903

2)

(9)



1 119909

2isin 119883 and 119903

1 119903

2gt 119877

119883 120601(119909

1 119903

1) = 120601(119909

2 119903

2))


120601119898120572(119861) = 120583 (119861)

120572

120601119903120572(119861) = 119903(119861)

120572

(10)


120583 (119861) ≃ 119903(119861)119899

(11)


120601119898120572(119861) ≃ 120601

119903120572(119861)

119899

119861 isinB120588 (12)


119901120601120588acting


119872119901120601120588119891 (119909) = sup

119909isin119861isinB120588

1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(13)



119872119888

119901120601120588119891 (119909) = sup

1205880(119909)lt119903lt120588(119909)

1

120601 (119861 (119909 119903))

times (1

120583 (119861 (119909 119903))int119861(119909119903)

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(14)



119872119901120601120588119891 (119909) = sup

119909isin119861isinB120588

inf119911isinC

1

120601 (119861)

times (1

120583 (119861)int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(15)




119901120601120588119891 (119909) = sup

119909isin119861isinB120588

1

120601 (119861)

times (1

120583 (119861)int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(16)



119901120601120588 Clearly

119872119901120601120588119891 (119909) ≃

119901120601120588119891 (119909) 119872

119888119901120601120588119891 (119909) ≃

119888119901120601120588119891 (119909)

(17)



119901120601120588instead of

119901120601120588


1198711



11987211120588 119872

11120588 and




1198721199011120601120588119891 (119909) le 119872

1199012120601120588119891 (119909) 119909 isin 119883 (18)




and 119901120601120588




119901120601120588119872

119901120601120588 and


is

119872119901120601120588119891 (119909) = sup

119909isin119861isinB120588

1

120601 (119861)

(1

120583 (119861)

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(19)




120583(119861(119909 120576)) gt 0 then 1205880(119909) = 120588




120581120588119891 (119909) = sup

119909isin119861isinB120588

1

120583 (120581119861)int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 119909 isin 119883 (20)





1120601120581120588 where 120601

120581(119861) = 120583(120581119861)120583(119861) See [7 p 493] where





119901120581120588119891 (119909) = (

120581120588(10038161003816100381610038161198911003816100381610038161003816

119901

) (119909))1119901

119909 isin 119883 (21)


119872(120572)

119891 (119909) = sup119909isin119861isinB

120588

1

120583(119861)1minus120572120591

int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 119909 isin 119883 (22)


120583 (119861 (119909 119903)) ≲ 119903120591

(23)




119872(120572)

120581119901119891 (119909) = sup

119909isin119861isinB120588

(1

120583(120581119861)1minus119901120572120591

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(24)


119889(infin)


119901120601120581

where

120601120581(119861) = 120601

120581119901120572(119861) = (

120583(120581119861)1minus119901120572120591

120583 (119861))

1119901

(25)




0is an




is usc and 120601119898120572



119901120601120588119891119872119888

119901120601120588119891119872

119901120601120588119891 and119872119888



119901120601120588119891

when 119891 isin 1198711

loc120588(119883)



120582(119891) =


0from this set


0isin 119861

and

1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

gt 120582 (27)


119901120601120588119891(119910) gt 120582 hence 119861 sub 119865





119865120582= 119909 isin 119883 119872


0from this


0isin 119861 and



119901120601120588119891(119910) gt 120582 and hence 119861 sub 119865



1205880 120588 and 120601 For119872119888


120582= 119865

119888

120582(119891) =

119909 isin 119883 119872119888


119865119888

120582= ⋃

119903gt0

119910 isin 119883 1

120601 (119861 (119910 119903))(⟨10038161003816100381610038161198911003816100381610038161003816

119901

⟩119861(119910119903)

)

1119901

gt 120582

cap 119910 isin 119883 120588 (119910) gt 119903 gt 1205880(119910)

(28)



119910 isin 119883 120588 (119910) gt 119903 gt 1205880(119910) = 119910 isin 119883 120588 (119910) gt 119903

cap 119910 isin 119883 1205880(119910) lt 119903

(29)



119883 ni 119910 997891997888rarr1

120601 (119861 (119910 119903))(⟨10038161003816100381610038161198911003816100381610038161003816

119901

⟩119861(119910119903)

)

1119901

(30)


119860 120583(119860) lt infin But then

119883 ni 119910 997891997888rarr1

120601 (119861 (119910 119903))

120583(119860 cap 119861 (119910 119903))1119901

120583 (119861 (119910 119903))

(31)









119888119901120601120588119891(119909) gt 120582 an argument


does the job


120588cupB




0(119910

0) lt

1199030lt 120588(119910

0)

120601 (119861 (1199100 119903

0)) = lim

119903rarr119903+

0

120601 (119861 (1199100 119903))

120601 (119861 (1199100 119903

0)) = lim


0

120601 (119861 (1199100 119903))

(32)

Note that 120601119898120572




119872119901120601120588119891 (119909) = 119872

119901120601120588119891 (119909) 119891 isin 119871

119901

119897119900119888120588(119883) 119909 isin 119883 (33)

119901120601120588119891 (119909) =

119901120601120588119891 (119909) 119891 isin 119871

119901

119897119900119888120588(119883) 119909 isin 119883 (34)


119897119900119888120588(119883) the functions 119872

119901120601120588119891

119872119888

119901120601120588119891

119901120601120588119891 and


measurable


0gt 0 we have

119861 (1199100 119903

0) = ⋂

119903gt1199030

119861 (1199100 119903) 119861 (119910

0 119903

0) = ⋃

0lt119903lt1199030

119861 (1199100 119903)

(35)


119861(1199100 119903

0) isinB

120588 such that 119909

0isin 119861


1

120601 (1198610)(

1

120583 (1198610)int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901


120588

1

120601 (119861)

(1

120583 (119861)

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(36)

Let 119903119899rarr 119903

minus

0and 119903

119899gt 120588

0(119910



1

120601 (119861 (1199100 119903

119899))

(1

120583 (119861 (1199100 119903

119899))

int119861(1199100119903119899)

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

997888rarr1

120601 (1198610)(

1

120583 (1198610)int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(37)


0= 119861(119910

0 119903

0) isinB

120588 such that 119909

0isin 119861


1

120601 (1198610)

(1

120583 (1198610)

int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901


120588

1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(38)

Let 119903119899rarr 119903

+

0and 119903

119899lt 120588(119910



1

120601 (119861 (1199100 119903

119899))(

1

120583 (119861 (1199100 119903

119899))int119861(1199100119903119899)

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

997888rarr1

120601 (1198610)

(1

120583 (1198610)

int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(39)


⟨119891⟩119861(11991001199030)= lim

119899rarrinfin

⟨119891⟩119861(1199100119903119899)

⟨119891⟩119861(11991001199030)= lim

119899rarrinfin

⟨119891⟩119861(1199100119903119899)

(40)

(note that 119871119901

loc120588(119883) sub 1198711






119870= 119870(3119870+2) suffices



0equiv 0


120583 (120591119861) ≲ 120583 (119861) 119861 isinB120588 (41)


0equiv 0





120588 and (119883 119889)


Then1198721120601120588

maps1198711

(119883 120583) into1198711infin


1120601120588is bounded on 119871119901



1120601120588≲ 119872


implies 1198721120601120588







11120588 and 1120581119870120588



1120601120588follows




119901





119888





119871119901120601120588

(119883) = 119871119901120601120588

(119883 119889 120583)

L119901120601120588

(119883) =L119901120601120588

(119883 119889 120583)

(42)


= sup119909isin119883

119872119901120601120588119891 (119909) lt infin (43)

10038171003817100381710038171198911003817100381710038171003817L119901120601120588

= sup119909isin119883

119872119901120601120588119891 (119909) lt infin (44)


sup119909isin119883

119872119888

119901120601120588119891 (119909) = sup

119909isin119883

119872119901120601120588119891 (119909)

sup119909isin119883

119872119888119901120601120588119891 (119909) = sup

119909isin119883

119872119901120601120588119891 (119909)

(45)


119901120601120588and119872

119901120601120588in (43) and (44)


sup119909isin119883

119888119901120601120588119891 (119909) = sup

119909isin119883

119901120601120588119891 (119909) (46)


or 119901120601120588

in place of 119872119901120601120588


119901120601120588(119883) and L

119901120601120588(119883) a priori



119901120601120588(119883) sub 119871

119901

loc120588(119883) andL119901120601120588(119883) sub 119871

119901


119872119901120601120588119891 (119909) le 119872

119901120601120588119891 (119909) 119909 isin 119883 (47)


le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119871119901120601120588

(119883) subL119901120601120588

(119883) (48)


1198711199011120601120588(119883) sub 119871

1199012120601120588(119883) L

1199011120601120588(119883) subL

1199012120601120588(119883)

(49)

for 1 le 1199011lt 119901


119872119901120601120588

When 120588



BMO120588(119883) = BMO



Since sdot L119901120601120588


119901120601120588(119883)119882

0120588


is

1198820120588= 119891 isinL

119901120601120588(119883)

10038171003817100381710038171198911003817100381710038171003817L119901120601120588

= 0 (50)






120588and


1 119861

119898 sub B

120588such

that 119909 isin 1198611 119910 isin 119861

119898 and 119861

119894cap 119861

119894+1= 0 119894 = 1 119898 minus 1



119901120601120588and 119872



119901120601120588(119883)



119871119901120601120588

(119883) = 119871119901120601120588

(119883) L119901120601120588

(119883) =L119901120601120588

(119883) (51)



119901120582andL


to the choice of 120601 = 120601119898120572


119871119901120582= 119871

119901120582(R

119899

)

= 119891 10038171003817100381710038171198911003817100381710038171003817(119901120582)

= sup119909isinR119899119903gt0

(1

119903120582int119861(119909119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

L119901120582=L

119901120582(R

119899

)

= 119891 10038171003817100381710038171198911003817100381710038171003817119901120582

= sup119909isinR119899119903gt0

inf119911isinC(1

119903120582int119861(119909119903)

1003816100381610038161003816119891 (119910) minus 1199111003816100381610038161003816

119901

119889119910)

1119901

lt infin

(52)


1199010= 119871

119901

(R119899


119901120582C cong

119871119901120582

for 120582 = 119899 L119901119899= BMO(R119899

) and 119871119901119899= 119871

infin

(R119899

) andfor 119899 lt 120582 le 119899+119901L

119901120582= Lip

120572(R119899




120583 (119861 (119909 2119903)) ≲ 120583 (119861 (119909 119903)) (53)


0equiv 0






loc120588(119883) there exists119909


119901120601120588119891(119909




119901120601119891(119909


119891 isin 119871119901

loc(R119899


119901120582(R119899


119901120601120588(119883 119889 120583) is





loc120588(119883) 1198820120588


119901120601120588119891(119909




119901120601119891(119909

0) = infin for

every 119891 isin 119871119901

loc(R119899

) C(R119899


L119901120582(R119899

) = C(R119899


119901120582(R119899


119901120601(R119899

)




0lt119905le119903120601(119905) gt 0 for every


) = 119871119901120601(R119899


119903le119905ltinfin120601(119905)119905

119899119901


119903le119905ltinfin120601(119905)119905


119901120601(R119899

) = 119871119901

120601(R119899





with these of (119883 1198891015840

120583 1205881015840

1206011015840



1205881015840

1206011015840




1 119861

1015840

1198990


12058810158401198891015840 such that

1206011015840

(1198611015840

119895) 120583(119861

1015840

119895)1119901

≲ 120601 (119861) 120583(119861)1119901

119895 = 1 1198990 (54)

Then10038171003817100381710038171198911003817100381710038171003817119871119901120601120588119889

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601101584012058810158401198891015840

119891 isin 119871119901120601101584012058810158401198891015840 (119883) (55)


119901120601120588119889(119883) Similarly if for


isin B12058810158401198891015840 such that

119861 sub 1198611015840 and

1206011015840

(1198611015840

) 120583(1198611015840

)1119901

≲ 120601 (119861) 120583(119861)1119901

(56)

then10038171003817100381710038171198911003817100381710038171003817L119901120601120588119889

≲10038171003817100381710038171198911003817100381710038171003817L119901120601101584012058810158401198891015840

119891 isinL119901120601101584012058810158401198891015840 (119883) (57)

and henceL119901120601101584012058810158401198891015840(119883) subL

119901120601120588119889(119883)



0

119895=1119861

1015840



1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le

1198990

sum

119895=1

1

120601 (119861) 120583(119861)1119901

(int1198611015840

119895

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲ sum

119895

1

1206011015840 (1198611015840

119895) 120583(119861

1015840

119895)1119901

(int1198611015840

119895

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le 1198990

10038171003817100381710038171198911003817100381710038171003817119871119901120601101584012058810158401198891015840

(58)




119901120601101584012058810158401198891015840(119883) and 119861 isin B

120588119889 and consider 1198611015840 119861 sub 1198611015840


inf119911isinC

1

120601 (119861)(1

120583 (119861)int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

≲ inf119911isinC

1

1206011015840 (1198611015840) 120583(1198611015840)1119901

(int1198611015840

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817L119901120601101584012058810158401198891015840

(59)



1015840

1205881015840

1206011015840

) switched we have

119871119901120601120588119889

(119883) = 119871119901120601101584012058810158401198891015840 (119883)

L119901120601120588119889

(119883) =L119901120601101584012058810158401198891015840 (119883)

(60)



(119861) ≲ 120601(119861)



0= 1 and the




1015840


119898120572and 1206011015840

= 120601(1198891015840

)

119898120572 Then for


119872119901120601119889119891 (119909) ≃ 119872

11990112060110158401198891015840119891 (119909)

119872119901120601119889119891 (119909) ≃ 119872

11990112060110158401198891015840119891 (119909)

(61)


119871119901120601119889

(119883) = 11987111990112060110158401198891015840 (119883) L

119901120601119889(119883) =L

11990112060110158401198891015840 (119883)

(62)


1198891015840



119861 (119909 119903) sub 1198611015840

(119909 119888119903) sub 119861 (119909 1198882

119903) (63)




120583 (1198611015840

(119909 119888119903)) le 120583 (119861 (119909 1198882

119903)) le 119862120583 (119861 (119909 119903)) (64)



119889(infin)






sup120583(119876)gt0

120583(119896119876)1119902minus1119901

(int119876

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (65)



space 119871119901120601(R119899

119889(infin)


120601 (119876) = 120583(119876)minus1119901

120583(119896119876)1119901minus1119902

(66)


119901(119896 120583) does not depend


= 119889(infin) 120588 = 1205881015840


(119876) =

120583(119876)minus1119901

120583(1198961015840






119889(infin)




satisfying

sup120583(119876)gt0

(1

120595 (120583 (119896119876))int119876

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (67)



119889(infin)

120583) where

120601 (119876) = (120595 (120583 (119896119876))

120583 (119876))

1119901

(68)


(119896 120583) =M119902




(119876) = (120595(120583(1198961015840

119876))120583(119876))1119901



119886





B119886



120583(119861)1119901

(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (69)


119889(2)

120574119899) where 120574


119899(119909) =

120587minus1198992 exp(minus1199092





119889(2)


space E119901119902

B119886


119883 satisfying

100381710038171003817100381711989110038171003817100381710038171198711(120574

119899)+ sup

119861isinB119886

120583(119861)1119901

(1

120583 (119861)int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816

119901

119889120583)

1119901

lt infin

(70)




(R2

120583) (= L11(R

2

119889(2)


(R2

120583) (=L

11(R2

119889(infin)

120583)) differ



119899 where 119886

119899gt 0



loc(N) = ℓ119901


119895isinN isin

ℓ119901

(N 120583)1198721199011119891 and119872119888


1198721199011119891 (119899) = 119872

1198881199011119891 (119899) = (sum

119895

1003816100381610038161003816119891 (119895) minus ⟨119891⟩N

1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(71)

where ⟨119891⟩N = sum119895119891(119895)119886

119895 HenceL

1199011(N) = ℓ119901

(N 120583) 119891L1199011

≃ 119891 minus ⟨119891⟩Nℓ119901(N120583) 119882

0= C and L




1198721199011119891 (119899) = 119872

119888

1199011119891 (119899)

= max

1003816100381610038161003816119891 (119899)1003816100381610038161003816 (sum

119895

1003816100381610038161003816119891 (119895)1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(72)

and hence 1198711199011(N) = ℓ

infin



of the form 119861 = 119899 119899 isin N 119871119901



1199011120588(N) = 119904(N) and119891L




hence 1198711199011120588(N) = ℓ

infin


ℓinfin



loc(Ω) such that

10038171003817100381710038171198911003817100381710038171003817BMO(Ω)

= sup119861subΩ

1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816 lt infin (73)


120588Ω(119909) = 119889 (119909 120597Ω) 119909 isin Ω (74)


119889 (119909 120597Ω) = inf 119889 (119909 119910) 119910 isin 120597Ω (75)




10038171003817100381710038171198911003817100381710038171003817119871119901120582(Ω)

= sup119909isinΩ

sup0lt119903lt119903

0

(1

119903120582int119861(119909119903)capΩ

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

(76)









119901120601(Ω) =

119871119901120601120588Ω

(Ω) andL119901120601(Ω) =L

119901120601120588Ω


Ω

119871119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817119871119901120601

(Ω)

= sup119861isinB120588(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

L119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817L119901120601

(Ω)

= sup119861isinB120588(Ω)

inf119911isinC

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910) minus 1199111003816100381610038161003816

119901

119889119910)

1119901

lt infin

(77)



function 120588119896= 120588

119896Ωas

120588119896(119909) = 119896 sdot 119889 (119909 120597Ω) 119909 isin Ω (78)

so that 1205881= 120588


119871119901120601119896(Ω) = 119871

119901120601120588119896

(Ω) and L119901120601119896(Ω) = L

119901120601120588119896


119901120601119896(Ω) and L

119901120601119896(Ω)


119901120601119896(Ω)Ω = ⋃

119895isin119869Ω

119895(119869 is finite



119895 then

10038171003817100381710038171198911003817100381710038171003817119871119901120601119896

(Ω)= sup

119909isinΩ


119896(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901


= sup119895isin119869



119896(Ω119895)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

= sup119895isin119869

10038171003817100381710038171003817119891

119895

10038171003817100381710038171003817119871119901120601119895119896(Ω119895)

(79)



119896and 120588(infin)

119896 119871B

119901120601119896(Ω) and 119871Q

119901120601119896(Ω) and so



BΩ



by QΩ

119896


119898120572 Clearly

120601119898120572


119901120572119896(Ω) in place of 119871Q

119901120601119898120572







119901120572119896(Ω)


(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817119871Q119901120572119896

(Ω) 119891 isin 119871

Q119901120572119896

(Ω) (80)10038171003817100381710038171198911003817100381710038171003817LQ119901120572119898

(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) 119891 isinL

Q119901120572119896

(Ω) (81)


Q119901120572119898

(Ω) andLQ

119901120572119896(Ω) sub LQ







of them in QΩ

119896 Thus

1

|119876|120572(1

|119876|int119876

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le1

|119876|120572sum

119895

(1

|119876|int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le sum

119895

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572(1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le 21198991198731003817100381710038171003817119891

1003817100381710038171003817119871Q119901120572119896

(Ω)

(82)



LQ119901120572119896(Ω) and 119876 isin QΩ

119898 QΩ

119896 Then

inf119911isinC

1

|119876|120572(1

|119876|int119876

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

=1

|119876|120572+1119901

10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

(83)

where 119871119901




1199011015840


=

1 and 1198711199011015840



119876


10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

= sup10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

120595 isin 1198711199011015840

0(119876)

100381710038171003817100381712059510038171003817100381710038171199011015840le 1

(84)


119862(119896119898 1199011015840


119898



119895isin

QΩ


119895isin 119871

1199011015840

0(119876) 1 le


119895and 120595

1198951199011015840le 119862120595

1199011015840 for

119895 = 1 119873 and 120595 = sum119873

119895=1120595

119895 Take 120595 isin 119871119901

1015840



subcubes 119876119895119873


119895119873


Then

1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816100381610038161003816

int119876

119891(

119873

sum

119895=1

120595119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

119891120595119895

100381610038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

(119891 minus 119891119876119895

)120595119895

100381610038161003816100381610038161003816100381610038161003816

le 119862

119873

sum

119895=1

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572+1119901

(int119876119895

100381610038161003816100381610038161003816119891 minus 119891

119876119895

100381610038161003816100381610038161003816

119901

)

1119901

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω)

(85)

Hence

1

|119876|120572+1119901

100381710038171003817100381711989110038171003817100381710038171198711(119876)C

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) (86)


(Ω)le 119862119873119891LQ

119901120572119896(Ω)




119901120572119896(Ω)



119898



119896




1015840

0(119861)




1199011015840

0(119861) 1 le 119895 le 119873 such

that supp(120595119895) sub 119861

119895and 120595

1198951199011015840le 119862120595

1199011015840 for 119895 = 1 119873

and 120595 = sum119873

119895=1120595





lt 119877 and 120576 gt1 minus 119877

1015840

119877 there exists 1198731015840

= 1198731015840

(1198771015840


0 119877) = 119909 isin R119899


points 1199091 119909


119861 (1199090 2119877 minus 119877

1015840

) 119861 (1199090 119877

1015840

) sub cup1198731015840

119895=1119861 (119909

119895 120576119877) (87)

(if 1198771015840

= 0 then we set 119861(1199090 0) = 0)




0)

with 1199030= 119898119889

(2)

(1199090 120597Ω) Let 119877

119895= (119895119904)119903



119895= 119878(119909

0 119877




0 119877

119895+1)

119861(1199090 119877



0and 1198771015840

= 119877119895minus1=

((119895 minus 1)119904)1199030(so that 1198771015840


1015840


0 119903


119904minus1

119895=1119873

1015840

((119895 minus 1)119895 2119895 119899)


119896 take

119861(119909lowast

0 119903

lowast


0 119877



(120596 1199090) le

119889(2)

(120596 119909lowast

0) + 119889

(2)

(119909lowast

0 119909

0) and 119889(2)

(119909lowast

0 119909

0) = ((119904 minus 1)119904)119903

0=

((119904 minus 1)119904)119898119889(2)


119889(2)

(119909lowast

0 120597Ω) ge 119889

(2)

(1199090 120597Ω) (1 minus 119898

119904 minus 1

119904) (88)


119903lowast

0=2

1199041199030=2

119904119898119889

(2)

(1199090 120597Ω) lt

2

119904

1

1 minus 119898119889

(2)

(119909lowast

0 120597Ω)

le 119896119889(2)

(119909lowast

0 120597Ω)

(89)



Q11990112057212

(Ω) and sdot 119871Q119901120572

(Ω)= sdot

119871Q11990112057212

(Ω)(the


119901120572(Ω) LB




119871Q119901120572(Ω) = 119871

B119901120572(Ω) L

Q119901120572(Ω) =L

B119901120572(Ω) (90)



119876or 119876



infinle sdot



119861119876isinBΩ

radic119899119896 if119876 isin QΩ

119896 and119876

119861isin QΩ

radic119899119896 if 119861 isinBΩ

119896 Moreover

|119861119876| = 119888

1|119876| and |119876

119861| = 119888

2|119861| where 119888

1and 119888

2 depend on the


119901120572radic119899119896(Ω) For any 119861


inf119911isinC

1

|119861|120572(1

|119861|int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

le 1198881015840

2inf119911isinC

1

1003816100381610038161003816119876119861

1003816100381610038161003816

120572(1

1003816100381610038161003816119876119861

1003816100381610038161003816

int119876119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

(91)

Consequently 119891LB119901120572119896

(Ω)le 119888

1015840

2119891LQ

119901120572radic119889119896


that LQ119901120572radic119899119896



119901120572(Ω) and sdot LB

119901120572(Ω)

le

119862 sdot LQ119901120572


in an analogous way


Ω)





119870= 119870(2119870 + 1)









120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ 120601 (119861) 120583(119861)1119901

(92)


11120588 is bounded on 119871119901


119901120601120591119870120588(119883) to 119871

119901120601120588(119883)


11120588 thatis

119872120588119891 (119909) = sup

119909isin119861isinB120588

1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 (93)



splitting119891 = 1198911+119891


1= 119891120594

120591119870119861


119872120588119891

2(119909) le sup

1198611015840

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 (94)



119909 isin 1198611015840 and 119903(1198611015840


1

120601 (119861)

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 le

1

120593 (119903 (119861))(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le1

120601 (1198611015840)(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

(95)

Consequently

1

120601 (119861)119872

120588119891

2(119909) le

10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119909 isin 119861 (96)


119872120588is bounded on 119871119901

(119883 120583) give

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891)

119901

119889120583)

1119901

le sum

119894=12

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891

119894)119901

119889120583)

1119901

le 119888119901

1

120601 (119861) 120583(119861)1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

le 1198881015840

119901

1

120601 (120591119870119861) 120583(120591

119870119861)

1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

le 11988810158401015840

119901

10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

(97)


119871119901120601120588

≲ 119891119871119901120601120591119870120588


119901120582(R119899



(R119899


120593(119903) = 119903120572 120572 le 0 so that 120601 = 120601




119901120601119903120572

into itself


120573by letting

119868120573119891 (119909) = int

119883

119891 (119910)

119889(119909 119910)119899minus120573

119889120583 (119910) 119909 isin 119883 (98)


120593 (119903)


int

infin

119903

119905120573minus1

120593 (119905) 119889119905 ≲ 120595 (119903) 119903 gt 0

(99)


120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ Ψ (119861) 120583(119861)1119902

(100)



(119883 120583) to119871119902


119901120601(119883) to 119871

119902Ψ120588(119883)


le sdot 119871119902Ψ


1

Ψ (119861)(1

120583 (119861)int119861

1003816100381610038161003816100381611986812057311989110038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(101)



1+ 119891

2

1198911= 119891120594

120591119870119861 and 120591





119868120573and (100) we write

1

Ψ (119861)(1

120583 (119861)int119861

10038161003816100381610038161003816119868120573119891

1

10038161003816100381610038161003816

119902

119889120583)

1119902

le1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171003817119868120573119891

1

10038171003817100381710038171003817119871119902(120583)

≲1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171198911

1003817100381710038171003817119871119901(120583)

≲1

120601 (120591119870119861)(

1

120583 (120591119870119861)int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(102)


119870119903)

119888

sub 119861(119909 119903)119888


10038161003816100381610038161003816119868120573119891

2(119909)10038161003816100381610038161003816le int

(120591119870119861)119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le int119861(119909119903)

119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

=

infin

sum

119895=0

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le

infin

sum

119895=0

(2119895

119903)120573minus119899

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲

infin

sum

119895=0

(2119895

119903)120573 1

120583 (119861 (119909 2119895+1119903))int119861(1199092119895+1

119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲ (

infin

sum

119895=0

(2119895

119903)120573

120601 (119861 (119909 2119895+1

119903)))10038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ int

infin

119903

119905120573minus1

120593 (119905) 11988911990510038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ 120595 (119903)10038171003817100381710038171198911003817100381710038171003817119871119901120601

(103)


1

Ψ (119861 (119886 119903))(

1

120583 (119861 (119886 119903))int119861(119886119903)

10038161003816100381610038161003816119868120573119891

2

10038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(104)



(120583) to 119871119902



120573is bounded

from 119871119901120601(119883) to 119871

119902Ψ(119883)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













+rarr R

+and then to




120601 (1199091 119903

1) = 120601 (119909

2 119903

2) whenever 119861 (119909

1 119903

1) = 119861 (119909

2 119903

2)

(9)



1 119909

2isin 119883 and 119903

1 119903

2gt 119877

119883 120601(119909

1 119903

1) = 120601(119909

2 119903

2))


120601119898120572(119861) = 120583 (119861)

120572

120601119903120572(119861) = 119903(119861)

120572

(10)


120583 (119861) ≃ 119903(119861)119899

(11)


120601119898120572(119861) ≃ 120601

119903120572(119861)

119899

119861 isinB120588 (12)


119901120601120588acting


119872119901120601120588119891 (119909) = sup

119909isin119861isinB120588

1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(13)



119872119888

119901120601120588119891 (119909) = sup

1205880(119909)lt119903lt120588(119909)

1

120601 (119861 (119909 119903))

times (1

120583 (119861 (119909 119903))int119861(119909119903)

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(14)



119872119901120601120588119891 (119909) = sup

119909isin119861isinB120588

inf119911isinC

1

120601 (119861)

times (1

120583 (119861)int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(15)




119901120601120588119891 (119909) = sup

119909isin119861isinB120588

1

120601 (119861)

times (1

120583 (119861)int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(16)



119901120601120588 Clearly

119872119901120601120588119891 (119909) ≃

119901120601120588119891 (119909) 119872

119888119901120601120588119891 (119909) ≃

119888119901120601120588119891 (119909)

(17)



119901120601120588instead of

119901120601120588


1198711



11987211120588 119872

11120588 and




1198721199011120601120588119891 (119909) le 119872

1199012120601120588119891 (119909) 119909 isin 119883 (18)




and 119901120601120588




119901120601120588119872

119901120601120588 and


is

119872119901120601120588119891 (119909) = sup

119909isin119861isinB120588

1

120601 (119861)

(1

120583 (119861)

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(19)




120583(119861(119909 120576)) gt 0 then 1205880(119909) = 120588




120581120588119891 (119909) = sup

119909isin119861isinB120588

1

120583 (120581119861)int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 119909 isin 119883 (20)





1120601120581120588 where 120601

120581(119861) = 120583(120581119861)120583(119861) See [7 p 493] where





119901120581120588119891 (119909) = (

120581120588(10038161003816100381610038161198911003816100381610038161003816

119901

) (119909))1119901

119909 isin 119883 (21)


119872(120572)

119891 (119909) = sup119909isin119861isinB

120588

1

120583(119861)1minus120572120591

int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 119909 isin 119883 (22)


120583 (119861 (119909 119903)) ≲ 119903120591

(23)




119872(120572)

120581119901119891 (119909) = sup

119909isin119861isinB120588

(1

120583(120581119861)1minus119901120572120591

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(24)


119889(infin)


119901120601120581

where

120601120581(119861) = 120601

120581119901120572(119861) = (

120583(120581119861)1minus119901120572120591

120583 (119861))

1119901

(25)




0is an




is usc and 120601119898120572



119901120601120588119891119872119888

119901120601120588119891119872

119901120601120588119891 and119872119888



119901120601120588119891

when 119891 isin 1198711

loc120588(119883)



120582(119891) =


0from this set


0isin 119861

and

1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

gt 120582 (27)


119901120601120588119891(119910) gt 120582 hence 119861 sub 119865





119865120582= 119909 isin 119883 119872


0from this


0isin 119861 and



119901120601120588119891(119910) gt 120582 and hence 119861 sub 119865



1205880 120588 and 120601 For119872119888


120582= 119865

119888

120582(119891) =

119909 isin 119883 119872119888


119865119888

120582= ⋃

119903gt0

119910 isin 119883 1

120601 (119861 (119910 119903))(⟨10038161003816100381610038161198911003816100381610038161003816

119901

⟩119861(119910119903)

)

1119901

gt 120582

cap 119910 isin 119883 120588 (119910) gt 119903 gt 1205880(119910)

(28)



119910 isin 119883 120588 (119910) gt 119903 gt 1205880(119910) = 119910 isin 119883 120588 (119910) gt 119903

cap 119910 isin 119883 1205880(119910) lt 119903

(29)



119883 ni 119910 997891997888rarr1

120601 (119861 (119910 119903))(⟨10038161003816100381610038161198911003816100381610038161003816

119901

⟩119861(119910119903)

)

1119901

(30)


119860 120583(119860) lt infin But then

119883 ni 119910 997891997888rarr1

120601 (119861 (119910 119903))

120583(119860 cap 119861 (119910 119903))1119901

120583 (119861 (119910 119903))

(31)









119888119901120601120588119891(119909) gt 120582 an argument


does the job


120588cupB




0(119910

0) lt

1199030lt 120588(119910

0)

120601 (119861 (1199100 119903

0)) = lim

119903rarr119903+

0

120601 (119861 (1199100 119903))

120601 (119861 (1199100 119903

0)) = lim


0

120601 (119861 (1199100 119903))

(32)

Note that 120601119898120572




119872119901120601120588119891 (119909) = 119872

119901120601120588119891 (119909) 119891 isin 119871

119901

119897119900119888120588(119883) 119909 isin 119883 (33)

119901120601120588119891 (119909) =

119901120601120588119891 (119909) 119891 isin 119871

119901

119897119900119888120588(119883) 119909 isin 119883 (34)


119897119900119888120588(119883) the functions 119872

119901120601120588119891

119872119888

119901120601120588119891

119901120601120588119891 and


measurable


0gt 0 we have

119861 (1199100 119903

0) = ⋂

119903gt1199030

119861 (1199100 119903) 119861 (119910

0 119903

0) = ⋃

0lt119903lt1199030

119861 (1199100 119903)

(35)


119861(1199100 119903

0) isinB

120588 such that 119909

0isin 119861


1

120601 (1198610)(

1

120583 (1198610)int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901


120588

1

120601 (119861)

(1

120583 (119861)

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(36)

Let 119903119899rarr 119903

minus

0and 119903

119899gt 120588

0(119910



1

120601 (119861 (1199100 119903

119899))

(1

120583 (119861 (1199100 119903

119899))

int119861(1199100119903119899)

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

997888rarr1

120601 (1198610)(

1

120583 (1198610)int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(37)


0= 119861(119910

0 119903

0) isinB

120588 such that 119909

0isin 119861


1

120601 (1198610)

(1

120583 (1198610)

int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901


120588

1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(38)

Let 119903119899rarr 119903

+

0and 119903

119899lt 120588(119910



1

120601 (119861 (1199100 119903

119899))(

1

120583 (119861 (1199100 119903

119899))int119861(1199100119903119899)

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

997888rarr1

120601 (1198610)

(1

120583 (1198610)

int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(39)


⟨119891⟩119861(11991001199030)= lim

119899rarrinfin

⟨119891⟩119861(1199100119903119899)

⟨119891⟩119861(11991001199030)= lim

119899rarrinfin

⟨119891⟩119861(1199100119903119899)

(40)

(note that 119871119901

loc120588(119883) sub 1198711






119870= 119870(3119870+2) suffices



0equiv 0


120583 (120591119861) ≲ 120583 (119861) 119861 isinB120588 (41)


0equiv 0





120588 and (119883 119889)


Then1198721120601120588

maps1198711

(119883 120583) into1198711infin


1120601120588is bounded on 119871119901



1120601120588≲ 119872


implies 1198721120601120588







11120588 and 1120581119870120588



1120601120588follows




119901





119888





119871119901120601120588

(119883) = 119871119901120601120588

(119883 119889 120583)

L119901120601120588

(119883) =L119901120601120588

(119883 119889 120583)

(42)


= sup119909isin119883

119872119901120601120588119891 (119909) lt infin (43)

10038171003817100381710038171198911003817100381710038171003817L119901120601120588

= sup119909isin119883

119872119901120601120588119891 (119909) lt infin (44)


sup119909isin119883

119872119888

119901120601120588119891 (119909) = sup

119909isin119883

119872119901120601120588119891 (119909)

sup119909isin119883

119872119888119901120601120588119891 (119909) = sup

119909isin119883

119872119901120601120588119891 (119909)

(45)


119901120601120588and119872

119901120601120588in (43) and (44)


sup119909isin119883

119888119901120601120588119891 (119909) = sup

119909isin119883

119901120601120588119891 (119909) (46)


or 119901120601120588

in place of 119872119901120601120588


119901120601120588(119883) and L

119901120601120588(119883) a priori



119901120601120588(119883) sub 119871

119901

loc120588(119883) andL119901120601120588(119883) sub 119871

119901


119872119901120601120588119891 (119909) le 119872

119901120601120588119891 (119909) 119909 isin 119883 (47)


le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119871119901120601120588

(119883) subL119901120601120588

(119883) (48)


1198711199011120601120588(119883) sub 119871

1199012120601120588(119883) L

1199011120601120588(119883) subL

1199012120601120588(119883)

(49)

for 1 le 1199011lt 119901


119872119901120601120588

When 120588



BMO120588(119883) = BMO



Since sdot L119901120601120588


119901120601120588(119883)119882

0120588


is

1198820120588= 119891 isinL

119901120601120588(119883)

10038171003817100381710038171198911003817100381710038171003817L119901120601120588

= 0 (50)






120588and


1 119861

119898 sub B

120588such

that 119909 isin 1198611 119910 isin 119861

119898 and 119861

119894cap 119861

119894+1= 0 119894 = 1 119898 minus 1



119901120601120588and 119872



119901120601120588(119883)



119871119901120601120588

(119883) = 119871119901120601120588

(119883) L119901120601120588

(119883) =L119901120601120588

(119883) (51)



119901120582andL


to the choice of 120601 = 120601119898120572


119871119901120582= 119871

119901120582(R

119899

)

= 119891 10038171003817100381710038171198911003817100381710038171003817(119901120582)

= sup119909isinR119899119903gt0

(1

119903120582int119861(119909119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

L119901120582=L

119901120582(R

119899

)

= 119891 10038171003817100381710038171198911003817100381710038171003817119901120582

= sup119909isinR119899119903gt0

inf119911isinC(1

119903120582int119861(119909119903)

1003816100381610038161003816119891 (119910) minus 1199111003816100381610038161003816

119901

119889119910)

1119901

lt infin

(52)


1199010= 119871

119901

(R119899


119901120582C cong

119871119901120582

for 120582 = 119899 L119901119899= BMO(R119899

) and 119871119901119899= 119871

infin

(R119899

) andfor 119899 lt 120582 le 119899+119901L

119901120582= Lip

120572(R119899




120583 (119861 (119909 2119903)) ≲ 120583 (119861 (119909 119903)) (53)


0equiv 0






loc120588(119883) there exists119909


119901120601120588119891(119909




119901120601119891(119909


119891 isin 119871119901

loc(R119899


119901120582(R119899


119901120601120588(119883 119889 120583) is





loc120588(119883) 1198820120588


119901120601120588119891(119909




119901120601119891(119909

0) = infin for

every 119891 isin 119871119901

loc(R119899

) C(R119899


L119901120582(R119899

) = C(R119899


119901120582(R119899


119901120601(R119899

)




0lt119905le119903120601(119905) gt 0 for every


) = 119871119901120601(R119899


119903le119905ltinfin120601(119905)119905

119899119901


119903le119905ltinfin120601(119905)119905


119901120601(R119899

) = 119871119901

120601(R119899





with these of (119883 1198891015840

120583 1205881015840

1206011015840



1205881015840

1206011015840




1 119861

1015840

1198990


12058810158401198891015840 such that

1206011015840

(1198611015840

119895) 120583(119861

1015840

119895)1119901

≲ 120601 (119861) 120583(119861)1119901

119895 = 1 1198990 (54)

Then10038171003817100381710038171198911003817100381710038171003817119871119901120601120588119889

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601101584012058810158401198891015840

119891 isin 119871119901120601101584012058810158401198891015840 (119883) (55)


119901120601120588119889(119883) Similarly if for


isin B12058810158401198891015840 such that

119861 sub 1198611015840 and

1206011015840

(1198611015840

) 120583(1198611015840

)1119901

≲ 120601 (119861) 120583(119861)1119901

(56)

then10038171003817100381710038171198911003817100381710038171003817L119901120601120588119889

≲10038171003817100381710038171198911003817100381710038171003817L119901120601101584012058810158401198891015840

119891 isinL119901120601101584012058810158401198891015840 (119883) (57)

and henceL119901120601101584012058810158401198891015840(119883) subL

119901120601120588119889(119883)



0

119895=1119861

1015840



1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le

1198990

sum

119895=1

1

120601 (119861) 120583(119861)1119901

(int1198611015840

119895

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲ sum

119895

1

1206011015840 (1198611015840

119895) 120583(119861

1015840

119895)1119901

(int1198611015840

119895

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le 1198990

10038171003817100381710038171198911003817100381710038171003817119871119901120601101584012058810158401198891015840

(58)




119901120601101584012058810158401198891015840(119883) and 119861 isin B

120588119889 and consider 1198611015840 119861 sub 1198611015840


inf119911isinC

1

120601 (119861)(1

120583 (119861)int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

≲ inf119911isinC

1

1206011015840 (1198611015840) 120583(1198611015840)1119901

(int1198611015840

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817L119901120601101584012058810158401198891015840

(59)



1015840

1205881015840

1206011015840

) switched we have

119871119901120601120588119889

(119883) = 119871119901120601101584012058810158401198891015840 (119883)

L119901120601120588119889

(119883) =L119901120601101584012058810158401198891015840 (119883)

(60)



(119861) ≲ 120601(119861)



0= 1 and the




1015840


119898120572and 1206011015840

= 120601(1198891015840

)

119898120572 Then for


119872119901120601119889119891 (119909) ≃ 119872

11990112060110158401198891015840119891 (119909)

119872119901120601119889119891 (119909) ≃ 119872

11990112060110158401198891015840119891 (119909)

(61)


119871119901120601119889

(119883) = 11987111990112060110158401198891015840 (119883) L

119901120601119889(119883) =L

11990112060110158401198891015840 (119883)

(62)


1198891015840



119861 (119909 119903) sub 1198611015840

(119909 119888119903) sub 119861 (119909 1198882

119903) (63)




120583 (1198611015840

(119909 119888119903)) le 120583 (119861 (119909 1198882

119903)) le 119862120583 (119861 (119909 119903)) (64)



119889(infin)






sup120583(119876)gt0

120583(119896119876)1119902minus1119901

(int119876

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (65)



space 119871119901120601(R119899

119889(infin)


120601 (119876) = 120583(119876)minus1119901

120583(119896119876)1119901minus1119902

(66)


119901(119896 120583) does not depend


= 119889(infin) 120588 = 1205881015840


(119876) =

120583(119876)minus1119901

120583(1198961015840






119889(infin)




satisfying

sup120583(119876)gt0

(1

120595 (120583 (119896119876))int119876

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (67)



119889(infin)

120583) where

120601 (119876) = (120595 (120583 (119896119876))

120583 (119876))

1119901

(68)


(119896 120583) =M119902




(119876) = (120595(120583(1198961015840

119876))120583(119876))1119901



119886





B119886



120583(119861)1119901

(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (69)


119889(2)

120574119899) where 120574


119899(119909) =

120587minus1198992 exp(minus1199092





119889(2)


space E119901119902

B119886


119883 satisfying

100381710038171003817100381711989110038171003817100381710038171198711(120574

119899)+ sup

119861isinB119886

120583(119861)1119901

(1

120583 (119861)int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816

119901

119889120583)

1119901

lt infin

(70)




(R2

120583) (= L11(R

2

119889(2)


(R2

120583) (=L

11(R2

119889(infin)

120583)) differ



119899 where 119886

119899gt 0



loc(N) = ℓ119901


119895isinN isin

ℓ119901

(N 120583)1198721199011119891 and119872119888


1198721199011119891 (119899) = 119872

1198881199011119891 (119899) = (sum

119895

1003816100381610038161003816119891 (119895) minus ⟨119891⟩N

1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(71)

where ⟨119891⟩N = sum119895119891(119895)119886

119895 HenceL

1199011(N) = ℓ119901

(N 120583) 119891L1199011

≃ 119891 minus ⟨119891⟩Nℓ119901(N120583) 119882

0= C and L




1198721199011119891 (119899) = 119872

119888

1199011119891 (119899)

= max

1003816100381610038161003816119891 (119899)1003816100381610038161003816 (sum

119895

1003816100381610038161003816119891 (119895)1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(72)

and hence 1198711199011(N) = ℓ

infin



of the form 119861 = 119899 119899 isin N 119871119901



1199011120588(N) = 119904(N) and119891L




hence 1198711199011120588(N) = ℓ

infin


ℓinfin



loc(Ω) such that

10038171003817100381710038171198911003817100381710038171003817BMO(Ω)

= sup119861subΩ

1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816 lt infin (73)


120588Ω(119909) = 119889 (119909 120597Ω) 119909 isin Ω (74)


119889 (119909 120597Ω) = inf 119889 (119909 119910) 119910 isin 120597Ω (75)




10038171003817100381710038171198911003817100381710038171003817119871119901120582(Ω)

= sup119909isinΩ

sup0lt119903lt119903

0

(1

119903120582int119861(119909119903)capΩ

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

(76)









119901120601(Ω) =

119871119901120601120588Ω

(Ω) andL119901120601(Ω) =L

119901120601120588Ω


Ω

119871119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817119871119901120601

(Ω)

= sup119861isinB120588(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

L119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817L119901120601

(Ω)

= sup119861isinB120588(Ω)

inf119911isinC

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910) minus 1199111003816100381610038161003816

119901

119889119910)

1119901

lt infin

(77)



function 120588119896= 120588

119896Ωas

120588119896(119909) = 119896 sdot 119889 (119909 120597Ω) 119909 isin Ω (78)

so that 1205881= 120588


119871119901120601119896(Ω) = 119871

119901120601120588119896

(Ω) and L119901120601119896(Ω) = L

119901120601120588119896


119901120601119896(Ω) and L

119901120601119896(Ω)


119901120601119896(Ω)Ω = ⋃

119895isin119869Ω

119895(119869 is finite



119895 then

10038171003817100381710038171198911003817100381710038171003817119871119901120601119896

(Ω)= sup

119909isinΩ


119896(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901


= sup119895isin119869



119896(Ω119895)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

= sup119895isin119869

10038171003817100381710038171003817119891

119895

10038171003817100381710038171003817119871119901120601119895119896(Ω119895)

(79)



119896and 120588(infin)

119896 119871B

119901120601119896(Ω) and 119871Q

119901120601119896(Ω) and so



BΩ



by QΩ

119896


119898120572 Clearly

120601119898120572


119901120572119896(Ω) in place of 119871Q

119901120601119898120572







119901120572119896(Ω)


(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817119871Q119901120572119896

(Ω) 119891 isin 119871

Q119901120572119896

(Ω) (80)10038171003817100381710038171198911003817100381710038171003817LQ119901120572119898

(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) 119891 isinL

Q119901120572119896

(Ω) (81)


Q119901120572119898

(Ω) andLQ

119901120572119896(Ω) sub LQ







of them in QΩ

119896 Thus

1

|119876|120572(1

|119876|int119876

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le1

|119876|120572sum

119895

(1

|119876|int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le sum

119895

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572(1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le 21198991198731003817100381710038171003817119891

1003817100381710038171003817119871Q119901120572119896

(Ω)

(82)



LQ119901120572119896(Ω) and 119876 isin QΩ

119898 QΩ

119896 Then

inf119911isinC

1

|119876|120572(1

|119876|int119876

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

=1

|119876|120572+1119901

10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

(83)

where 119871119901




1199011015840


=

1 and 1198711199011015840



119876


10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

= sup10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

120595 isin 1198711199011015840

0(119876)

100381710038171003817100381712059510038171003817100381710038171199011015840le 1

(84)


119862(119896119898 1199011015840


119898



119895isin

QΩ


119895isin 119871

1199011015840

0(119876) 1 le


119895and 120595

1198951199011015840le 119862120595

1199011015840 for

119895 = 1 119873 and 120595 = sum119873

119895=1120595

119895 Take 120595 isin 119871119901

1015840



subcubes 119876119895119873


119895119873


Then

1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816100381610038161003816

int119876

119891(

119873

sum

119895=1

120595119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

119891120595119895

100381610038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

(119891 minus 119891119876119895

)120595119895

100381610038161003816100381610038161003816100381610038161003816

le 119862

119873

sum

119895=1

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572+1119901

(int119876119895

100381610038161003816100381610038161003816119891 minus 119891

119876119895

100381610038161003816100381610038161003816

119901

)

1119901

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω)

(85)

Hence

1

|119876|120572+1119901

100381710038171003817100381711989110038171003817100381710038171198711(119876)C

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) (86)


(Ω)le 119862119873119891LQ

119901120572119896(Ω)




119901120572119896(Ω)



119898



119896




1015840

0(119861)




1199011015840

0(119861) 1 le 119895 le 119873 such

that supp(120595119895) sub 119861

119895and 120595

1198951199011015840le 119862120595

1199011015840 for 119895 = 1 119873

and 120595 = sum119873

119895=1120595





lt 119877 and 120576 gt1 minus 119877

1015840

119877 there exists 1198731015840

= 1198731015840

(1198771015840


0 119877) = 119909 isin R119899


points 1199091 119909


119861 (1199090 2119877 minus 119877

1015840

) 119861 (1199090 119877

1015840

) sub cup1198731015840

119895=1119861 (119909

119895 120576119877) (87)

(if 1198771015840

= 0 then we set 119861(1199090 0) = 0)




0)

with 1199030= 119898119889

(2)

(1199090 120597Ω) Let 119877

119895= (119895119904)119903



119895= 119878(119909

0 119877




0 119877

119895+1)

119861(1199090 119877



0and 1198771015840

= 119877119895minus1=

((119895 minus 1)119904)1199030(so that 1198771015840


1015840


0 119903


119904minus1

119895=1119873

1015840

((119895 minus 1)119895 2119895 119899)


119896 take

119861(119909lowast

0 119903

lowast


0 119877



(120596 1199090) le

119889(2)

(120596 119909lowast

0) + 119889

(2)

(119909lowast

0 119909

0) and 119889(2)

(119909lowast

0 119909

0) = ((119904 minus 1)119904)119903

0=

((119904 minus 1)119904)119898119889(2)


119889(2)

(119909lowast

0 120597Ω) ge 119889

(2)

(1199090 120597Ω) (1 minus 119898

119904 minus 1

119904) (88)


119903lowast

0=2

1199041199030=2

119904119898119889

(2)

(1199090 120597Ω) lt

2

119904

1

1 minus 119898119889

(2)

(119909lowast

0 120597Ω)

le 119896119889(2)

(119909lowast

0 120597Ω)

(89)



Q11990112057212

(Ω) and sdot 119871Q119901120572

(Ω)= sdot

119871Q11990112057212

(Ω)(the


119901120572(Ω) LB




119871Q119901120572(Ω) = 119871

B119901120572(Ω) L

Q119901120572(Ω) =L

B119901120572(Ω) (90)



119876or 119876



infinle sdot



119861119876isinBΩ

radic119899119896 if119876 isin QΩ

119896 and119876

119861isin QΩ

radic119899119896 if 119861 isinBΩ

119896 Moreover

|119861119876| = 119888

1|119876| and |119876

119861| = 119888

2|119861| where 119888

1and 119888

2 depend on the


119901120572radic119899119896(Ω) For any 119861


inf119911isinC

1

|119861|120572(1

|119861|int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

le 1198881015840

2inf119911isinC

1

1003816100381610038161003816119876119861

1003816100381610038161003816

120572(1

1003816100381610038161003816119876119861

1003816100381610038161003816

int119876119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

(91)

Consequently 119891LB119901120572119896

(Ω)le 119888

1015840

2119891LQ

119901120572radic119889119896


that LQ119901120572radic119899119896



119901120572(Ω) and sdot LB

119901120572(Ω)

le

119862 sdot LQ119901120572


in an analogous way


Ω)





119870= 119870(2119870 + 1)









120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ 120601 (119861) 120583(119861)1119901

(92)


11120588 is bounded on 119871119901


119901120601120591119870120588(119883) to 119871

119901120601120588(119883)


11120588 thatis

119872120588119891 (119909) = sup

119909isin119861isinB120588

1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 (93)



splitting119891 = 1198911+119891


1= 119891120594

120591119870119861


119872120588119891

2(119909) le sup

1198611015840

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 (94)



119909 isin 1198611015840 and 119903(1198611015840


1

120601 (119861)

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 le

1

120593 (119903 (119861))(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le1

120601 (1198611015840)(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

(95)

Consequently

1

120601 (119861)119872

120588119891

2(119909) le

10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119909 isin 119861 (96)


119872120588is bounded on 119871119901

(119883 120583) give

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891)

119901

119889120583)

1119901

le sum

119894=12

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891

119894)119901

119889120583)

1119901

le 119888119901

1

120601 (119861) 120583(119861)1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

le 1198881015840

119901

1

120601 (120591119870119861) 120583(120591

119870119861)

1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

le 11988810158401015840

119901

10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

(97)


119871119901120601120588

≲ 119891119871119901120601120591119870120588


119901120582(R119899



(R119899


120593(119903) = 119903120572 120572 le 0 so that 120601 = 120601




119901120601119903120572

into itself


120573by letting

119868120573119891 (119909) = int

119883

119891 (119910)

119889(119909 119910)119899minus120573

119889120583 (119910) 119909 isin 119883 (98)


120593 (119903)


int

infin

119903

119905120573minus1

120593 (119905) 119889119905 ≲ 120595 (119903) 119903 gt 0

(99)


120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ Ψ (119861) 120583(119861)1119902

(100)



(119883 120583) to119871119902


119901120601(119883) to 119871

119902Ψ120588(119883)


le sdot 119871119902Ψ


1

Ψ (119861)(1

120583 (119861)int119861

1003816100381610038161003816100381611986812057311989110038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(101)



1+ 119891

2

1198911= 119891120594

120591119870119861 and 120591





119868120573and (100) we write

1

Ψ (119861)(1

120583 (119861)int119861

10038161003816100381610038161003816119868120573119891

1

10038161003816100381610038161003816

119902

119889120583)

1119902

le1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171003817119868120573119891

1

10038171003817100381710038171003817119871119902(120583)

≲1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171198911

1003817100381710038171003817119871119901(120583)

≲1

120601 (120591119870119861)(

1

120583 (120591119870119861)int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(102)


119870119903)

119888

sub 119861(119909 119903)119888


10038161003816100381610038161003816119868120573119891

2(119909)10038161003816100381610038161003816le int

(120591119870119861)119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le int119861(119909119903)

119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

=

infin

sum

119895=0

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le

infin

sum

119895=0

(2119895

119903)120573minus119899

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲

infin

sum

119895=0

(2119895

119903)120573 1

120583 (119861 (119909 2119895+1119903))int119861(1199092119895+1

119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲ (

infin

sum

119895=0

(2119895

119903)120573

120601 (119861 (119909 2119895+1

119903)))10038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ int

infin

119903

119905120573minus1

120593 (119905) 11988911990510038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ 120595 (119903)10038171003817100381710038171198911003817100381710038171003817119871119901120601

(103)


1

Ψ (119861 (119886 119903))(

1

120583 (119861 (119886 119903))int119861(119886119903)

10038161003816100381610038161003816119868120573119891

2

10038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(104)



(120583) to 119871119902



120573is bounded

from 119871119901120601(119883) to 119871

119902Ψ(119883)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119901120601120588119872

119901120601120588 and


is

119872119901120601120588119891 (119909) = sup

119909isin119861isinB120588

1

120601 (119861)

(1

120583 (119861)

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(19)




120583(119861(119909 120576)) gt 0 then 1205880(119909) = 120588




120581120588119891 (119909) = sup

119909isin119861isinB120588

1

120583 (120581119861)int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 119909 isin 119883 (20)





1120601120581120588 where 120601

120581(119861) = 120583(120581119861)120583(119861) See [7 p 493] where





119901120581120588119891 (119909) = (

120581120588(10038161003816100381610038161198911003816100381610038161003816

119901

) (119909))1119901

119909 isin 119883 (21)


119872(120572)

119891 (119909) = sup119909isin119861isinB

120588

1

120583(119861)1minus120572120591

int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 119909 isin 119883 (22)


120583 (119861 (119909 119903)) ≲ 119903120591

(23)




119872(120572)

120581119901119891 (119909) = sup

119909isin119861isinB120588

(1

120583(120581119861)1minus119901120572120591

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

119909 isin 119883

(24)


119889(infin)


119901120601120581

where

120601120581(119861) = 120601

120581119901120572(119861) = (

120583(120581119861)1minus119901120572120591

120583 (119861))

1119901

(25)




0is an




is usc and 120601119898120572



119901120601120588119891119872119888

119901120601120588119891119872

119901120601120588119891 and119872119888



119901120601120588119891

when 119891 isin 1198711

loc120588(119883)



120582(119891) =


0from this set


0isin 119861

and

1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

gt 120582 (27)


119901120601120588119891(119910) gt 120582 hence 119861 sub 119865





119865120582= 119909 isin 119883 119872


0from this


0isin 119861 and



119901120601120588119891(119910) gt 120582 and hence 119861 sub 119865



1205880 120588 and 120601 For119872119888


120582= 119865

119888

120582(119891) =

119909 isin 119883 119872119888


119865119888

120582= ⋃

119903gt0

119910 isin 119883 1

120601 (119861 (119910 119903))(⟨10038161003816100381610038161198911003816100381610038161003816

119901

⟩119861(119910119903)

)

1119901

gt 120582

cap 119910 isin 119883 120588 (119910) gt 119903 gt 1205880(119910)

(28)



119910 isin 119883 120588 (119910) gt 119903 gt 1205880(119910) = 119910 isin 119883 120588 (119910) gt 119903

cap 119910 isin 119883 1205880(119910) lt 119903

(29)



119883 ni 119910 997891997888rarr1

120601 (119861 (119910 119903))(⟨10038161003816100381610038161198911003816100381610038161003816

119901

⟩119861(119910119903)

)

1119901

(30)


119860 120583(119860) lt infin But then

119883 ni 119910 997891997888rarr1

120601 (119861 (119910 119903))

120583(119860 cap 119861 (119910 119903))1119901

120583 (119861 (119910 119903))

(31)









119888119901120601120588119891(119909) gt 120582 an argument


does the job


120588cupB




0(119910

0) lt

1199030lt 120588(119910

0)

120601 (119861 (1199100 119903

0)) = lim

119903rarr119903+

0

120601 (119861 (1199100 119903))

120601 (119861 (1199100 119903

0)) = lim


0

120601 (119861 (1199100 119903))

(32)

Note that 120601119898120572




119872119901120601120588119891 (119909) = 119872

119901120601120588119891 (119909) 119891 isin 119871

119901

119897119900119888120588(119883) 119909 isin 119883 (33)

119901120601120588119891 (119909) =

119901120601120588119891 (119909) 119891 isin 119871

119901

119897119900119888120588(119883) 119909 isin 119883 (34)


119897119900119888120588(119883) the functions 119872

119901120601120588119891

119872119888

119901120601120588119891

119901120601120588119891 and


measurable


0gt 0 we have

119861 (1199100 119903

0) = ⋂

119903gt1199030

119861 (1199100 119903) 119861 (119910

0 119903

0) = ⋃

0lt119903lt1199030

119861 (1199100 119903)

(35)


119861(1199100 119903

0) isinB

120588 such that 119909

0isin 119861


1

120601 (1198610)(

1

120583 (1198610)int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901


120588

1

120601 (119861)

(1

120583 (119861)

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(36)

Let 119903119899rarr 119903

minus

0and 119903

119899gt 120588

0(119910



1

120601 (119861 (1199100 119903

119899))

(1

120583 (119861 (1199100 119903

119899))

int119861(1199100119903119899)

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

997888rarr1

120601 (1198610)(

1

120583 (1198610)int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(37)


0= 119861(119910

0 119903

0) isinB

120588 such that 119909

0isin 119861


1

120601 (1198610)

(1

120583 (1198610)

int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901


120588

1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(38)

Let 119903119899rarr 119903

+

0and 119903

119899lt 120588(119910



1

120601 (119861 (1199100 119903

119899))(

1

120583 (119861 (1199100 119903

119899))int119861(1199100119903119899)

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

997888rarr1

120601 (1198610)

(1

120583 (1198610)

int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(39)


⟨119891⟩119861(11991001199030)= lim

119899rarrinfin

⟨119891⟩119861(1199100119903119899)

⟨119891⟩119861(11991001199030)= lim

119899rarrinfin

⟨119891⟩119861(1199100119903119899)

(40)

(note that 119871119901

loc120588(119883) sub 1198711






119870= 119870(3119870+2) suffices



0equiv 0


120583 (120591119861) ≲ 120583 (119861) 119861 isinB120588 (41)


0equiv 0





120588 and (119883 119889)


Then1198721120601120588

maps1198711

(119883 120583) into1198711infin


1120601120588is bounded on 119871119901



1120601120588≲ 119872


implies 1198721120601120588







11120588 and 1120581119870120588



1120601120588follows




119901





119888





119871119901120601120588

(119883) = 119871119901120601120588

(119883 119889 120583)

L119901120601120588

(119883) =L119901120601120588

(119883 119889 120583)

(42)


= sup119909isin119883

119872119901120601120588119891 (119909) lt infin (43)

10038171003817100381710038171198911003817100381710038171003817L119901120601120588

= sup119909isin119883

119872119901120601120588119891 (119909) lt infin (44)


sup119909isin119883

119872119888

119901120601120588119891 (119909) = sup

119909isin119883

119872119901120601120588119891 (119909)

sup119909isin119883

119872119888119901120601120588119891 (119909) = sup

119909isin119883

119872119901120601120588119891 (119909)

(45)


119901120601120588and119872

119901120601120588in (43) and (44)


sup119909isin119883

119888119901120601120588119891 (119909) = sup

119909isin119883

119901120601120588119891 (119909) (46)


or 119901120601120588

in place of 119872119901120601120588


119901120601120588(119883) and L

119901120601120588(119883) a priori



119901120601120588(119883) sub 119871

119901

loc120588(119883) andL119901120601120588(119883) sub 119871

119901


119872119901120601120588119891 (119909) le 119872

119901120601120588119891 (119909) 119909 isin 119883 (47)


le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119871119901120601120588

(119883) subL119901120601120588

(119883) (48)


1198711199011120601120588(119883) sub 119871

1199012120601120588(119883) L

1199011120601120588(119883) subL

1199012120601120588(119883)

(49)

for 1 le 1199011lt 119901


119872119901120601120588

When 120588



BMO120588(119883) = BMO



Since sdot L119901120601120588


119901120601120588(119883)119882

0120588


is

1198820120588= 119891 isinL

119901120601120588(119883)

10038171003817100381710038171198911003817100381710038171003817L119901120601120588

= 0 (50)






120588and


1 119861

119898 sub B

120588such

that 119909 isin 1198611 119910 isin 119861

119898 and 119861

119894cap 119861

119894+1= 0 119894 = 1 119898 minus 1



119901120601120588and 119872



119901120601120588(119883)



119871119901120601120588

(119883) = 119871119901120601120588

(119883) L119901120601120588

(119883) =L119901120601120588

(119883) (51)



119901120582andL


to the choice of 120601 = 120601119898120572


119871119901120582= 119871

119901120582(R

119899

)

= 119891 10038171003817100381710038171198911003817100381710038171003817(119901120582)

= sup119909isinR119899119903gt0

(1

119903120582int119861(119909119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

L119901120582=L

119901120582(R

119899

)

= 119891 10038171003817100381710038171198911003817100381710038171003817119901120582

= sup119909isinR119899119903gt0

inf119911isinC(1

119903120582int119861(119909119903)

1003816100381610038161003816119891 (119910) minus 1199111003816100381610038161003816

119901

119889119910)

1119901

lt infin

(52)


1199010= 119871

119901

(R119899


119901120582C cong

119871119901120582

for 120582 = 119899 L119901119899= BMO(R119899

) and 119871119901119899= 119871

infin

(R119899

) andfor 119899 lt 120582 le 119899+119901L

119901120582= Lip

120572(R119899




120583 (119861 (119909 2119903)) ≲ 120583 (119861 (119909 119903)) (53)


0equiv 0






loc120588(119883) there exists119909


119901120601120588119891(119909




119901120601119891(119909


119891 isin 119871119901

loc(R119899


119901120582(R119899


119901120601120588(119883 119889 120583) is





loc120588(119883) 1198820120588


119901120601120588119891(119909




119901120601119891(119909

0) = infin for

every 119891 isin 119871119901

loc(R119899

) C(R119899


L119901120582(R119899

) = C(R119899


119901120582(R119899


119901120601(R119899

)




0lt119905le119903120601(119905) gt 0 for every


) = 119871119901120601(R119899


119903le119905ltinfin120601(119905)119905

119899119901


119903le119905ltinfin120601(119905)119905


119901120601(R119899

) = 119871119901

120601(R119899





with these of (119883 1198891015840

120583 1205881015840

1206011015840



1205881015840

1206011015840




1 119861

1015840

1198990


12058810158401198891015840 such that

1206011015840

(1198611015840

119895) 120583(119861

1015840

119895)1119901

≲ 120601 (119861) 120583(119861)1119901

119895 = 1 1198990 (54)

Then10038171003817100381710038171198911003817100381710038171003817119871119901120601120588119889

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601101584012058810158401198891015840

119891 isin 119871119901120601101584012058810158401198891015840 (119883) (55)


119901120601120588119889(119883) Similarly if for


isin B12058810158401198891015840 such that

119861 sub 1198611015840 and

1206011015840

(1198611015840

) 120583(1198611015840

)1119901

≲ 120601 (119861) 120583(119861)1119901

(56)

then10038171003817100381710038171198911003817100381710038171003817L119901120601120588119889

≲10038171003817100381710038171198911003817100381710038171003817L119901120601101584012058810158401198891015840

119891 isinL119901120601101584012058810158401198891015840 (119883) (57)

and henceL119901120601101584012058810158401198891015840(119883) subL

119901120601120588119889(119883)



0

119895=1119861

1015840



1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le

1198990

sum

119895=1

1

120601 (119861) 120583(119861)1119901

(int1198611015840

119895

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲ sum

119895

1

1206011015840 (1198611015840

119895) 120583(119861

1015840

119895)1119901

(int1198611015840

119895

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le 1198990

10038171003817100381710038171198911003817100381710038171003817119871119901120601101584012058810158401198891015840

(58)




119901120601101584012058810158401198891015840(119883) and 119861 isin B

120588119889 and consider 1198611015840 119861 sub 1198611015840


inf119911isinC

1

120601 (119861)(1

120583 (119861)int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

≲ inf119911isinC

1

1206011015840 (1198611015840) 120583(1198611015840)1119901

(int1198611015840

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817L119901120601101584012058810158401198891015840

(59)



1015840

1205881015840

1206011015840

) switched we have

119871119901120601120588119889

(119883) = 119871119901120601101584012058810158401198891015840 (119883)

L119901120601120588119889

(119883) =L119901120601101584012058810158401198891015840 (119883)

(60)



(119861) ≲ 120601(119861)



0= 1 and the




1015840


119898120572and 1206011015840

= 120601(1198891015840

)

119898120572 Then for


119872119901120601119889119891 (119909) ≃ 119872

11990112060110158401198891015840119891 (119909)

119872119901120601119889119891 (119909) ≃ 119872

11990112060110158401198891015840119891 (119909)

(61)


119871119901120601119889

(119883) = 11987111990112060110158401198891015840 (119883) L

119901120601119889(119883) =L

11990112060110158401198891015840 (119883)

(62)


1198891015840



119861 (119909 119903) sub 1198611015840

(119909 119888119903) sub 119861 (119909 1198882

119903) (63)




120583 (1198611015840

(119909 119888119903)) le 120583 (119861 (119909 1198882

119903)) le 119862120583 (119861 (119909 119903)) (64)



119889(infin)






sup120583(119876)gt0

120583(119896119876)1119902minus1119901

(int119876

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (65)



space 119871119901120601(R119899

119889(infin)


120601 (119876) = 120583(119876)minus1119901

120583(119896119876)1119901minus1119902

(66)


119901(119896 120583) does not depend


= 119889(infin) 120588 = 1205881015840


(119876) =

120583(119876)minus1119901

120583(1198961015840






119889(infin)




satisfying

sup120583(119876)gt0

(1

120595 (120583 (119896119876))int119876

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (67)



119889(infin)

120583) where

120601 (119876) = (120595 (120583 (119896119876))

120583 (119876))

1119901

(68)


(119896 120583) =M119902




(119876) = (120595(120583(1198961015840

119876))120583(119876))1119901



119886





B119886



120583(119861)1119901

(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (69)


119889(2)

120574119899) where 120574


119899(119909) =

120587minus1198992 exp(minus1199092





119889(2)


space E119901119902

B119886


119883 satisfying

100381710038171003817100381711989110038171003817100381710038171198711(120574

119899)+ sup

119861isinB119886

120583(119861)1119901

(1

120583 (119861)int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816

119901

119889120583)

1119901

lt infin

(70)




(R2

120583) (= L11(R

2

119889(2)


(R2

120583) (=L

11(R2

119889(infin)

120583)) differ



119899 where 119886

119899gt 0



loc(N) = ℓ119901


119895isinN isin

ℓ119901

(N 120583)1198721199011119891 and119872119888


1198721199011119891 (119899) = 119872

1198881199011119891 (119899) = (sum

119895

1003816100381610038161003816119891 (119895) minus ⟨119891⟩N

1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(71)

where ⟨119891⟩N = sum119895119891(119895)119886

119895 HenceL

1199011(N) = ℓ119901

(N 120583) 119891L1199011

≃ 119891 minus ⟨119891⟩Nℓ119901(N120583) 119882

0= C and L




1198721199011119891 (119899) = 119872

119888

1199011119891 (119899)

= max

1003816100381610038161003816119891 (119899)1003816100381610038161003816 (sum

119895

1003816100381610038161003816119891 (119895)1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(72)

and hence 1198711199011(N) = ℓ

infin



of the form 119861 = 119899 119899 isin N 119871119901



1199011120588(N) = 119904(N) and119891L




hence 1198711199011120588(N) = ℓ

infin


ℓinfin



loc(Ω) such that

10038171003817100381710038171198911003817100381710038171003817BMO(Ω)

= sup119861subΩ

1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816 lt infin (73)


120588Ω(119909) = 119889 (119909 120597Ω) 119909 isin Ω (74)


119889 (119909 120597Ω) = inf 119889 (119909 119910) 119910 isin 120597Ω (75)




10038171003817100381710038171198911003817100381710038171003817119871119901120582(Ω)

= sup119909isinΩ

sup0lt119903lt119903

0

(1

119903120582int119861(119909119903)capΩ

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

(76)









119901120601(Ω) =

119871119901120601120588Ω

(Ω) andL119901120601(Ω) =L

119901120601120588Ω


Ω

119871119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817119871119901120601

(Ω)

= sup119861isinB120588(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

L119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817L119901120601

(Ω)

= sup119861isinB120588(Ω)

inf119911isinC

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910) minus 1199111003816100381610038161003816

119901

119889119910)

1119901

lt infin

(77)



function 120588119896= 120588

119896Ωas

120588119896(119909) = 119896 sdot 119889 (119909 120597Ω) 119909 isin Ω (78)

so that 1205881= 120588


119871119901120601119896(Ω) = 119871

119901120601120588119896

(Ω) and L119901120601119896(Ω) = L

119901120601120588119896


119901120601119896(Ω) and L

119901120601119896(Ω)


119901120601119896(Ω)Ω = ⋃

119895isin119869Ω

119895(119869 is finite



119895 then

10038171003817100381710038171198911003817100381710038171003817119871119901120601119896

(Ω)= sup

119909isinΩ


119896(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901


= sup119895isin119869



119896(Ω119895)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

= sup119895isin119869

10038171003817100381710038171003817119891

119895

10038171003817100381710038171003817119871119901120601119895119896(Ω119895)

(79)



119896and 120588(infin)

119896 119871B

119901120601119896(Ω) and 119871Q

119901120601119896(Ω) and so



BΩ



by QΩ

119896


119898120572 Clearly

120601119898120572


119901120572119896(Ω) in place of 119871Q

119901120601119898120572







119901120572119896(Ω)


(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817119871Q119901120572119896

(Ω) 119891 isin 119871

Q119901120572119896

(Ω) (80)10038171003817100381710038171198911003817100381710038171003817LQ119901120572119898

(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) 119891 isinL

Q119901120572119896

(Ω) (81)


Q119901120572119898

(Ω) andLQ

119901120572119896(Ω) sub LQ







of them in QΩ

119896 Thus

1

|119876|120572(1

|119876|int119876

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le1

|119876|120572sum

119895

(1

|119876|int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le sum

119895

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572(1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le 21198991198731003817100381710038171003817119891

1003817100381710038171003817119871Q119901120572119896

(Ω)

(82)



LQ119901120572119896(Ω) and 119876 isin QΩ

119898 QΩ

119896 Then

inf119911isinC

1

|119876|120572(1

|119876|int119876

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

=1

|119876|120572+1119901

10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

(83)

where 119871119901




1199011015840


=

1 and 1198711199011015840



119876


10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

= sup10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

120595 isin 1198711199011015840

0(119876)

100381710038171003817100381712059510038171003817100381710038171199011015840le 1

(84)


119862(119896119898 1199011015840


119898



119895isin

QΩ


119895isin 119871

1199011015840

0(119876) 1 le


119895and 120595

1198951199011015840le 119862120595

1199011015840 for

119895 = 1 119873 and 120595 = sum119873

119895=1120595

119895 Take 120595 isin 119871119901

1015840



subcubes 119876119895119873


119895119873


Then

1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816100381610038161003816

int119876

119891(

119873

sum

119895=1

120595119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

119891120595119895

100381610038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

(119891 minus 119891119876119895

)120595119895

100381610038161003816100381610038161003816100381610038161003816

le 119862

119873

sum

119895=1

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572+1119901

(int119876119895

100381610038161003816100381610038161003816119891 minus 119891

119876119895

100381610038161003816100381610038161003816

119901

)

1119901

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω)

(85)

Hence

1

|119876|120572+1119901

100381710038171003817100381711989110038171003817100381710038171198711(119876)C

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) (86)


(Ω)le 119862119873119891LQ

119901120572119896(Ω)




119901120572119896(Ω)



119898



119896




1015840

0(119861)




1199011015840

0(119861) 1 le 119895 le 119873 such

that supp(120595119895) sub 119861

119895and 120595

1198951199011015840le 119862120595

1199011015840 for 119895 = 1 119873

and 120595 = sum119873

119895=1120595





lt 119877 and 120576 gt1 minus 119877

1015840

119877 there exists 1198731015840

= 1198731015840

(1198771015840


0 119877) = 119909 isin R119899


points 1199091 119909


119861 (1199090 2119877 minus 119877

1015840

) 119861 (1199090 119877

1015840

) sub cup1198731015840

119895=1119861 (119909

119895 120576119877) (87)

(if 1198771015840

= 0 then we set 119861(1199090 0) = 0)




0)

with 1199030= 119898119889

(2)

(1199090 120597Ω) Let 119877

119895= (119895119904)119903



119895= 119878(119909

0 119877




0 119877

119895+1)

119861(1199090 119877



0and 1198771015840

= 119877119895minus1=

((119895 minus 1)119904)1199030(so that 1198771015840


1015840


0 119903


119904minus1

119895=1119873

1015840

((119895 minus 1)119895 2119895 119899)


119896 take

119861(119909lowast

0 119903

lowast


0 119877



(120596 1199090) le

119889(2)

(120596 119909lowast

0) + 119889

(2)

(119909lowast

0 119909

0) and 119889(2)

(119909lowast

0 119909

0) = ((119904 minus 1)119904)119903

0=

((119904 minus 1)119904)119898119889(2)


119889(2)

(119909lowast

0 120597Ω) ge 119889

(2)

(1199090 120597Ω) (1 minus 119898

119904 minus 1

119904) (88)


119903lowast

0=2

1199041199030=2

119904119898119889

(2)

(1199090 120597Ω) lt

2

119904

1

1 minus 119898119889

(2)

(119909lowast

0 120597Ω)

le 119896119889(2)

(119909lowast

0 120597Ω)

(89)



Q11990112057212

(Ω) and sdot 119871Q119901120572

(Ω)= sdot

119871Q11990112057212

(Ω)(the


119901120572(Ω) LB




119871Q119901120572(Ω) = 119871

B119901120572(Ω) L

Q119901120572(Ω) =L

B119901120572(Ω) (90)



119876or 119876



infinle sdot



119861119876isinBΩ

radic119899119896 if119876 isin QΩ

119896 and119876

119861isin QΩ

radic119899119896 if 119861 isinBΩ

119896 Moreover

|119861119876| = 119888

1|119876| and |119876

119861| = 119888

2|119861| where 119888

1and 119888

2 depend on the


119901120572radic119899119896(Ω) For any 119861


inf119911isinC

1

|119861|120572(1

|119861|int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

le 1198881015840

2inf119911isinC

1

1003816100381610038161003816119876119861

1003816100381610038161003816

120572(1

1003816100381610038161003816119876119861

1003816100381610038161003816

int119876119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

(91)

Consequently 119891LB119901120572119896

(Ω)le 119888

1015840

2119891LQ

119901120572radic119889119896


that LQ119901120572radic119899119896



119901120572(Ω) and sdot LB

119901120572(Ω)

le

119862 sdot LQ119901120572


in an analogous way


Ω)





119870= 119870(2119870 + 1)









120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ 120601 (119861) 120583(119861)1119901

(92)


11120588 is bounded on 119871119901


119901120601120591119870120588(119883) to 119871

119901120601120588(119883)


11120588 thatis

119872120588119891 (119909) = sup

119909isin119861isinB120588

1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 (93)



splitting119891 = 1198911+119891


1= 119891120594

120591119870119861


119872120588119891

2(119909) le sup

1198611015840

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 (94)



119909 isin 1198611015840 and 119903(1198611015840


1

120601 (119861)

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 le

1

120593 (119903 (119861))(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le1

120601 (1198611015840)(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

(95)

Consequently

1

120601 (119861)119872

120588119891

2(119909) le

10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119909 isin 119861 (96)


119872120588is bounded on 119871119901

(119883 120583) give

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891)

119901

119889120583)

1119901

le sum

119894=12

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891

119894)119901

119889120583)

1119901

le 119888119901

1

120601 (119861) 120583(119861)1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

le 1198881015840

119901

1

120601 (120591119870119861) 120583(120591

119870119861)

1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

le 11988810158401015840

119901

10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

(97)


119871119901120601120588

≲ 119891119871119901120601120591119870120588


119901120582(R119899



(R119899


120593(119903) = 119903120572 120572 le 0 so that 120601 = 120601




119901120601119903120572

into itself


120573by letting

119868120573119891 (119909) = int

119883

119891 (119910)

119889(119909 119910)119899minus120573

119889120583 (119910) 119909 isin 119883 (98)


120593 (119903)


int

infin

119903

119905120573minus1

120593 (119905) 119889119905 ≲ 120595 (119903) 119903 gt 0

(99)


120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ Ψ (119861) 120583(119861)1119902

(100)



(119883 120583) to119871119902


119901120601(119883) to 119871

119902Ψ120588(119883)


le sdot 119871119902Ψ


1

Ψ (119861)(1

120583 (119861)int119861

1003816100381610038161003816100381611986812057311989110038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(101)



1+ 119891

2

1198911= 119891120594

120591119870119861 and 120591





119868120573and (100) we write

1

Ψ (119861)(1

120583 (119861)int119861

10038161003816100381610038161003816119868120573119891

1

10038161003816100381610038161003816

119902

119889120583)

1119902

le1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171003817119868120573119891

1

10038171003817100381710038171003817119871119902(120583)

≲1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171198911

1003817100381710038171003817119871119901(120583)

≲1

120601 (120591119870119861)(

1

120583 (120591119870119861)int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(102)


119870119903)

119888

sub 119861(119909 119903)119888


10038161003816100381610038161003816119868120573119891

2(119909)10038161003816100381610038161003816le int

(120591119870119861)119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le int119861(119909119903)

119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

=

infin

sum

119895=0

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le

infin

sum

119895=0

(2119895

119903)120573minus119899

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲

infin

sum

119895=0

(2119895

119903)120573 1

120583 (119861 (119909 2119895+1119903))int119861(1199092119895+1

119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲ (

infin

sum

119895=0

(2119895

119903)120573

120601 (119861 (119909 2119895+1

119903)))10038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ int

infin

119903

119905120573minus1

120593 (119905) 11988911990510038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ 120595 (119903)10038171003817100381710038171198911003817100381710038171003817119871119901120601

(103)


1

Ψ (119861 (119886 119903))(

1

120583 (119861 (119886 119903))int119861(119886119903)

10038161003816100381610038161003816119868120573119891

2

10038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(104)



(120583) to 119871119902



120573is bounded

from 119871119901120601(119883) to 119871

119902Ψ(119883)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119910 isin 119883 120588 (119910) gt 119903 gt 1205880(119910) = 119910 isin 119883 120588 (119910) gt 119903

cap 119910 isin 119883 1205880(119910) lt 119903

(29)



119883 ni 119910 997891997888rarr1

120601 (119861 (119910 119903))(⟨10038161003816100381610038161198911003816100381610038161003816

119901

⟩119861(119910119903)

)

1119901

(30)


119860 120583(119860) lt infin But then

119883 ni 119910 997891997888rarr1

120601 (119861 (119910 119903))

120583(119860 cap 119861 (119910 119903))1119901

120583 (119861 (119910 119903))

(31)









119888119901120601120588119891(119909) gt 120582 an argument


does the job


120588cupB




0(119910

0) lt

1199030lt 120588(119910

0)

120601 (119861 (1199100 119903

0)) = lim

119903rarr119903+

0

120601 (119861 (1199100 119903))

120601 (119861 (1199100 119903

0)) = lim


0

120601 (119861 (1199100 119903))

(32)

Note that 120601119898120572




119872119901120601120588119891 (119909) = 119872

119901120601120588119891 (119909) 119891 isin 119871

119901

119897119900119888120588(119883) 119909 isin 119883 (33)

119901120601120588119891 (119909) =

119901120601120588119891 (119909) 119891 isin 119871

119901

119897119900119888120588(119883) 119909 isin 119883 (34)


119897119900119888120588(119883) the functions 119872

119901120601120588119891

119872119888

119901120601120588119891

119901120601120588119891 and


measurable


0gt 0 we have

119861 (1199100 119903

0) = ⋂

119903gt1199030

119861 (1199100 119903) 119861 (119910

0 119903

0) = ⋃

0lt119903lt1199030

119861 (1199100 119903)

(35)


119861(1199100 119903

0) isinB

120588 such that 119909

0isin 119861


1

120601 (1198610)(

1

120583 (1198610)int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901


120588

1

120601 (119861)

(1

120583 (119861)

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(36)

Let 119903119899rarr 119903

minus

0and 119903

119899gt 120588

0(119910



1

120601 (119861 (1199100 119903

119899))

(1

120583 (119861 (1199100 119903

119899))

int119861(1199100119903119899)

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

997888rarr1

120601 (1198610)(

1

120583 (1198610)int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(37)


0= 119861(119910

0 119903

0) isinB

120588 such that 119909

0isin 119861


1

120601 (1198610)

(1

120583 (1198610)

int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901


120588

1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(38)

Let 119903119899rarr 119903

+

0and 119903

119899lt 120588(119910



1

120601 (119861 (1199100 119903

119899))(

1

120583 (119861 (1199100 119903

119899))int119861(1199100119903119899)

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

997888rarr1

120601 (1198610)

(1

120583 (1198610)

int1198610

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

(39)


⟨119891⟩119861(11991001199030)= lim

119899rarrinfin

⟨119891⟩119861(1199100119903119899)

⟨119891⟩119861(11991001199030)= lim

119899rarrinfin

⟨119891⟩119861(1199100119903119899)

(40)

(note that 119871119901

loc120588(119883) sub 1198711






119870= 119870(3119870+2) suffices



0equiv 0


120583 (120591119861) ≲ 120583 (119861) 119861 isinB120588 (41)


0equiv 0





120588 and (119883 119889)


Then1198721120601120588

maps1198711

(119883 120583) into1198711infin


1120601120588is bounded on 119871119901



1120601120588≲ 119872


implies 1198721120601120588







11120588 and 1120581119870120588



1120601120588follows




119901





119888





119871119901120601120588

(119883) = 119871119901120601120588

(119883 119889 120583)

L119901120601120588

(119883) =L119901120601120588

(119883 119889 120583)

(42)


= sup119909isin119883

119872119901120601120588119891 (119909) lt infin (43)

10038171003817100381710038171198911003817100381710038171003817L119901120601120588

= sup119909isin119883

119872119901120601120588119891 (119909) lt infin (44)


sup119909isin119883

119872119888

119901120601120588119891 (119909) = sup

119909isin119883

119872119901120601120588119891 (119909)

sup119909isin119883

119872119888119901120601120588119891 (119909) = sup

119909isin119883

119872119901120601120588119891 (119909)

(45)


119901120601120588and119872

119901120601120588in (43) and (44)


sup119909isin119883

119888119901120601120588119891 (119909) = sup

119909isin119883

119901120601120588119891 (119909) (46)


or 119901120601120588

in place of 119872119901120601120588


119901120601120588(119883) and L

119901120601120588(119883) a priori



119901120601120588(119883) sub 119871

119901

loc120588(119883) andL119901120601120588(119883) sub 119871

119901


119872119901120601120588119891 (119909) le 119872

119901120601120588119891 (119909) 119909 isin 119883 (47)


le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119871119901120601120588

(119883) subL119901120601120588

(119883) (48)


1198711199011120601120588(119883) sub 119871

1199012120601120588(119883) L

1199011120601120588(119883) subL

1199012120601120588(119883)

(49)

for 1 le 1199011lt 119901


119872119901120601120588

When 120588



BMO120588(119883) = BMO



Since sdot L119901120601120588


119901120601120588(119883)119882

0120588


is

1198820120588= 119891 isinL

119901120601120588(119883)

10038171003817100381710038171198911003817100381710038171003817L119901120601120588

= 0 (50)






120588and


1 119861

119898 sub B

120588such

that 119909 isin 1198611 119910 isin 119861

119898 and 119861

119894cap 119861

119894+1= 0 119894 = 1 119898 minus 1



119901120601120588and 119872



119901120601120588(119883)



119871119901120601120588

(119883) = 119871119901120601120588

(119883) L119901120601120588

(119883) =L119901120601120588

(119883) (51)



119901120582andL


to the choice of 120601 = 120601119898120572


119871119901120582= 119871

119901120582(R

119899

)

= 119891 10038171003817100381710038171198911003817100381710038171003817(119901120582)

= sup119909isinR119899119903gt0

(1

119903120582int119861(119909119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

L119901120582=L

119901120582(R

119899

)

= 119891 10038171003817100381710038171198911003817100381710038171003817119901120582

= sup119909isinR119899119903gt0

inf119911isinC(1

119903120582int119861(119909119903)

1003816100381610038161003816119891 (119910) minus 1199111003816100381610038161003816

119901

119889119910)

1119901

lt infin

(52)


1199010= 119871

119901

(R119899


119901120582C cong

119871119901120582

for 120582 = 119899 L119901119899= BMO(R119899

) and 119871119901119899= 119871

infin

(R119899

) andfor 119899 lt 120582 le 119899+119901L

119901120582= Lip

120572(R119899




120583 (119861 (119909 2119903)) ≲ 120583 (119861 (119909 119903)) (53)


0equiv 0






loc120588(119883) there exists119909


119901120601120588119891(119909




119901120601119891(119909


119891 isin 119871119901

loc(R119899


119901120582(R119899


119901120601120588(119883 119889 120583) is





loc120588(119883) 1198820120588


119901120601120588119891(119909




119901120601119891(119909

0) = infin for

every 119891 isin 119871119901

loc(R119899

) C(R119899


L119901120582(R119899

) = C(R119899


119901120582(R119899


119901120601(R119899

)




0lt119905le119903120601(119905) gt 0 for every


) = 119871119901120601(R119899


119903le119905ltinfin120601(119905)119905

119899119901


119903le119905ltinfin120601(119905)119905


119901120601(R119899

) = 119871119901

120601(R119899





with these of (119883 1198891015840

120583 1205881015840

1206011015840



1205881015840

1206011015840




1 119861

1015840

1198990


12058810158401198891015840 such that

1206011015840

(1198611015840

119895) 120583(119861

1015840

119895)1119901

≲ 120601 (119861) 120583(119861)1119901

119895 = 1 1198990 (54)

Then10038171003817100381710038171198911003817100381710038171003817119871119901120601120588119889

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601101584012058810158401198891015840

119891 isin 119871119901120601101584012058810158401198891015840 (119883) (55)


119901120601120588119889(119883) Similarly if for


isin B12058810158401198891015840 such that

119861 sub 1198611015840 and

1206011015840

(1198611015840

) 120583(1198611015840

)1119901

≲ 120601 (119861) 120583(119861)1119901

(56)

then10038171003817100381710038171198911003817100381710038171003817L119901120601120588119889

≲10038171003817100381710038171198911003817100381710038171003817L119901120601101584012058810158401198891015840

119891 isinL119901120601101584012058810158401198891015840 (119883) (57)

and henceL119901120601101584012058810158401198891015840(119883) subL

119901120601120588119889(119883)



0

119895=1119861

1015840



1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le

1198990

sum

119895=1

1

120601 (119861) 120583(119861)1119901

(int1198611015840

119895

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲ sum

119895

1

1206011015840 (1198611015840

119895) 120583(119861

1015840

119895)1119901

(int1198611015840

119895

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le 1198990

10038171003817100381710038171198911003817100381710038171003817119871119901120601101584012058810158401198891015840

(58)




119901120601101584012058810158401198891015840(119883) and 119861 isin B

120588119889 and consider 1198611015840 119861 sub 1198611015840


inf119911isinC

1

120601 (119861)(1

120583 (119861)int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

≲ inf119911isinC

1

1206011015840 (1198611015840) 120583(1198611015840)1119901

(int1198611015840

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817L119901120601101584012058810158401198891015840

(59)



1015840

1205881015840

1206011015840

) switched we have

119871119901120601120588119889

(119883) = 119871119901120601101584012058810158401198891015840 (119883)

L119901120601120588119889

(119883) =L119901120601101584012058810158401198891015840 (119883)

(60)



(119861) ≲ 120601(119861)



0= 1 and the




1015840


119898120572and 1206011015840

= 120601(1198891015840

)

119898120572 Then for


119872119901120601119889119891 (119909) ≃ 119872

11990112060110158401198891015840119891 (119909)

119872119901120601119889119891 (119909) ≃ 119872

11990112060110158401198891015840119891 (119909)

(61)


119871119901120601119889

(119883) = 11987111990112060110158401198891015840 (119883) L

119901120601119889(119883) =L

11990112060110158401198891015840 (119883)

(62)


1198891015840



119861 (119909 119903) sub 1198611015840

(119909 119888119903) sub 119861 (119909 1198882

119903) (63)




120583 (1198611015840

(119909 119888119903)) le 120583 (119861 (119909 1198882

119903)) le 119862120583 (119861 (119909 119903)) (64)



119889(infin)






sup120583(119876)gt0

120583(119896119876)1119902minus1119901

(int119876

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (65)



space 119871119901120601(R119899

119889(infin)


120601 (119876) = 120583(119876)minus1119901

120583(119896119876)1119901minus1119902

(66)


119901(119896 120583) does not depend


= 119889(infin) 120588 = 1205881015840


(119876) =

120583(119876)minus1119901

120583(1198961015840






119889(infin)




satisfying

sup120583(119876)gt0

(1

120595 (120583 (119896119876))int119876

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (67)



119889(infin)

120583) where

120601 (119876) = (120595 (120583 (119896119876))

120583 (119876))

1119901

(68)


(119896 120583) =M119902




(119876) = (120595(120583(1198961015840

119876))120583(119876))1119901



119886





B119886



120583(119861)1119901

(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (69)


119889(2)

120574119899) where 120574


119899(119909) =

120587minus1198992 exp(minus1199092





119889(2)


space E119901119902

B119886


119883 satisfying

100381710038171003817100381711989110038171003817100381710038171198711(120574

119899)+ sup

119861isinB119886

120583(119861)1119901

(1

120583 (119861)int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816

119901

119889120583)

1119901

lt infin

(70)




(R2

120583) (= L11(R

2

119889(2)


(R2

120583) (=L

11(R2

119889(infin)

120583)) differ



119899 where 119886

119899gt 0



loc(N) = ℓ119901


119895isinN isin

ℓ119901

(N 120583)1198721199011119891 and119872119888


1198721199011119891 (119899) = 119872

1198881199011119891 (119899) = (sum

119895

1003816100381610038161003816119891 (119895) minus ⟨119891⟩N

1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(71)

where ⟨119891⟩N = sum119895119891(119895)119886

119895 HenceL

1199011(N) = ℓ119901

(N 120583) 119891L1199011

≃ 119891 minus ⟨119891⟩Nℓ119901(N120583) 119882

0= C and L




1198721199011119891 (119899) = 119872

119888

1199011119891 (119899)

= max

1003816100381610038161003816119891 (119899)1003816100381610038161003816 (sum

119895

1003816100381610038161003816119891 (119895)1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(72)

and hence 1198711199011(N) = ℓ

infin



of the form 119861 = 119899 119899 isin N 119871119901



1199011120588(N) = 119904(N) and119891L




hence 1198711199011120588(N) = ℓ

infin


ℓinfin



loc(Ω) such that

10038171003817100381710038171198911003817100381710038171003817BMO(Ω)

= sup119861subΩ

1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816 lt infin (73)


120588Ω(119909) = 119889 (119909 120597Ω) 119909 isin Ω (74)


119889 (119909 120597Ω) = inf 119889 (119909 119910) 119910 isin 120597Ω (75)




10038171003817100381710038171198911003817100381710038171003817119871119901120582(Ω)

= sup119909isinΩ

sup0lt119903lt119903

0

(1

119903120582int119861(119909119903)capΩ

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

(76)









119901120601(Ω) =

119871119901120601120588Ω

(Ω) andL119901120601(Ω) =L

119901120601120588Ω


Ω

119871119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817119871119901120601

(Ω)

= sup119861isinB120588(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

L119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817L119901120601

(Ω)

= sup119861isinB120588(Ω)

inf119911isinC

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910) minus 1199111003816100381610038161003816

119901

119889119910)

1119901

lt infin

(77)



function 120588119896= 120588

119896Ωas

120588119896(119909) = 119896 sdot 119889 (119909 120597Ω) 119909 isin Ω (78)

so that 1205881= 120588


119871119901120601119896(Ω) = 119871

119901120601120588119896

(Ω) and L119901120601119896(Ω) = L

119901120601120588119896


119901120601119896(Ω) and L

119901120601119896(Ω)


119901120601119896(Ω)Ω = ⋃

119895isin119869Ω

119895(119869 is finite



119895 then

10038171003817100381710038171198911003817100381710038171003817119871119901120601119896

(Ω)= sup

119909isinΩ


119896(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901


= sup119895isin119869



119896(Ω119895)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

= sup119895isin119869

10038171003817100381710038171003817119891

119895

10038171003817100381710038171003817119871119901120601119895119896(Ω119895)

(79)



119896and 120588(infin)

119896 119871B

119901120601119896(Ω) and 119871Q

119901120601119896(Ω) and so



BΩ



by QΩ

119896


119898120572 Clearly

120601119898120572


119901120572119896(Ω) in place of 119871Q

119901120601119898120572







119901120572119896(Ω)


(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817119871Q119901120572119896

(Ω) 119891 isin 119871

Q119901120572119896

(Ω) (80)10038171003817100381710038171198911003817100381710038171003817LQ119901120572119898

(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) 119891 isinL

Q119901120572119896

(Ω) (81)


Q119901120572119898

(Ω) andLQ

119901120572119896(Ω) sub LQ







of them in QΩ

119896 Thus

1

|119876|120572(1

|119876|int119876

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le1

|119876|120572sum

119895

(1

|119876|int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le sum

119895

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572(1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le 21198991198731003817100381710038171003817119891

1003817100381710038171003817119871Q119901120572119896

(Ω)

(82)



LQ119901120572119896(Ω) and 119876 isin QΩ

119898 QΩ

119896 Then

inf119911isinC

1

|119876|120572(1

|119876|int119876

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

=1

|119876|120572+1119901

10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

(83)

where 119871119901




1199011015840


=

1 and 1198711199011015840



119876


10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

= sup10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

120595 isin 1198711199011015840

0(119876)

100381710038171003817100381712059510038171003817100381710038171199011015840le 1

(84)


119862(119896119898 1199011015840


119898



119895isin

QΩ


119895isin 119871

1199011015840

0(119876) 1 le


119895and 120595

1198951199011015840le 119862120595

1199011015840 for

119895 = 1 119873 and 120595 = sum119873

119895=1120595

119895 Take 120595 isin 119871119901

1015840



subcubes 119876119895119873


119895119873


Then

1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816100381610038161003816

int119876

119891(

119873

sum

119895=1

120595119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

119891120595119895

100381610038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

(119891 minus 119891119876119895

)120595119895

100381610038161003816100381610038161003816100381610038161003816

le 119862

119873

sum

119895=1

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572+1119901

(int119876119895

100381610038161003816100381610038161003816119891 minus 119891

119876119895

100381610038161003816100381610038161003816

119901

)

1119901

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω)

(85)

Hence

1

|119876|120572+1119901

100381710038171003817100381711989110038171003817100381710038171198711(119876)C

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) (86)


(Ω)le 119862119873119891LQ

119901120572119896(Ω)




119901120572119896(Ω)



119898



119896




1015840

0(119861)




1199011015840

0(119861) 1 le 119895 le 119873 such

that supp(120595119895) sub 119861

119895and 120595

1198951199011015840le 119862120595

1199011015840 for 119895 = 1 119873

and 120595 = sum119873

119895=1120595





lt 119877 and 120576 gt1 minus 119877

1015840

119877 there exists 1198731015840

= 1198731015840

(1198771015840


0 119877) = 119909 isin R119899


points 1199091 119909


119861 (1199090 2119877 minus 119877

1015840

) 119861 (1199090 119877

1015840

) sub cup1198731015840

119895=1119861 (119909

119895 120576119877) (87)

(if 1198771015840

= 0 then we set 119861(1199090 0) = 0)




0)

with 1199030= 119898119889

(2)

(1199090 120597Ω) Let 119877

119895= (119895119904)119903



119895= 119878(119909

0 119877




0 119877

119895+1)

119861(1199090 119877



0and 1198771015840

= 119877119895minus1=

((119895 minus 1)119904)1199030(so that 1198771015840


1015840


0 119903


119904minus1

119895=1119873

1015840

((119895 minus 1)119895 2119895 119899)


119896 take

119861(119909lowast

0 119903

lowast


0 119877



(120596 1199090) le

119889(2)

(120596 119909lowast

0) + 119889

(2)

(119909lowast

0 119909

0) and 119889(2)

(119909lowast

0 119909

0) = ((119904 minus 1)119904)119903

0=

((119904 minus 1)119904)119898119889(2)


119889(2)

(119909lowast

0 120597Ω) ge 119889

(2)

(1199090 120597Ω) (1 minus 119898

119904 minus 1

119904) (88)


119903lowast

0=2

1199041199030=2

119904119898119889

(2)

(1199090 120597Ω) lt

2

119904

1

1 minus 119898119889

(2)

(119909lowast

0 120597Ω)

le 119896119889(2)

(119909lowast

0 120597Ω)

(89)



Q11990112057212

(Ω) and sdot 119871Q119901120572

(Ω)= sdot

119871Q11990112057212

(Ω)(the


119901120572(Ω) LB




119871Q119901120572(Ω) = 119871

B119901120572(Ω) L

Q119901120572(Ω) =L

B119901120572(Ω) (90)



119876or 119876



infinle sdot



119861119876isinBΩ

radic119899119896 if119876 isin QΩ

119896 and119876

119861isin QΩ

radic119899119896 if 119861 isinBΩ

119896 Moreover

|119861119876| = 119888

1|119876| and |119876

119861| = 119888

2|119861| where 119888

1and 119888

2 depend on the


119901120572radic119899119896(Ω) For any 119861


inf119911isinC

1

|119861|120572(1

|119861|int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

le 1198881015840

2inf119911isinC

1

1003816100381610038161003816119876119861

1003816100381610038161003816

120572(1

1003816100381610038161003816119876119861

1003816100381610038161003816

int119876119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

(91)

Consequently 119891LB119901120572119896

(Ω)le 119888

1015840

2119891LQ

119901120572radic119889119896


that LQ119901120572radic119899119896



119901120572(Ω) and sdot LB

119901120572(Ω)

le

119862 sdot LQ119901120572


in an analogous way


Ω)





119870= 119870(2119870 + 1)









120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ 120601 (119861) 120583(119861)1119901

(92)


11120588 is bounded on 119871119901


119901120601120591119870120588(119883) to 119871

119901120601120588(119883)


11120588 thatis

119872120588119891 (119909) = sup

119909isin119861isinB120588

1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 (93)



splitting119891 = 1198911+119891


1= 119891120594

120591119870119861


119872120588119891

2(119909) le sup

1198611015840

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 (94)



119909 isin 1198611015840 and 119903(1198611015840


1

120601 (119861)

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 le

1

120593 (119903 (119861))(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le1

120601 (1198611015840)(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

(95)

Consequently

1

120601 (119861)119872

120588119891

2(119909) le

10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119909 isin 119861 (96)


119872120588is bounded on 119871119901

(119883 120583) give

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891)

119901

119889120583)

1119901

le sum

119894=12

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891

119894)119901

119889120583)

1119901

le 119888119901

1

120601 (119861) 120583(119861)1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

le 1198881015840

119901

1

120601 (120591119870119861) 120583(120591

119870119861)

1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

le 11988810158401015840

119901

10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

(97)


119871119901120601120588

≲ 119891119871119901120601120591119870120588


119901120582(R119899



(R119899


120593(119903) = 119903120572 120572 le 0 so that 120601 = 120601




119901120601119903120572

into itself


120573by letting

119868120573119891 (119909) = int

119883

119891 (119910)

119889(119909 119910)119899minus120573

119889120583 (119910) 119909 isin 119883 (98)


120593 (119903)


int

infin

119903

119905120573minus1

120593 (119905) 119889119905 ≲ 120595 (119903) 119903 gt 0

(99)


120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ Ψ (119861) 120583(119861)1119902

(100)



(119883 120583) to119871119902


119901120601(119883) to 119871

119902Ψ120588(119883)


le sdot 119871119902Ψ


1

Ψ (119861)(1

120583 (119861)int119861

1003816100381610038161003816100381611986812057311989110038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(101)



1+ 119891

2

1198911= 119891120594

120591119870119861 and 120591





119868120573and (100) we write

1

Ψ (119861)(1

120583 (119861)int119861

10038161003816100381610038161003816119868120573119891

1

10038161003816100381610038161003816

119902

119889120583)

1119902

le1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171003817119868120573119891

1

10038171003817100381710038171003817119871119902(120583)

≲1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171198911

1003817100381710038171003817119871119901(120583)

≲1

120601 (120591119870119861)(

1

120583 (120591119870119861)int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(102)


119870119903)

119888

sub 119861(119909 119903)119888


10038161003816100381610038161003816119868120573119891

2(119909)10038161003816100381610038161003816le int

(120591119870119861)119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le int119861(119909119903)

119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

=

infin

sum

119895=0

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le

infin

sum

119895=0

(2119895

119903)120573minus119899

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲

infin

sum

119895=0

(2119895

119903)120573 1

120583 (119861 (119909 2119895+1119903))int119861(1199092119895+1

119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲ (

infin

sum

119895=0

(2119895

119903)120573

120601 (119861 (119909 2119895+1

119903)))10038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ int

infin

119903

119905120573minus1

120593 (119905) 11988911990510038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ 120595 (119903)10038171003817100381710038171198911003817100381710038171003817119871119901120601

(103)


1

Ψ (119861 (119886 119903))(

1

120583 (119861 (119886 119903))int119861(119886119903)

10038161003816100381610038161003816119868120573119891

2

10038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(104)



(120583) to 119871119902



120573is bounded

from 119871119901120601(119883) to 119871

119902Ψ(119883)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119870= 119870(3119870+2) suffices



0equiv 0


120583 (120591119861) ≲ 120583 (119861) 119861 isinB120588 (41)


0equiv 0





120588 and (119883 119889)


Then1198721120601120588

maps1198711

(119883 120583) into1198711infin


1120601120588is bounded on 119871119901



1120601120588≲ 119872


implies 1198721120601120588







11120588 and 1120581119870120588



1120601120588follows




119901





119888





119871119901120601120588

(119883) = 119871119901120601120588

(119883 119889 120583)

L119901120601120588

(119883) =L119901120601120588

(119883 119889 120583)

(42)


= sup119909isin119883

119872119901120601120588119891 (119909) lt infin (43)

10038171003817100381710038171198911003817100381710038171003817L119901120601120588

= sup119909isin119883

119872119901120601120588119891 (119909) lt infin (44)


sup119909isin119883

119872119888

119901120601120588119891 (119909) = sup

119909isin119883

119872119901120601120588119891 (119909)

sup119909isin119883

119872119888119901120601120588119891 (119909) = sup

119909isin119883

119872119901120601120588119891 (119909)

(45)


119901120601120588and119872

119901120601120588in (43) and (44)


sup119909isin119883

119888119901120601120588119891 (119909) = sup

119909isin119883

119901120601120588119891 (119909) (46)


or 119901120601120588

in place of 119872119901120601120588


119901120601120588(119883) and L

119901120601120588(119883) a priori



119901120601120588(119883) sub 119871

119901

loc120588(119883) andL119901120601120588(119883) sub 119871

119901


119872119901120601120588119891 (119909) le 119872

119901120601120588119891 (119909) 119909 isin 119883 (47)


le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119871119901120601120588

(119883) subL119901120601120588

(119883) (48)


1198711199011120601120588(119883) sub 119871

1199012120601120588(119883) L

1199011120601120588(119883) subL

1199012120601120588(119883)

(49)

for 1 le 1199011lt 119901


119872119901120601120588

When 120588



BMO120588(119883) = BMO



Since sdot L119901120601120588


119901120601120588(119883)119882

0120588


is

1198820120588= 119891 isinL

119901120601120588(119883)

10038171003817100381710038171198911003817100381710038171003817L119901120601120588

= 0 (50)






120588and


1 119861

119898 sub B

120588such

that 119909 isin 1198611 119910 isin 119861

119898 and 119861

119894cap 119861

119894+1= 0 119894 = 1 119898 minus 1



119901120601120588and 119872



119901120601120588(119883)



119871119901120601120588

(119883) = 119871119901120601120588

(119883) L119901120601120588

(119883) =L119901120601120588

(119883) (51)



119901120582andL


to the choice of 120601 = 120601119898120572


119871119901120582= 119871

119901120582(R

119899

)

= 119891 10038171003817100381710038171198911003817100381710038171003817(119901120582)

= sup119909isinR119899119903gt0

(1

119903120582int119861(119909119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

L119901120582=L

119901120582(R

119899

)

= 119891 10038171003817100381710038171198911003817100381710038171003817119901120582

= sup119909isinR119899119903gt0

inf119911isinC(1

119903120582int119861(119909119903)

1003816100381610038161003816119891 (119910) minus 1199111003816100381610038161003816

119901

119889119910)

1119901

lt infin

(52)


1199010= 119871

119901

(R119899


119901120582C cong

119871119901120582

for 120582 = 119899 L119901119899= BMO(R119899

) and 119871119901119899= 119871

infin

(R119899

) andfor 119899 lt 120582 le 119899+119901L

119901120582= Lip

120572(R119899




120583 (119861 (119909 2119903)) ≲ 120583 (119861 (119909 119903)) (53)


0equiv 0






loc120588(119883) there exists119909


119901120601120588119891(119909




119901120601119891(119909


119891 isin 119871119901

loc(R119899


119901120582(R119899


119901120601120588(119883 119889 120583) is





loc120588(119883) 1198820120588


119901120601120588119891(119909




119901120601119891(119909

0) = infin for

every 119891 isin 119871119901

loc(R119899

) C(R119899


L119901120582(R119899

) = C(R119899


119901120582(R119899


119901120601(R119899

)




0lt119905le119903120601(119905) gt 0 for every


) = 119871119901120601(R119899


119903le119905ltinfin120601(119905)119905

119899119901


119903le119905ltinfin120601(119905)119905


119901120601(R119899

) = 119871119901

120601(R119899





with these of (119883 1198891015840

120583 1205881015840

1206011015840



1205881015840

1206011015840




1 119861

1015840

1198990


12058810158401198891015840 such that

1206011015840

(1198611015840

119895) 120583(119861

1015840

119895)1119901

≲ 120601 (119861) 120583(119861)1119901

119895 = 1 1198990 (54)

Then10038171003817100381710038171198911003817100381710038171003817119871119901120601120588119889

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601101584012058810158401198891015840

119891 isin 119871119901120601101584012058810158401198891015840 (119883) (55)


119901120601120588119889(119883) Similarly if for


isin B12058810158401198891015840 such that

119861 sub 1198611015840 and

1206011015840

(1198611015840

) 120583(1198611015840

)1119901

≲ 120601 (119861) 120583(119861)1119901

(56)

then10038171003817100381710038171198911003817100381710038171003817L119901120601120588119889

≲10038171003817100381710038171198911003817100381710038171003817L119901120601101584012058810158401198891015840

119891 isinL119901120601101584012058810158401198891015840 (119883) (57)

and henceL119901120601101584012058810158401198891015840(119883) subL

119901120601120588119889(119883)



0

119895=1119861

1015840



1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le

1198990

sum

119895=1

1

120601 (119861) 120583(119861)1119901

(int1198611015840

119895

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲ sum

119895

1

1206011015840 (1198611015840

119895) 120583(119861

1015840

119895)1119901

(int1198611015840

119895

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le 1198990

10038171003817100381710038171198911003817100381710038171003817119871119901120601101584012058810158401198891015840

(58)




119901120601101584012058810158401198891015840(119883) and 119861 isin B

120588119889 and consider 1198611015840 119861 sub 1198611015840


inf119911isinC

1

120601 (119861)(1

120583 (119861)int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

≲ inf119911isinC

1

1206011015840 (1198611015840) 120583(1198611015840)1119901

(int1198611015840

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817L119901120601101584012058810158401198891015840

(59)



1015840

1205881015840

1206011015840

) switched we have

119871119901120601120588119889

(119883) = 119871119901120601101584012058810158401198891015840 (119883)

L119901120601120588119889

(119883) =L119901120601101584012058810158401198891015840 (119883)

(60)



(119861) ≲ 120601(119861)



0= 1 and the




1015840


119898120572and 1206011015840

= 120601(1198891015840

)

119898120572 Then for


119872119901120601119889119891 (119909) ≃ 119872

11990112060110158401198891015840119891 (119909)

119872119901120601119889119891 (119909) ≃ 119872

11990112060110158401198891015840119891 (119909)

(61)


119871119901120601119889

(119883) = 11987111990112060110158401198891015840 (119883) L

119901120601119889(119883) =L

11990112060110158401198891015840 (119883)

(62)


1198891015840



119861 (119909 119903) sub 1198611015840

(119909 119888119903) sub 119861 (119909 1198882

119903) (63)




120583 (1198611015840

(119909 119888119903)) le 120583 (119861 (119909 1198882

119903)) le 119862120583 (119861 (119909 119903)) (64)



119889(infin)






sup120583(119876)gt0

120583(119896119876)1119902minus1119901

(int119876

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (65)



space 119871119901120601(R119899

119889(infin)


120601 (119876) = 120583(119876)minus1119901

120583(119896119876)1119901minus1119902

(66)


119901(119896 120583) does not depend


= 119889(infin) 120588 = 1205881015840


(119876) =

120583(119876)minus1119901

120583(1198961015840






119889(infin)




satisfying

sup120583(119876)gt0

(1

120595 (120583 (119896119876))int119876

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (67)



119889(infin)

120583) where

120601 (119876) = (120595 (120583 (119896119876))

120583 (119876))

1119901

(68)


(119896 120583) =M119902




(119876) = (120595(120583(1198961015840

119876))120583(119876))1119901



119886





B119886



120583(119861)1119901

(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (69)


119889(2)

120574119899) where 120574


119899(119909) =

120587minus1198992 exp(minus1199092





119889(2)


space E119901119902

B119886


119883 satisfying

100381710038171003817100381711989110038171003817100381710038171198711(120574

119899)+ sup

119861isinB119886

120583(119861)1119901

(1

120583 (119861)int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816

119901

119889120583)

1119901

lt infin

(70)




(R2

120583) (= L11(R

2

119889(2)


(R2

120583) (=L

11(R2

119889(infin)

120583)) differ



119899 where 119886

119899gt 0



loc(N) = ℓ119901


119895isinN isin

ℓ119901

(N 120583)1198721199011119891 and119872119888


1198721199011119891 (119899) = 119872

1198881199011119891 (119899) = (sum

119895

1003816100381610038161003816119891 (119895) minus ⟨119891⟩N

1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(71)

where ⟨119891⟩N = sum119895119891(119895)119886

119895 HenceL

1199011(N) = ℓ119901

(N 120583) 119891L1199011

≃ 119891 minus ⟨119891⟩Nℓ119901(N120583) 119882

0= C and L




1198721199011119891 (119899) = 119872

119888

1199011119891 (119899)

= max

1003816100381610038161003816119891 (119899)1003816100381610038161003816 (sum

119895

1003816100381610038161003816119891 (119895)1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(72)

and hence 1198711199011(N) = ℓ

infin



of the form 119861 = 119899 119899 isin N 119871119901



1199011120588(N) = 119904(N) and119891L




hence 1198711199011120588(N) = ℓ

infin


ℓinfin



loc(Ω) such that

10038171003817100381710038171198911003817100381710038171003817BMO(Ω)

= sup119861subΩ

1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816 lt infin (73)


120588Ω(119909) = 119889 (119909 120597Ω) 119909 isin Ω (74)


119889 (119909 120597Ω) = inf 119889 (119909 119910) 119910 isin 120597Ω (75)




10038171003817100381710038171198911003817100381710038171003817119871119901120582(Ω)

= sup119909isinΩ

sup0lt119903lt119903

0

(1

119903120582int119861(119909119903)capΩ

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

(76)









119901120601(Ω) =

119871119901120601120588Ω

(Ω) andL119901120601(Ω) =L

119901120601120588Ω


Ω

119871119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817119871119901120601

(Ω)

= sup119861isinB120588(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

L119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817L119901120601

(Ω)

= sup119861isinB120588(Ω)

inf119911isinC

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910) minus 1199111003816100381610038161003816

119901

119889119910)

1119901

lt infin

(77)



function 120588119896= 120588

119896Ωas

120588119896(119909) = 119896 sdot 119889 (119909 120597Ω) 119909 isin Ω (78)

so that 1205881= 120588


119871119901120601119896(Ω) = 119871

119901120601120588119896

(Ω) and L119901120601119896(Ω) = L

119901120601120588119896


119901120601119896(Ω) and L

119901120601119896(Ω)


119901120601119896(Ω)Ω = ⋃

119895isin119869Ω

119895(119869 is finite



119895 then

10038171003817100381710038171198911003817100381710038171003817119871119901120601119896

(Ω)= sup

119909isinΩ


119896(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901


= sup119895isin119869



119896(Ω119895)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

= sup119895isin119869

10038171003817100381710038171003817119891

119895

10038171003817100381710038171003817119871119901120601119895119896(Ω119895)

(79)



119896and 120588(infin)

119896 119871B

119901120601119896(Ω) and 119871Q

119901120601119896(Ω) and so



BΩ



by QΩ

119896


119898120572 Clearly

120601119898120572


119901120572119896(Ω) in place of 119871Q

119901120601119898120572







119901120572119896(Ω)


(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817119871Q119901120572119896

(Ω) 119891 isin 119871

Q119901120572119896

(Ω) (80)10038171003817100381710038171198911003817100381710038171003817LQ119901120572119898

(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) 119891 isinL

Q119901120572119896

(Ω) (81)


Q119901120572119898

(Ω) andLQ

119901120572119896(Ω) sub LQ







of them in QΩ

119896 Thus

1

|119876|120572(1

|119876|int119876

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le1

|119876|120572sum

119895

(1

|119876|int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le sum

119895

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572(1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le 21198991198731003817100381710038171003817119891

1003817100381710038171003817119871Q119901120572119896

(Ω)

(82)



LQ119901120572119896(Ω) and 119876 isin QΩ

119898 QΩ

119896 Then

inf119911isinC

1

|119876|120572(1

|119876|int119876

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

=1

|119876|120572+1119901

10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

(83)

where 119871119901




1199011015840


=

1 and 1198711199011015840



119876


10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

= sup10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

120595 isin 1198711199011015840

0(119876)

100381710038171003817100381712059510038171003817100381710038171199011015840le 1

(84)


119862(119896119898 1199011015840


119898



119895isin

QΩ


119895isin 119871

1199011015840

0(119876) 1 le


119895and 120595

1198951199011015840le 119862120595

1199011015840 for

119895 = 1 119873 and 120595 = sum119873

119895=1120595

119895 Take 120595 isin 119871119901

1015840



subcubes 119876119895119873


119895119873


Then

1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816100381610038161003816

int119876

119891(

119873

sum

119895=1

120595119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

119891120595119895

100381610038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

(119891 minus 119891119876119895

)120595119895

100381610038161003816100381610038161003816100381610038161003816

le 119862

119873

sum

119895=1

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572+1119901

(int119876119895

100381610038161003816100381610038161003816119891 minus 119891

119876119895

100381610038161003816100381610038161003816

119901

)

1119901

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω)

(85)

Hence

1

|119876|120572+1119901

100381710038171003817100381711989110038171003817100381710038171198711(119876)C

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) (86)


(Ω)le 119862119873119891LQ

119901120572119896(Ω)




119901120572119896(Ω)



119898



119896




1015840

0(119861)




1199011015840

0(119861) 1 le 119895 le 119873 such

that supp(120595119895) sub 119861

119895and 120595

1198951199011015840le 119862120595

1199011015840 for 119895 = 1 119873

and 120595 = sum119873

119895=1120595





lt 119877 and 120576 gt1 minus 119877

1015840

119877 there exists 1198731015840

= 1198731015840

(1198771015840


0 119877) = 119909 isin R119899


points 1199091 119909


119861 (1199090 2119877 minus 119877

1015840

) 119861 (1199090 119877

1015840

) sub cup1198731015840

119895=1119861 (119909

119895 120576119877) (87)

(if 1198771015840

= 0 then we set 119861(1199090 0) = 0)




0)

with 1199030= 119898119889

(2)

(1199090 120597Ω) Let 119877

119895= (119895119904)119903



119895= 119878(119909

0 119877




0 119877

119895+1)

119861(1199090 119877



0and 1198771015840

= 119877119895minus1=

((119895 minus 1)119904)1199030(so that 1198771015840


1015840


0 119903


119904minus1

119895=1119873

1015840

((119895 minus 1)119895 2119895 119899)


119896 take

119861(119909lowast

0 119903

lowast


0 119877



(120596 1199090) le

119889(2)

(120596 119909lowast

0) + 119889

(2)

(119909lowast

0 119909

0) and 119889(2)

(119909lowast

0 119909

0) = ((119904 minus 1)119904)119903

0=

((119904 minus 1)119904)119898119889(2)


119889(2)

(119909lowast

0 120597Ω) ge 119889

(2)

(1199090 120597Ω) (1 minus 119898

119904 minus 1

119904) (88)


119903lowast

0=2

1199041199030=2

119904119898119889

(2)

(1199090 120597Ω) lt

2

119904

1

1 minus 119898119889

(2)

(119909lowast

0 120597Ω)

le 119896119889(2)

(119909lowast

0 120597Ω)

(89)



Q11990112057212

(Ω) and sdot 119871Q119901120572

(Ω)= sdot

119871Q11990112057212

(Ω)(the


119901120572(Ω) LB




119871Q119901120572(Ω) = 119871

B119901120572(Ω) L

Q119901120572(Ω) =L

B119901120572(Ω) (90)



119876or 119876



infinle sdot



119861119876isinBΩ

radic119899119896 if119876 isin QΩ

119896 and119876

119861isin QΩ

radic119899119896 if 119861 isinBΩ

119896 Moreover

|119861119876| = 119888

1|119876| and |119876

119861| = 119888

2|119861| where 119888

1and 119888

2 depend on the


119901120572radic119899119896(Ω) For any 119861


inf119911isinC

1

|119861|120572(1

|119861|int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

le 1198881015840

2inf119911isinC

1

1003816100381610038161003816119876119861

1003816100381610038161003816

120572(1

1003816100381610038161003816119876119861

1003816100381610038161003816

int119876119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

(91)

Consequently 119891LB119901120572119896

(Ω)le 119888

1015840

2119891LQ

119901120572radic119889119896


that LQ119901120572radic119899119896



119901120572(Ω) and sdot LB

119901120572(Ω)

le

119862 sdot LQ119901120572


in an analogous way


Ω)





119870= 119870(2119870 + 1)









120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ 120601 (119861) 120583(119861)1119901

(92)


11120588 is bounded on 119871119901


119901120601120591119870120588(119883) to 119871

119901120601120588(119883)


11120588 thatis

119872120588119891 (119909) = sup

119909isin119861isinB120588

1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 (93)



splitting119891 = 1198911+119891


1= 119891120594

120591119870119861


119872120588119891

2(119909) le sup

1198611015840

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 (94)



119909 isin 1198611015840 and 119903(1198611015840


1

120601 (119861)

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 le

1

120593 (119903 (119861))(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le1

120601 (1198611015840)(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

(95)

Consequently

1

120601 (119861)119872

120588119891

2(119909) le

10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119909 isin 119861 (96)


119872120588is bounded on 119871119901

(119883 120583) give

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891)

119901

119889120583)

1119901

le sum

119894=12

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891

119894)119901

119889120583)

1119901

le 119888119901

1

120601 (119861) 120583(119861)1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

le 1198881015840

119901

1

120601 (120591119870119861) 120583(120591

119870119861)

1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

le 11988810158401015840

119901

10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

(97)


119871119901120601120588

≲ 119891119871119901120601120591119870120588


119901120582(R119899



(R119899


120593(119903) = 119903120572 120572 le 0 so that 120601 = 120601




119901120601119903120572

into itself


120573by letting

119868120573119891 (119909) = int

119883

119891 (119910)

119889(119909 119910)119899minus120573

119889120583 (119910) 119909 isin 119883 (98)


120593 (119903)


int

infin

119903

119905120573minus1

120593 (119905) 119889119905 ≲ 120595 (119903) 119903 gt 0

(99)


120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ Ψ (119861) 120583(119861)1119902

(100)



(119883 120583) to119871119902


119901120601(119883) to 119871

119902Ψ120588(119883)


le sdot 119871119902Ψ


1

Ψ (119861)(1

120583 (119861)int119861

1003816100381610038161003816100381611986812057311989110038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(101)



1+ 119891

2

1198911= 119891120594

120591119870119861 and 120591





119868120573and (100) we write

1

Ψ (119861)(1

120583 (119861)int119861

10038161003816100381610038161003816119868120573119891

1

10038161003816100381610038161003816

119902

119889120583)

1119902

le1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171003817119868120573119891

1

10038171003817100381710038171003817119871119902(120583)

≲1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171198911

1003817100381710038171003817119871119901(120583)

≲1

120601 (120591119870119861)(

1

120583 (120591119870119861)int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(102)


119870119903)

119888

sub 119861(119909 119903)119888


10038161003816100381610038161003816119868120573119891

2(119909)10038161003816100381610038161003816le int

(120591119870119861)119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le int119861(119909119903)

119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

=

infin

sum

119895=0

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le

infin

sum

119895=0

(2119895

119903)120573minus119899

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲

infin

sum

119895=0

(2119895

119903)120573 1

120583 (119861 (119909 2119895+1119903))int119861(1199092119895+1

119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲ (

infin

sum

119895=0

(2119895

119903)120573

120601 (119861 (119909 2119895+1

119903)))10038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ int

infin

119903

119905120573minus1

120593 (119905) 11988911990510038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ 120595 (119903)10038171003817100381710038171198911003817100381710038171003817119871119901120601

(103)


1

Ψ (119861 (119886 119903))(

1

120583 (119861 (119886 119903))int119861(119886119903)

10038161003816100381610038161003816119868120573119891

2

10038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(104)



(120583) to 119871119902



120573is bounded

from 119871119901120601(119883) to 119871

119902Ψ(119883)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














120588and


1 119861

119898 sub B

120588such

that 119909 isin 1198611 119910 isin 119861

119898 and 119861

119894cap 119861

119894+1= 0 119894 = 1 119898 minus 1



119901120601120588and 119872



119901120601120588(119883)



119871119901120601120588

(119883) = 119871119901120601120588

(119883) L119901120601120588

(119883) =L119901120601120588

(119883) (51)



119901120582andL


to the choice of 120601 = 120601119898120572


119871119901120582= 119871

119901120582(R

119899

)

= 119891 10038171003817100381710038171198911003817100381710038171003817(119901120582)

= sup119909isinR119899119903gt0

(1

119903120582int119861(119909119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

L119901120582=L

119901120582(R

119899

)

= 119891 10038171003817100381710038171198911003817100381710038171003817119901120582

= sup119909isinR119899119903gt0

inf119911isinC(1

119903120582int119861(119909119903)

1003816100381610038161003816119891 (119910) minus 1199111003816100381610038161003816

119901

119889119910)

1119901

lt infin

(52)


1199010= 119871

119901

(R119899


119901120582C cong

119871119901120582

for 120582 = 119899 L119901119899= BMO(R119899

) and 119871119901119899= 119871

infin

(R119899

) andfor 119899 lt 120582 le 119899+119901L

119901120582= Lip

120572(R119899




120583 (119861 (119909 2119903)) ≲ 120583 (119861 (119909 119903)) (53)


0equiv 0






loc120588(119883) there exists119909


119901120601120588119891(119909




119901120601119891(119909


119891 isin 119871119901

loc(R119899


119901120582(R119899


119901120601120588(119883 119889 120583) is





loc120588(119883) 1198820120588


119901120601120588119891(119909




119901120601119891(119909

0) = infin for

every 119891 isin 119871119901

loc(R119899

) C(R119899


L119901120582(R119899

) = C(R119899


119901120582(R119899


119901120601(R119899

)




0lt119905le119903120601(119905) gt 0 for every


) = 119871119901120601(R119899


119903le119905ltinfin120601(119905)119905

119899119901


119903le119905ltinfin120601(119905)119905


119901120601(R119899

) = 119871119901

120601(R119899





with these of (119883 1198891015840

120583 1205881015840

1206011015840



1205881015840

1206011015840




1 119861

1015840

1198990


12058810158401198891015840 such that

1206011015840

(1198611015840

119895) 120583(119861

1015840

119895)1119901

≲ 120601 (119861) 120583(119861)1119901

119895 = 1 1198990 (54)

Then10038171003817100381710038171198911003817100381710038171003817119871119901120601120588119889

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601101584012058810158401198891015840

119891 isin 119871119901120601101584012058810158401198891015840 (119883) (55)


119901120601120588119889(119883) Similarly if for


isin B12058810158401198891015840 such that

119861 sub 1198611015840 and

1206011015840

(1198611015840

) 120583(1198611015840

)1119901

≲ 120601 (119861) 120583(119861)1119901

(56)

then10038171003817100381710038171198911003817100381710038171003817L119901120601120588119889

≲10038171003817100381710038171198911003817100381710038171003817L119901120601101584012058810158401198891015840

119891 isinL119901120601101584012058810158401198891015840 (119883) (57)

and henceL119901120601101584012058810158401198891015840(119883) subL

119901120601120588119889(119883)



0

119895=1119861

1015840



1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le

1198990

sum

119895=1

1

120601 (119861) 120583(119861)1119901

(int1198611015840

119895

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲ sum

119895

1

1206011015840 (1198611015840

119895) 120583(119861

1015840

119895)1119901

(int1198611015840

119895

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le 1198990

10038171003817100381710038171198911003817100381710038171003817119871119901120601101584012058810158401198891015840

(58)




119901120601101584012058810158401198891015840(119883) and 119861 isin B

120588119889 and consider 1198611015840 119861 sub 1198611015840


inf119911isinC

1

120601 (119861)(1

120583 (119861)int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

≲ inf119911isinC

1

1206011015840 (1198611015840) 120583(1198611015840)1119901

(int1198611015840

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817L119901120601101584012058810158401198891015840

(59)



1015840

1205881015840

1206011015840

) switched we have

119871119901120601120588119889

(119883) = 119871119901120601101584012058810158401198891015840 (119883)

L119901120601120588119889

(119883) =L119901120601101584012058810158401198891015840 (119883)

(60)



(119861) ≲ 120601(119861)



0= 1 and the




1015840


119898120572and 1206011015840

= 120601(1198891015840

)

119898120572 Then for


119872119901120601119889119891 (119909) ≃ 119872

11990112060110158401198891015840119891 (119909)

119872119901120601119889119891 (119909) ≃ 119872

11990112060110158401198891015840119891 (119909)

(61)


119871119901120601119889

(119883) = 11987111990112060110158401198891015840 (119883) L

119901120601119889(119883) =L

11990112060110158401198891015840 (119883)

(62)


1198891015840



119861 (119909 119903) sub 1198611015840

(119909 119888119903) sub 119861 (119909 1198882

119903) (63)




120583 (1198611015840

(119909 119888119903)) le 120583 (119861 (119909 1198882

119903)) le 119862120583 (119861 (119909 119903)) (64)



119889(infin)






sup120583(119876)gt0

120583(119896119876)1119902minus1119901

(int119876

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (65)



space 119871119901120601(R119899

119889(infin)


120601 (119876) = 120583(119876)minus1119901

120583(119896119876)1119901minus1119902

(66)


119901(119896 120583) does not depend


= 119889(infin) 120588 = 1205881015840


(119876) =

120583(119876)minus1119901

120583(1198961015840






119889(infin)




satisfying

sup120583(119876)gt0

(1

120595 (120583 (119896119876))int119876

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (67)



119889(infin)

120583) where

120601 (119876) = (120595 (120583 (119896119876))

120583 (119876))

1119901

(68)


(119896 120583) =M119902




(119876) = (120595(120583(1198961015840

119876))120583(119876))1119901



119886





B119886



120583(119861)1119901

(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (69)


119889(2)

120574119899) where 120574


119899(119909) =

120587minus1198992 exp(minus1199092





119889(2)


space E119901119902

B119886


119883 satisfying

100381710038171003817100381711989110038171003817100381710038171198711(120574

119899)+ sup

119861isinB119886

120583(119861)1119901

(1

120583 (119861)int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816

119901

119889120583)

1119901

lt infin

(70)




(R2

120583) (= L11(R

2

119889(2)


(R2

120583) (=L

11(R2

119889(infin)

120583)) differ



119899 where 119886

119899gt 0



loc(N) = ℓ119901


119895isinN isin

ℓ119901

(N 120583)1198721199011119891 and119872119888


1198721199011119891 (119899) = 119872

1198881199011119891 (119899) = (sum

119895

1003816100381610038161003816119891 (119895) minus ⟨119891⟩N

1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(71)

where ⟨119891⟩N = sum119895119891(119895)119886

119895 HenceL

1199011(N) = ℓ119901

(N 120583) 119891L1199011

≃ 119891 minus ⟨119891⟩Nℓ119901(N120583) 119882

0= C and L




1198721199011119891 (119899) = 119872

119888

1199011119891 (119899)

= max

1003816100381610038161003816119891 (119899)1003816100381610038161003816 (sum

119895

1003816100381610038161003816119891 (119895)1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(72)

and hence 1198711199011(N) = ℓ

infin



of the form 119861 = 119899 119899 isin N 119871119901



1199011120588(N) = 119904(N) and119891L




hence 1198711199011120588(N) = ℓ

infin


ℓinfin



loc(Ω) such that

10038171003817100381710038171198911003817100381710038171003817BMO(Ω)

= sup119861subΩ

1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816 lt infin (73)


120588Ω(119909) = 119889 (119909 120597Ω) 119909 isin Ω (74)


119889 (119909 120597Ω) = inf 119889 (119909 119910) 119910 isin 120597Ω (75)




10038171003817100381710038171198911003817100381710038171003817119871119901120582(Ω)

= sup119909isinΩ

sup0lt119903lt119903

0

(1

119903120582int119861(119909119903)capΩ

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

(76)









119901120601(Ω) =

119871119901120601120588Ω

(Ω) andL119901120601(Ω) =L

119901120601120588Ω


Ω

119871119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817119871119901120601

(Ω)

= sup119861isinB120588(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

L119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817L119901120601

(Ω)

= sup119861isinB120588(Ω)

inf119911isinC

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910) minus 1199111003816100381610038161003816

119901

119889119910)

1119901

lt infin

(77)



function 120588119896= 120588

119896Ωas

120588119896(119909) = 119896 sdot 119889 (119909 120597Ω) 119909 isin Ω (78)

so that 1205881= 120588


119871119901120601119896(Ω) = 119871

119901120601120588119896

(Ω) and L119901120601119896(Ω) = L

119901120601120588119896


119901120601119896(Ω) and L

119901120601119896(Ω)


119901120601119896(Ω)Ω = ⋃

119895isin119869Ω

119895(119869 is finite



119895 then

10038171003817100381710038171198911003817100381710038171003817119871119901120601119896

(Ω)= sup

119909isinΩ


119896(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901


= sup119895isin119869



119896(Ω119895)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

= sup119895isin119869

10038171003817100381710038171003817119891

119895

10038171003817100381710038171003817119871119901120601119895119896(Ω119895)

(79)



119896and 120588(infin)

119896 119871B

119901120601119896(Ω) and 119871Q

119901120601119896(Ω) and so



BΩ



by QΩ

119896


119898120572 Clearly

120601119898120572


119901120572119896(Ω) in place of 119871Q

119901120601119898120572







119901120572119896(Ω)


(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817119871Q119901120572119896

(Ω) 119891 isin 119871

Q119901120572119896

(Ω) (80)10038171003817100381710038171198911003817100381710038171003817LQ119901120572119898

(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) 119891 isinL

Q119901120572119896

(Ω) (81)


Q119901120572119898

(Ω) andLQ

119901120572119896(Ω) sub LQ







of them in QΩ

119896 Thus

1

|119876|120572(1

|119876|int119876

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le1

|119876|120572sum

119895

(1

|119876|int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le sum

119895

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572(1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le 21198991198731003817100381710038171003817119891

1003817100381710038171003817119871Q119901120572119896

(Ω)

(82)



LQ119901120572119896(Ω) and 119876 isin QΩ

119898 QΩ

119896 Then

inf119911isinC

1

|119876|120572(1

|119876|int119876

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

=1

|119876|120572+1119901

10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

(83)

where 119871119901




1199011015840


=

1 and 1198711199011015840



119876


10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

= sup10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

120595 isin 1198711199011015840

0(119876)

100381710038171003817100381712059510038171003817100381710038171199011015840le 1

(84)


119862(119896119898 1199011015840


119898



119895isin

QΩ


119895isin 119871

1199011015840

0(119876) 1 le


119895and 120595

1198951199011015840le 119862120595

1199011015840 for

119895 = 1 119873 and 120595 = sum119873

119895=1120595

119895 Take 120595 isin 119871119901

1015840



subcubes 119876119895119873


119895119873


Then

1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816100381610038161003816

int119876

119891(

119873

sum

119895=1

120595119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

119891120595119895

100381610038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

(119891 minus 119891119876119895

)120595119895

100381610038161003816100381610038161003816100381610038161003816

le 119862

119873

sum

119895=1

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572+1119901

(int119876119895

100381610038161003816100381610038161003816119891 minus 119891

119876119895

100381610038161003816100381610038161003816

119901

)

1119901

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω)

(85)

Hence

1

|119876|120572+1119901

100381710038171003817100381711989110038171003817100381710038171198711(119876)C

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) (86)


(Ω)le 119862119873119891LQ

119901120572119896(Ω)




119901120572119896(Ω)



119898



119896




1015840

0(119861)




1199011015840

0(119861) 1 le 119895 le 119873 such

that supp(120595119895) sub 119861

119895and 120595

1198951199011015840le 119862120595

1199011015840 for 119895 = 1 119873

and 120595 = sum119873

119895=1120595





lt 119877 and 120576 gt1 minus 119877

1015840

119877 there exists 1198731015840

= 1198731015840

(1198771015840


0 119877) = 119909 isin R119899


points 1199091 119909


119861 (1199090 2119877 minus 119877

1015840

) 119861 (1199090 119877

1015840

) sub cup1198731015840

119895=1119861 (119909

119895 120576119877) (87)

(if 1198771015840

= 0 then we set 119861(1199090 0) = 0)




0)

with 1199030= 119898119889

(2)

(1199090 120597Ω) Let 119877

119895= (119895119904)119903



119895= 119878(119909

0 119877




0 119877

119895+1)

119861(1199090 119877



0and 1198771015840

= 119877119895minus1=

((119895 minus 1)119904)1199030(so that 1198771015840


1015840


0 119903


119904minus1

119895=1119873

1015840

((119895 minus 1)119895 2119895 119899)


119896 take

119861(119909lowast

0 119903

lowast


0 119877



(120596 1199090) le

119889(2)

(120596 119909lowast

0) + 119889

(2)

(119909lowast

0 119909

0) and 119889(2)

(119909lowast

0 119909

0) = ((119904 minus 1)119904)119903

0=

((119904 minus 1)119904)119898119889(2)


119889(2)

(119909lowast

0 120597Ω) ge 119889

(2)

(1199090 120597Ω) (1 minus 119898

119904 minus 1

119904) (88)


119903lowast

0=2

1199041199030=2

119904119898119889

(2)

(1199090 120597Ω) lt

2

119904

1

1 minus 119898119889

(2)

(119909lowast

0 120597Ω)

le 119896119889(2)

(119909lowast

0 120597Ω)

(89)



Q11990112057212

(Ω) and sdot 119871Q119901120572

(Ω)= sdot

119871Q11990112057212

(Ω)(the


119901120572(Ω) LB




119871Q119901120572(Ω) = 119871

B119901120572(Ω) L

Q119901120572(Ω) =L

B119901120572(Ω) (90)



119876or 119876



infinle sdot



119861119876isinBΩ

radic119899119896 if119876 isin QΩ

119896 and119876

119861isin QΩ

radic119899119896 if 119861 isinBΩ

119896 Moreover

|119861119876| = 119888

1|119876| and |119876

119861| = 119888

2|119861| where 119888

1and 119888

2 depend on the


119901120572radic119899119896(Ω) For any 119861


inf119911isinC

1

|119861|120572(1

|119861|int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

le 1198881015840

2inf119911isinC

1

1003816100381610038161003816119876119861

1003816100381610038161003816

120572(1

1003816100381610038161003816119876119861

1003816100381610038161003816

int119876119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

(91)

Consequently 119891LB119901120572119896

(Ω)le 119888

1015840

2119891LQ

119901120572radic119889119896


that LQ119901120572radic119899119896



119901120572(Ω) and sdot LB

119901120572(Ω)

le

119862 sdot LQ119901120572


in an analogous way


Ω)





119870= 119870(2119870 + 1)









120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ 120601 (119861) 120583(119861)1119901

(92)


11120588 is bounded on 119871119901


119901120601120591119870120588(119883) to 119871

119901120601120588(119883)


11120588 thatis

119872120588119891 (119909) = sup

119909isin119861isinB120588

1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 (93)



splitting119891 = 1198911+119891


1= 119891120594

120591119870119861


119872120588119891

2(119909) le sup

1198611015840

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 (94)



119909 isin 1198611015840 and 119903(1198611015840


1

120601 (119861)

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 le

1

120593 (119903 (119861))(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le1

120601 (1198611015840)(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

(95)

Consequently

1

120601 (119861)119872

120588119891

2(119909) le

10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119909 isin 119861 (96)


119872120588is bounded on 119871119901

(119883 120583) give

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891)

119901

119889120583)

1119901

le sum

119894=12

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891

119894)119901

119889120583)

1119901

le 119888119901

1

120601 (119861) 120583(119861)1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

le 1198881015840

119901

1

120601 (120591119870119861) 120583(120591

119870119861)

1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

le 11988810158401015840

119901

10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

(97)


119871119901120601120588

≲ 119891119871119901120601120591119870120588


119901120582(R119899



(R119899


120593(119903) = 119903120572 120572 le 0 so that 120601 = 120601




119901120601119903120572

into itself


120573by letting

119868120573119891 (119909) = int

119883

119891 (119910)

119889(119909 119910)119899minus120573

119889120583 (119910) 119909 isin 119883 (98)


120593 (119903)


int

infin

119903

119905120573minus1

120593 (119905) 119889119905 ≲ 120595 (119903) 119903 gt 0

(99)


120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ Ψ (119861) 120583(119861)1119902

(100)



(119883 120583) to119871119902


119901120601(119883) to 119871

119902Ψ120588(119883)


le sdot 119871119902Ψ


1

Ψ (119861)(1

120583 (119861)int119861

1003816100381610038161003816100381611986812057311989110038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(101)



1+ 119891

2

1198911= 119891120594

120591119870119861 and 120591





119868120573and (100) we write

1

Ψ (119861)(1

120583 (119861)int119861

10038161003816100381610038161003816119868120573119891

1

10038161003816100381610038161003816

119902

119889120583)

1119902

le1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171003817119868120573119891

1

10038171003817100381710038171003817119871119902(120583)

≲1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171198911

1003817100381710038171003817119871119901(120583)

≲1

120601 (120591119870119861)(

1

120583 (120591119870119861)int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(102)


119870119903)

119888

sub 119861(119909 119903)119888


10038161003816100381610038161003816119868120573119891

2(119909)10038161003816100381610038161003816le int

(120591119870119861)119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le int119861(119909119903)

119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

=

infin

sum

119895=0

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le

infin

sum

119895=0

(2119895

119903)120573minus119899

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲

infin

sum

119895=0

(2119895

119903)120573 1

120583 (119861 (119909 2119895+1119903))int119861(1199092119895+1

119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲ (

infin

sum

119895=0

(2119895

119903)120573

120601 (119861 (119909 2119895+1

119903)))10038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ int

infin

119903

119905120573minus1

120593 (119905) 11988911990510038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ 120595 (119903)10038171003817100381710038171198911003817100381710038171003817119871119901120601

(103)


1

Ψ (119861 (119886 119903))(

1

120583 (119861 (119886 119903))int119861(119886119903)

10038161003816100381610038161003816119868120573119891

2

10038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(104)



(120583) to 119871119902



120573is bounded

from 119871119901120601(119883) to 119871

119902Ψ(119883)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










with these of (119883 1198891015840

120583 1205881015840

1206011015840



1205881015840

1206011015840




1 119861

1015840

1198990


12058810158401198891015840 such that

1206011015840

(1198611015840

119895) 120583(119861

1015840

119895)1119901

≲ 120601 (119861) 120583(119861)1119901

119895 = 1 1198990 (54)

Then10038171003817100381710038171198911003817100381710038171003817119871119901120601120588119889

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601101584012058810158401198891015840

119891 isin 119871119901120601101584012058810158401198891015840 (119883) (55)


119901120601120588119889(119883) Similarly if for


isin B12058810158401198891015840 such that

119861 sub 1198611015840 and

1206011015840

(1198611015840

) 120583(1198611015840

)1119901

≲ 120601 (119861) 120583(119861)1119901

(56)

then10038171003817100381710038171198911003817100381710038171003817L119901120601120588119889

≲10038171003817100381710038171198911003817100381710038171003817L119901120601101584012058810158401198891015840

119891 isinL119901120601101584012058810158401198891015840 (119883) (57)

and henceL119901120601101584012058810158401198891015840(119883) subL

119901120601120588119889(119883)



0

119895=1119861

1015840



1

120601 (119861)(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le

1198990

sum

119895=1

1

120601 (119861) 120583(119861)1119901

(int1198611015840

119895

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲ sum

119895

1

1206011015840 (1198611015840

119895) 120583(119861

1015840

119895)1119901

(int1198611015840

119895

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le 1198990

10038171003817100381710038171198911003817100381710038171003817119871119901120601101584012058810158401198891015840

(58)




119901120601101584012058810158401198891015840(119883) and 119861 isin B

120588119889 and consider 1198611015840 119861 sub 1198611015840


inf119911isinC

1

120601 (119861)(1

120583 (119861)int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

≲ inf119911isinC

1

1206011015840 (1198611015840) 120583(1198611015840)1119901

(int1198611015840

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817L119901120601101584012058810158401198891015840

(59)



1015840

1205881015840

1206011015840

) switched we have

119871119901120601120588119889

(119883) = 119871119901120601101584012058810158401198891015840 (119883)

L119901120601120588119889

(119883) =L119901120601101584012058810158401198891015840 (119883)

(60)



(119861) ≲ 120601(119861)



0= 1 and the




1015840


119898120572and 1206011015840

= 120601(1198891015840

)

119898120572 Then for


119872119901120601119889119891 (119909) ≃ 119872

11990112060110158401198891015840119891 (119909)

119872119901120601119889119891 (119909) ≃ 119872

11990112060110158401198891015840119891 (119909)

(61)


119871119901120601119889

(119883) = 11987111990112060110158401198891015840 (119883) L

119901120601119889(119883) =L

11990112060110158401198891015840 (119883)

(62)


1198891015840



119861 (119909 119903) sub 1198611015840

(119909 119888119903) sub 119861 (119909 1198882

119903) (63)




120583 (1198611015840

(119909 119888119903)) le 120583 (119861 (119909 1198882

119903)) le 119862120583 (119861 (119909 119903)) (64)



119889(infin)






sup120583(119876)gt0

120583(119896119876)1119902minus1119901

(int119876

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (65)



space 119871119901120601(R119899

119889(infin)


120601 (119876) = 120583(119876)minus1119901

120583(119896119876)1119901minus1119902

(66)


119901(119896 120583) does not depend


= 119889(infin) 120588 = 1205881015840


(119876) =

120583(119876)minus1119901

120583(1198961015840






119889(infin)




satisfying

sup120583(119876)gt0

(1

120595 (120583 (119896119876))int119876

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (67)



119889(infin)

120583) where

120601 (119876) = (120595 (120583 (119896119876))

120583 (119876))

1119901

(68)


(119896 120583) =M119902




(119876) = (120595(120583(1198961015840

119876))120583(119876))1119901



119886





B119886



120583(119861)1119901

(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (69)


119889(2)

120574119899) where 120574


119899(119909) =

120587minus1198992 exp(minus1199092





119889(2)


space E119901119902

B119886


119883 satisfying

100381710038171003817100381711989110038171003817100381710038171198711(120574

119899)+ sup

119861isinB119886

120583(119861)1119901

(1

120583 (119861)int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816

119901

119889120583)

1119901

lt infin

(70)




(R2

120583) (= L11(R

2

119889(2)


(R2

120583) (=L

11(R2

119889(infin)

120583)) differ



119899 where 119886

119899gt 0



loc(N) = ℓ119901


119895isinN isin

ℓ119901

(N 120583)1198721199011119891 and119872119888


1198721199011119891 (119899) = 119872

1198881199011119891 (119899) = (sum

119895

1003816100381610038161003816119891 (119895) minus ⟨119891⟩N

1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(71)

where ⟨119891⟩N = sum119895119891(119895)119886

119895 HenceL

1199011(N) = ℓ119901

(N 120583) 119891L1199011

≃ 119891 minus ⟨119891⟩Nℓ119901(N120583) 119882

0= C and L




1198721199011119891 (119899) = 119872

119888

1199011119891 (119899)

= max

1003816100381610038161003816119891 (119899)1003816100381610038161003816 (sum

119895

1003816100381610038161003816119891 (119895)1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(72)

and hence 1198711199011(N) = ℓ

infin



of the form 119861 = 119899 119899 isin N 119871119901



1199011120588(N) = 119904(N) and119891L




hence 1198711199011120588(N) = ℓ

infin


ℓinfin



loc(Ω) such that

10038171003817100381710038171198911003817100381710038171003817BMO(Ω)

= sup119861subΩ

1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816 lt infin (73)


120588Ω(119909) = 119889 (119909 120597Ω) 119909 isin Ω (74)


119889 (119909 120597Ω) = inf 119889 (119909 119910) 119910 isin 120597Ω (75)




10038171003817100381710038171198911003817100381710038171003817119871119901120582(Ω)

= sup119909isinΩ

sup0lt119903lt119903

0

(1

119903120582int119861(119909119903)capΩ

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

(76)









119901120601(Ω) =

119871119901120601120588Ω

(Ω) andL119901120601(Ω) =L

119901120601120588Ω


Ω

119871119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817119871119901120601

(Ω)

= sup119861isinB120588(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

L119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817L119901120601

(Ω)

= sup119861isinB120588(Ω)

inf119911isinC

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910) minus 1199111003816100381610038161003816

119901

119889119910)

1119901

lt infin

(77)



function 120588119896= 120588

119896Ωas

120588119896(119909) = 119896 sdot 119889 (119909 120597Ω) 119909 isin Ω (78)

so that 1205881= 120588


119871119901120601119896(Ω) = 119871

119901120601120588119896

(Ω) and L119901120601119896(Ω) = L

119901120601120588119896


119901120601119896(Ω) and L

119901120601119896(Ω)


119901120601119896(Ω)Ω = ⋃

119895isin119869Ω

119895(119869 is finite



119895 then

10038171003817100381710038171198911003817100381710038171003817119871119901120601119896

(Ω)= sup

119909isinΩ


119896(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901


= sup119895isin119869



119896(Ω119895)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

= sup119895isin119869

10038171003817100381710038171003817119891

119895

10038171003817100381710038171003817119871119901120601119895119896(Ω119895)

(79)



119896and 120588(infin)

119896 119871B

119901120601119896(Ω) and 119871Q

119901120601119896(Ω) and so



BΩ



by QΩ

119896


119898120572 Clearly

120601119898120572


119901120572119896(Ω) in place of 119871Q

119901120601119898120572







119901120572119896(Ω)


(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817119871Q119901120572119896

(Ω) 119891 isin 119871

Q119901120572119896

(Ω) (80)10038171003817100381710038171198911003817100381710038171003817LQ119901120572119898

(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) 119891 isinL

Q119901120572119896

(Ω) (81)


Q119901120572119898

(Ω) andLQ

119901120572119896(Ω) sub LQ







of them in QΩ

119896 Thus

1

|119876|120572(1

|119876|int119876

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le1

|119876|120572sum

119895

(1

|119876|int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le sum

119895

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572(1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le 21198991198731003817100381710038171003817119891

1003817100381710038171003817119871Q119901120572119896

(Ω)

(82)



LQ119901120572119896(Ω) and 119876 isin QΩ

119898 QΩ

119896 Then

inf119911isinC

1

|119876|120572(1

|119876|int119876

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

=1

|119876|120572+1119901

10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

(83)

where 119871119901




1199011015840


=

1 and 1198711199011015840



119876


10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

= sup10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

120595 isin 1198711199011015840

0(119876)

100381710038171003817100381712059510038171003817100381710038171199011015840le 1

(84)


119862(119896119898 1199011015840


119898



119895isin

QΩ


119895isin 119871

1199011015840

0(119876) 1 le


119895and 120595

1198951199011015840le 119862120595

1199011015840 for

119895 = 1 119873 and 120595 = sum119873

119895=1120595

119895 Take 120595 isin 119871119901

1015840



subcubes 119876119895119873


119895119873


Then

1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816100381610038161003816

int119876

119891(

119873

sum

119895=1

120595119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

119891120595119895

100381610038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

(119891 minus 119891119876119895

)120595119895

100381610038161003816100381610038161003816100381610038161003816

le 119862

119873

sum

119895=1

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572+1119901

(int119876119895

100381610038161003816100381610038161003816119891 minus 119891

119876119895

100381610038161003816100381610038161003816

119901

)

1119901

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω)

(85)

Hence

1

|119876|120572+1119901

100381710038171003817100381711989110038171003817100381710038171198711(119876)C

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) (86)


(Ω)le 119862119873119891LQ

119901120572119896(Ω)




119901120572119896(Ω)



119898



119896




1015840

0(119861)




1199011015840

0(119861) 1 le 119895 le 119873 such

that supp(120595119895) sub 119861

119895and 120595

1198951199011015840le 119862120595

1199011015840 for 119895 = 1 119873

and 120595 = sum119873

119895=1120595





lt 119877 and 120576 gt1 minus 119877

1015840

119877 there exists 1198731015840

= 1198731015840

(1198771015840


0 119877) = 119909 isin R119899


points 1199091 119909


119861 (1199090 2119877 minus 119877

1015840

) 119861 (1199090 119877

1015840

) sub cup1198731015840

119895=1119861 (119909

119895 120576119877) (87)

(if 1198771015840

= 0 then we set 119861(1199090 0) = 0)




0)

with 1199030= 119898119889

(2)

(1199090 120597Ω) Let 119877

119895= (119895119904)119903



119895= 119878(119909

0 119877




0 119877

119895+1)

119861(1199090 119877



0and 1198771015840

= 119877119895minus1=

((119895 minus 1)119904)1199030(so that 1198771015840


1015840


0 119903


119904minus1

119895=1119873

1015840

((119895 minus 1)119895 2119895 119899)


119896 take

119861(119909lowast

0 119903

lowast


0 119877



(120596 1199090) le

119889(2)

(120596 119909lowast

0) + 119889

(2)

(119909lowast

0 119909

0) and 119889(2)

(119909lowast

0 119909

0) = ((119904 minus 1)119904)119903

0=

((119904 minus 1)119904)119898119889(2)


119889(2)

(119909lowast

0 120597Ω) ge 119889

(2)

(1199090 120597Ω) (1 minus 119898

119904 minus 1

119904) (88)


119903lowast

0=2

1199041199030=2

119904119898119889

(2)

(1199090 120597Ω) lt

2

119904

1

1 minus 119898119889

(2)

(119909lowast

0 120597Ω)

le 119896119889(2)

(119909lowast

0 120597Ω)

(89)



Q11990112057212

(Ω) and sdot 119871Q119901120572

(Ω)= sdot

119871Q11990112057212

(Ω)(the


119901120572(Ω) LB




119871Q119901120572(Ω) = 119871

B119901120572(Ω) L

Q119901120572(Ω) =L

B119901120572(Ω) (90)



119876or 119876



infinle sdot



119861119876isinBΩ

radic119899119896 if119876 isin QΩ

119896 and119876

119861isin QΩ

radic119899119896 if 119861 isinBΩ

119896 Moreover

|119861119876| = 119888

1|119876| and |119876

119861| = 119888

2|119861| where 119888

1and 119888

2 depend on the


119901120572radic119899119896(Ω) For any 119861


inf119911isinC

1

|119861|120572(1

|119861|int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

le 1198881015840

2inf119911isinC

1

1003816100381610038161003816119876119861

1003816100381610038161003816

120572(1

1003816100381610038161003816119876119861

1003816100381610038161003816

int119876119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

(91)

Consequently 119891LB119901120572119896

(Ω)le 119888

1015840

2119891LQ

119901120572radic119889119896


that LQ119901120572radic119899119896



119901120572(Ω) and sdot LB

119901120572(Ω)

le

119862 sdot LQ119901120572


in an analogous way


Ω)





119870= 119870(2119870 + 1)









120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ 120601 (119861) 120583(119861)1119901

(92)


11120588 is bounded on 119871119901


119901120601120591119870120588(119883) to 119871

119901120601120588(119883)


11120588 thatis

119872120588119891 (119909) = sup

119909isin119861isinB120588

1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 (93)



splitting119891 = 1198911+119891


1= 119891120594

120591119870119861


119872120588119891

2(119909) le sup

1198611015840

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 (94)



119909 isin 1198611015840 and 119903(1198611015840


1

120601 (119861)

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 le

1

120593 (119903 (119861))(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le1

120601 (1198611015840)(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

(95)

Consequently

1

120601 (119861)119872

120588119891

2(119909) le

10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119909 isin 119861 (96)


119872120588is bounded on 119871119901

(119883 120583) give

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891)

119901

119889120583)

1119901

le sum

119894=12

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891

119894)119901

119889120583)

1119901

le 119888119901

1

120601 (119861) 120583(119861)1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

le 1198881015840

119901

1

120601 (120591119870119861) 120583(120591

119870119861)

1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

le 11988810158401015840

119901

10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

(97)


119871119901120601120588

≲ 119891119871119901120601120591119870120588


119901120582(R119899



(R119899


120593(119903) = 119903120572 120572 le 0 so that 120601 = 120601




119901120601119903120572

into itself


120573by letting

119868120573119891 (119909) = int

119883

119891 (119910)

119889(119909 119910)119899minus120573

119889120583 (119910) 119909 isin 119883 (98)


120593 (119903)


int

infin

119903

119905120573minus1

120593 (119905) 119889119905 ≲ 120595 (119903) 119903 gt 0

(99)


120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ Ψ (119861) 120583(119861)1119902

(100)



(119883 120583) to119871119902


119901120601(119883) to 119871

119902Ψ120588(119883)


le sdot 119871119902Ψ


1

Ψ (119861)(1

120583 (119861)int119861

1003816100381610038161003816100381611986812057311989110038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(101)



1+ 119891

2

1198911= 119891120594

120591119870119861 and 120591





119868120573and (100) we write

1

Ψ (119861)(1

120583 (119861)int119861

10038161003816100381610038161003816119868120573119891

1

10038161003816100381610038161003816

119902

119889120583)

1119902

le1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171003817119868120573119891

1

10038171003817100381710038171003817119871119902(120583)

≲1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171198911

1003817100381710038171003817119871119901(120583)

≲1

120601 (120591119870119861)(

1

120583 (120591119870119861)int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(102)


119870119903)

119888

sub 119861(119909 119903)119888


10038161003816100381610038161003816119868120573119891

2(119909)10038161003816100381610038161003816le int

(120591119870119861)119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le int119861(119909119903)

119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

=

infin

sum

119895=0

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le

infin

sum

119895=0

(2119895

119903)120573minus119899

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲

infin

sum

119895=0

(2119895

119903)120573 1

120583 (119861 (119909 2119895+1119903))int119861(1199092119895+1

119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲ (

infin

sum

119895=0

(2119895

119903)120573

120601 (119861 (119909 2119895+1

119903)))10038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ int

infin

119903

119905120573minus1

120593 (119905) 11988911990510038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ 120595 (119903)10038171003817100381710038171198911003817100381710038171003817119871119901120601

(103)


1

Ψ (119861 (119886 119903))(

1

120583 (119861 (119886 119903))int119861(119886119903)

10038161003816100381610038161003816119868120573119891

2

10038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(104)



(120583) to 119871119902



120573is bounded

from 119871119901120601(119883) to 119871

119902Ψ(119883)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













sup120583(119876)gt0

120583(119896119876)1119902minus1119901

(int119876

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (65)



space 119871119901120601(R119899

119889(infin)


120601 (119876) = 120583(119876)minus1119901

120583(119896119876)1119901minus1119902

(66)


119901(119896 120583) does not depend


= 119889(infin) 120588 = 1205881015840


(119876) =

120583(119876)minus1119901

120583(1198961015840






119889(infin)




satisfying

sup120583(119876)gt0

(1

120595 (120583 (119896119876))int119876

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (67)



119889(infin)

120583) where

120601 (119876) = (120595 (120583 (119896119876))

120583 (119876))

1119901

(68)


(119896 120583) =M119902




(119876) = (120595(120583(1198961015840

119876))120583(119876))1119901



119886





B119886



120583(119861)1119901

(1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

lt infin (69)


119889(2)

120574119899) where 120574


119899(119909) =

120587minus1198992 exp(minus1199092





119889(2)


space E119901119902

B119886


119883 satisfying

100381710038171003817100381711989110038171003817100381710038171198711(120574

119899)+ sup

119861isinB119886

120583(119861)1119901

(1

120583 (119861)int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816

119901

119889120583)

1119901

lt infin

(70)




(R2

120583) (= L11(R

2

119889(2)


(R2

120583) (=L

11(R2

119889(infin)

120583)) differ



119899 where 119886

119899gt 0



loc(N) = ℓ119901


119895isinN isin

ℓ119901

(N 120583)1198721199011119891 and119872119888


1198721199011119891 (119899) = 119872

1198881199011119891 (119899) = (sum

119895

1003816100381610038161003816119891 (119895) minus ⟨119891⟩N

1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(71)

where ⟨119891⟩N = sum119895119891(119895)119886

119895 HenceL

1199011(N) = ℓ119901

(N 120583) 119891L1199011

≃ 119891 minus ⟨119891⟩Nℓ119901(N120583) 119882

0= C and L




1198721199011119891 (119899) = 119872

119888

1199011119891 (119899)

= max

1003816100381610038161003816119891 (119899)1003816100381610038161003816 (sum

119895

1003816100381610038161003816119891 (119895)1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(72)

and hence 1198711199011(N) = ℓ

infin



of the form 119861 = 119899 119899 isin N 119871119901



1199011120588(N) = 119904(N) and119891L




hence 1198711199011120588(N) = ℓ

infin


ℓinfin



loc(Ω) such that

10038171003817100381710038171198911003817100381710038171003817BMO(Ω)

= sup119861subΩ

1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816 lt infin (73)


120588Ω(119909) = 119889 (119909 120597Ω) 119909 isin Ω (74)


119889 (119909 120597Ω) = inf 119889 (119909 119910) 119910 isin 120597Ω (75)




10038171003817100381710038171198911003817100381710038171003817119871119901120582(Ω)

= sup119909isinΩ

sup0lt119903lt119903

0

(1

119903120582int119861(119909119903)capΩ

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

(76)









119901120601(Ω) =

119871119901120601120588Ω

(Ω) andL119901120601(Ω) =L

119901120601120588Ω


Ω

119871119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817119871119901120601

(Ω)

= sup119861isinB120588(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

L119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817L119901120601

(Ω)

= sup119861isinB120588(Ω)

inf119911isinC

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910) minus 1199111003816100381610038161003816

119901

119889119910)

1119901

lt infin

(77)



function 120588119896= 120588

119896Ωas

120588119896(119909) = 119896 sdot 119889 (119909 120597Ω) 119909 isin Ω (78)

so that 1205881= 120588


119871119901120601119896(Ω) = 119871

119901120601120588119896

(Ω) and L119901120601119896(Ω) = L

119901120601120588119896


119901120601119896(Ω) and L

119901120601119896(Ω)


119901120601119896(Ω)Ω = ⋃

119895isin119869Ω

119895(119869 is finite



119895 then

10038171003817100381710038171198911003817100381710038171003817119871119901120601119896

(Ω)= sup

119909isinΩ


119896(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901


= sup119895isin119869



119896(Ω119895)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

= sup119895isin119869

10038171003817100381710038171003817119891

119895

10038171003817100381710038171003817119871119901120601119895119896(Ω119895)

(79)



119896and 120588(infin)

119896 119871B

119901120601119896(Ω) and 119871Q

119901120601119896(Ω) and so



BΩ



by QΩ

119896


119898120572 Clearly

120601119898120572


119901120572119896(Ω) in place of 119871Q

119901120601119898120572







119901120572119896(Ω)


(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817119871Q119901120572119896

(Ω) 119891 isin 119871

Q119901120572119896

(Ω) (80)10038171003817100381710038171198911003817100381710038171003817LQ119901120572119898

(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) 119891 isinL

Q119901120572119896

(Ω) (81)


Q119901120572119898

(Ω) andLQ

119901120572119896(Ω) sub LQ







of them in QΩ

119896 Thus

1

|119876|120572(1

|119876|int119876

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le1

|119876|120572sum

119895

(1

|119876|int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le sum

119895

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572(1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le 21198991198731003817100381710038171003817119891

1003817100381710038171003817119871Q119901120572119896

(Ω)

(82)



LQ119901120572119896(Ω) and 119876 isin QΩ

119898 QΩ

119896 Then

inf119911isinC

1

|119876|120572(1

|119876|int119876

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

=1

|119876|120572+1119901

10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

(83)

where 119871119901




1199011015840


=

1 and 1198711199011015840



119876


10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

= sup10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

120595 isin 1198711199011015840

0(119876)

100381710038171003817100381712059510038171003817100381710038171199011015840le 1

(84)


119862(119896119898 1199011015840


119898



119895isin

QΩ


119895isin 119871

1199011015840

0(119876) 1 le


119895and 120595

1198951199011015840le 119862120595

1199011015840 for

119895 = 1 119873 and 120595 = sum119873

119895=1120595

119895 Take 120595 isin 119871119901

1015840



subcubes 119876119895119873


119895119873


Then

1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816100381610038161003816

int119876

119891(

119873

sum

119895=1

120595119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

119891120595119895

100381610038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

(119891 minus 119891119876119895

)120595119895

100381610038161003816100381610038161003816100381610038161003816

le 119862

119873

sum

119895=1

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572+1119901

(int119876119895

100381610038161003816100381610038161003816119891 minus 119891

119876119895

100381610038161003816100381610038161003816

119901

)

1119901

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω)

(85)

Hence

1

|119876|120572+1119901

100381710038171003817100381711989110038171003817100381710038171198711(119876)C

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) (86)


(Ω)le 119862119873119891LQ

119901120572119896(Ω)




119901120572119896(Ω)



119898



119896




1015840

0(119861)




1199011015840

0(119861) 1 le 119895 le 119873 such

that supp(120595119895) sub 119861

119895and 120595

1198951199011015840le 119862120595

1199011015840 for 119895 = 1 119873

and 120595 = sum119873

119895=1120595





lt 119877 and 120576 gt1 minus 119877

1015840

119877 there exists 1198731015840

= 1198731015840

(1198771015840


0 119877) = 119909 isin R119899


points 1199091 119909


119861 (1199090 2119877 minus 119877

1015840

) 119861 (1199090 119877

1015840

) sub cup1198731015840

119895=1119861 (119909

119895 120576119877) (87)

(if 1198771015840

= 0 then we set 119861(1199090 0) = 0)




0)

with 1199030= 119898119889

(2)

(1199090 120597Ω) Let 119877

119895= (119895119904)119903



119895= 119878(119909

0 119877




0 119877

119895+1)

119861(1199090 119877



0and 1198771015840

= 119877119895minus1=

((119895 minus 1)119904)1199030(so that 1198771015840


1015840


0 119903


119904minus1

119895=1119873

1015840

((119895 minus 1)119895 2119895 119899)


119896 take

119861(119909lowast

0 119903

lowast


0 119877



(120596 1199090) le

119889(2)

(120596 119909lowast

0) + 119889

(2)

(119909lowast

0 119909

0) and 119889(2)

(119909lowast

0 119909

0) = ((119904 minus 1)119904)119903

0=

((119904 minus 1)119904)119898119889(2)


119889(2)

(119909lowast

0 120597Ω) ge 119889

(2)

(1199090 120597Ω) (1 minus 119898

119904 minus 1

119904) (88)


119903lowast

0=2

1199041199030=2

119904119898119889

(2)

(1199090 120597Ω) lt

2

119904

1

1 minus 119898119889

(2)

(119909lowast

0 120597Ω)

le 119896119889(2)

(119909lowast

0 120597Ω)

(89)



Q11990112057212

(Ω) and sdot 119871Q119901120572

(Ω)= sdot

119871Q11990112057212

(Ω)(the


119901120572(Ω) LB




119871Q119901120572(Ω) = 119871

B119901120572(Ω) L

Q119901120572(Ω) =L

B119901120572(Ω) (90)



119876or 119876



infinle sdot



119861119876isinBΩ

radic119899119896 if119876 isin QΩ

119896 and119876

119861isin QΩ

radic119899119896 if 119861 isinBΩ

119896 Moreover

|119861119876| = 119888

1|119876| and |119876

119861| = 119888

2|119861| where 119888

1and 119888

2 depend on the


119901120572radic119899119896(Ω) For any 119861


inf119911isinC

1

|119861|120572(1

|119861|int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

le 1198881015840

2inf119911isinC

1

1003816100381610038161003816119876119861

1003816100381610038161003816

120572(1

1003816100381610038161003816119876119861

1003816100381610038161003816

int119876119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

(91)

Consequently 119891LB119901120572119896

(Ω)le 119888

1015840

2119891LQ

119901120572radic119889119896


that LQ119901120572radic119899119896



119901120572(Ω) and sdot LB

119901120572(Ω)

le

119862 sdot LQ119901120572


in an analogous way


Ω)





119870= 119870(2119870 + 1)









120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ 120601 (119861) 120583(119861)1119901

(92)


11120588 is bounded on 119871119901


119901120601120591119870120588(119883) to 119871

119901120601120588(119883)


11120588 thatis

119872120588119891 (119909) = sup

119909isin119861isinB120588

1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 (93)



splitting119891 = 1198911+119891


1= 119891120594

120591119870119861


119872120588119891

2(119909) le sup

1198611015840

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 (94)



119909 isin 1198611015840 and 119903(1198611015840


1

120601 (119861)

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 le

1

120593 (119903 (119861))(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le1

120601 (1198611015840)(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

(95)

Consequently

1

120601 (119861)119872

120588119891

2(119909) le

10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119909 isin 119861 (96)


119872120588is bounded on 119871119901

(119883 120583) give

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891)

119901

119889120583)

1119901

le sum

119894=12

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891

119894)119901

119889120583)

1119901

le 119888119901

1

120601 (119861) 120583(119861)1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

le 1198881015840

119901

1

120601 (120591119870119861) 120583(120591

119870119861)

1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

le 11988810158401015840

119901

10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

(97)


119871119901120601120588

≲ 119891119871119901120601120591119870120588


119901120582(R119899



(R119899


120593(119903) = 119903120572 120572 le 0 so that 120601 = 120601




119901120601119903120572

into itself


120573by letting

119868120573119891 (119909) = int

119883

119891 (119910)

119889(119909 119910)119899minus120573

119889120583 (119910) 119909 isin 119883 (98)


120593 (119903)


int

infin

119903

119905120573minus1

120593 (119905) 119889119905 ≲ 120595 (119903) 119903 gt 0

(99)


120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ Ψ (119861) 120583(119861)1119902

(100)



(119883 120583) to119871119902


119901120601(119883) to 119871

119902Ψ120588(119883)


le sdot 119871119902Ψ


1

Ψ (119861)(1

120583 (119861)int119861

1003816100381610038161003816100381611986812057311989110038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(101)



1+ 119891

2

1198911= 119891120594

120591119870119861 and 120591





119868120573and (100) we write

1

Ψ (119861)(1

120583 (119861)int119861

10038161003816100381610038161003816119868120573119891

1

10038161003816100381610038161003816

119902

119889120583)

1119902

le1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171003817119868120573119891

1

10038171003817100381710038171003817119871119902(120583)

≲1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171198911

1003817100381710038171003817119871119901(120583)

≲1

120601 (120591119870119861)(

1

120583 (120591119870119861)int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(102)


119870119903)

119888

sub 119861(119909 119903)119888


10038161003816100381610038161003816119868120573119891

2(119909)10038161003816100381610038161003816le int

(120591119870119861)119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le int119861(119909119903)

119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

=

infin

sum

119895=0

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le

infin

sum

119895=0

(2119895

119903)120573minus119899

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲

infin

sum

119895=0

(2119895

119903)120573 1

120583 (119861 (119909 2119895+1119903))int119861(1199092119895+1

119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲ (

infin

sum

119895=0

(2119895

119903)120573

120601 (119861 (119909 2119895+1

119903)))10038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ int

infin

119903

119905120573minus1

120593 (119905) 11988911990510038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ 120595 (119903)10038171003817100381710038171198911003817100381710038171003817119871119901120601

(103)


1

Ψ (119861 (119886 119903))(

1

120583 (119861 (119886 119903))int119861(119886119903)

10038161003816100381610038161003816119868120573119891

2

10038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(104)



(120583) to 119871119902



120573is bounded

from 119871119901120601(119883) to 119871

119902Ψ(119883)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198721199011119891 (119899) = 119872

119888

1199011119891 (119899)

= max

1003816100381610038161003816119891 (119899)1003816100381610038161003816 (sum

119895

1003816100381610038161003816119891 (119895)1003816100381610038161003816

119901

119886119895)

1119901

119899 isin N

(72)

and hence 1198711199011(N) = ℓ

infin



of the form 119861 = 119899 119899 isin N 119871119901



1199011120588(N) = 119904(N) and119891L




hence 1198711199011120588(N) = ℓ

infin


ℓinfin



loc(Ω) such that

10038171003817100381710038171198911003817100381710038171003817BMO(Ω)

= sup119861subΩ

1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 minus ⟨119891⟩119861

1003816100381610038161003816 lt infin (73)


120588Ω(119909) = 119889 (119909 120597Ω) 119909 isin Ω (74)


119889 (119909 120597Ω) = inf 119889 (119909 119910) 119910 isin 120597Ω (75)




10038171003817100381710038171198911003817100381710038171003817119871119901120582(Ω)

= sup119909isinΩ

sup0lt119903lt119903

0

(1

119903120582int119861(119909119903)capΩ

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

(76)









119901120601(Ω) =

119871119901120601120588Ω

(Ω) andL119901120601(Ω) =L

119901120601120588Ω


Ω

119871119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817119871119901120601

(Ω)

= sup119861isinB120588(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119901

119889119910)

1119901

lt infin

L119901120601(Ω)

=

119891 10038171003817100381710038171198911003817100381710038171003817L119901120601

(Ω)

= sup119861isinB120588(Ω)

inf119911isinC

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

1003816100381610038161003816119891 (119910) minus 1199111003816100381610038161003816

119901

119889119910)

1119901

lt infin

(77)



function 120588119896= 120588

119896Ωas

120588119896(119909) = 119896 sdot 119889 (119909 120597Ω) 119909 isin Ω (78)

so that 1205881= 120588


119871119901120601119896(Ω) = 119871

119901120601120588119896

(Ω) and L119901120601119896(Ω) = L

119901120601120588119896


119901120601119896(Ω) and L

119901120601119896(Ω)


119901120601119896(Ω)Ω = ⋃

119895isin119869Ω

119895(119869 is finite



119895 then

10038171003817100381710038171198911003817100381710038171003817119871119901120601119896

(Ω)= sup

119909isinΩ


119896(Ω)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901


= sup119895isin119869



119896(Ω119895)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

= sup119895isin119869

10038171003817100381710038171003817119891

119895

10038171003817100381710038171003817119871119901120601119895119896(Ω119895)

(79)



119896and 120588(infin)

119896 119871B

119901120601119896(Ω) and 119871Q

119901120601119896(Ω) and so



BΩ



by QΩ

119896


119898120572 Clearly

120601119898120572


119901120572119896(Ω) in place of 119871Q

119901120601119898120572







119901120572119896(Ω)


(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817119871Q119901120572119896

(Ω) 119891 isin 119871

Q119901120572119896

(Ω) (80)10038171003817100381710038171198911003817100381710038171003817LQ119901120572119898

(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) 119891 isinL

Q119901120572119896

(Ω) (81)


Q119901120572119898

(Ω) andLQ

119901120572119896(Ω) sub LQ







of them in QΩ

119896 Thus

1

|119876|120572(1

|119876|int119876

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le1

|119876|120572sum

119895

(1

|119876|int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le sum

119895

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572(1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le 21198991198731003817100381710038171003817119891

1003817100381710038171003817119871Q119901120572119896

(Ω)

(82)



LQ119901120572119896(Ω) and 119876 isin QΩ

119898 QΩ

119896 Then

inf119911isinC

1

|119876|120572(1

|119876|int119876

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

=1

|119876|120572+1119901

10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

(83)

where 119871119901




1199011015840


=

1 and 1198711199011015840



119876


10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

= sup10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

120595 isin 1198711199011015840

0(119876)

100381710038171003817100381712059510038171003817100381710038171199011015840le 1

(84)


119862(119896119898 1199011015840


119898



119895isin

QΩ


119895isin 119871

1199011015840

0(119876) 1 le


119895and 120595

1198951199011015840le 119862120595

1199011015840 for

119895 = 1 119873 and 120595 = sum119873

119895=1120595

119895 Take 120595 isin 119871119901

1015840



subcubes 119876119895119873


119895119873


Then

1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816100381610038161003816

int119876

119891(

119873

sum

119895=1

120595119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

119891120595119895

100381610038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

(119891 minus 119891119876119895

)120595119895

100381610038161003816100381610038161003816100381610038161003816

le 119862

119873

sum

119895=1

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572+1119901

(int119876119895

100381610038161003816100381610038161003816119891 minus 119891

119876119895

100381610038161003816100381610038161003816

119901

)

1119901

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω)

(85)

Hence

1

|119876|120572+1119901

100381710038171003817100381711989110038171003817100381710038171198711(119876)C

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) (86)


(Ω)le 119862119873119891LQ

119901120572119896(Ω)




119901120572119896(Ω)



119898



119896




1015840

0(119861)




1199011015840

0(119861) 1 le 119895 le 119873 such

that supp(120595119895) sub 119861

119895and 120595

1198951199011015840le 119862120595

1199011015840 for 119895 = 1 119873

and 120595 = sum119873

119895=1120595





lt 119877 and 120576 gt1 minus 119877

1015840

119877 there exists 1198731015840

= 1198731015840

(1198771015840


0 119877) = 119909 isin R119899


points 1199091 119909


119861 (1199090 2119877 minus 119877

1015840

) 119861 (1199090 119877

1015840

) sub cup1198731015840

119895=1119861 (119909

119895 120576119877) (87)

(if 1198771015840

= 0 then we set 119861(1199090 0) = 0)




0)

with 1199030= 119898119889

(2)

(1199090 120597Ω) Let 119877

119895= (119895119904)119903



119895= 119878(119909

0 119877




0 119877

119895+1)

119861(1199090 119877



0and 1198771015840

= 119877119895minus1=

((119895 minus 1)119904)1199030(so that 1198771015840


1015840


0 119903


119904minus1

119895=1119873

1015840

((119895 minus 1)119895 2119895 119899)


119896 take

119861(119909lowast

0 119903

lowast


0 119877



(120596 1199090) le

119889(2)

(120596 119909lowast

0) + 119889

(2)

(119909lowast

0 119909

0) and 119889(2)

(119909lowast

0 119909

0) = ((119904 minus 1)119904)119903

0=

((119904 minus 1)119904)119898119889(2)


119889(2)

(119909lowast

0 120597Ω) ge 119889

(2)

(1199090 120597Ω) (1 minus 119898

119904 minus 1

119904) (88)


119903lowast

0=2

1199041199030=2

119904119898119889

(2)

(1199090 120597Ω) lt

2

119904

1

1 minus 119898119889

(2)

(119909lowast

0 120597Ω)

le 119896119889(2)

(119909lowast

0 120597Ω)

(89)



Q11990112057212

(Ω) and sdot 119871Q119901120572

(Ω)= sdot

119871Q11990112057212

(Ω)(the


119901120572(Ω) LB




119871Q119901120572(Ω) = 119871

B119901120572(Ω) L

Q119901120572(Ω) =L

B119901120572(Ω) (90)



119876or 119876



infinle sdot



119861119876isinBΩ

radic119899119896 if119876 isin QΩ

119896 and119876

119861isin QΩ

radic119899119896 if 119861 isinBΩ

119896 Moreover

|119861119876| = 119888

1|119876| and |119876

119861| = 119888

2|119861| where 119888

1and 119888

2 depend on the


119901120572radic119899119896(Ω) For any 119861


inf119911isinC

1

|119861|120572(1

|119861|int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

le 1198881015840

2inf119911isinC

1

1003816100381610038161003816119876119861

1003816100381610038161003816

120572(1

1003816100381610038161003816119876119861

1003816100381610038161003816

int119876119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

(91)

Consequently 119891LB119901120572119896

(Ω)le 119888

1015840

2119891LQ

119901120572radic119889119896


that LQ119901120572radic119899119896



119901120572(Ω) and sdot LB

119901120572(Ω)

le

119862 sdot LQ119901120572


in an analogous way


Ω)





119870= 119870(2119870 + 1)









120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ 120601 (119861) 120583(119861)1119901

(92)


11120588 is bounded on 119871119901


119901120601120591119870120588(119883) to 119871

119901120601120588(119883)


11120588 thatis

119872120588119891 (119909) = sup

119909isin119861isinB120588

1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 (93)



splitting119891 = 1198911+119891


1= 119891120594

120591119870119861


119872120588119891

2(119909) le sup

1198611015840

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 (94)



119909 isin 1198611015840 and 119903(1198611015840


1

120601 (119861)

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 le

1

120593 (119903 (119861))(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le1

120601 (1198611015840)(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

(95)

Consequently

1

120601 (119861)119872

120588119891

2(119909) le

10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119909 isin 119861 (96)


119872120588is bounded on 119871119901

(119883 120583) give

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891)

119901

119889120583)

1119901

le sum

119894=12

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891

119894)119901

119889120583)

1119901

le 119888119901

1

120601 (119861) 120583(119861)1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

le 1198881015840

119901

1

120601 (120591119870119861) 120583(120591

119870119861)

1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

le 11988810158401015840

119901

10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

(97)


119871119901120601120588

≲ 119891119871119901120601120591119870120588


119901120582(R119899



(R119899


120593(119903) = 119903120572 120572 le 0 so that 120601 = 120601




119901120601119903120572

into itself


120573by letting

119868120573119891 (119909) = int

119883

119891 (119910)

119889(119909 119910)119899minus120573

119889120583 (119910) 119909 isin 119883 (98)


120593 (119903)


int

infin

119903

119905120573minus1

120593 (119905) 119889119905 ≲ 120595 (119903) 119903 gt 0

(99)


120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ Ψ (119861) 120583(119861)1119902

(100)



(119883 120583) to119871119902


119901120601(119883) to 119871

119902Ψ120588(119883)


le sdot 119871119902Ψ


1

Ψ (119861)(1

120583 (119861)int119861

1003816100381610038161003816100381611986812057311989110038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(101)



1+ 119891

2

1198911= 119891120594

120591119870119861 and 120591





119868120573and (100) we write

1

Ψ (119861)(1

120583 (119861)int119861

10038161003816100381610038161003816119868120573119891

1

10038161003816100381610038161003816

119902

119889120583)

1119902

le1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171003817119868120573119891

1

10038171003817100381710038171003817119871119902(120583)

≲1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171198911

1003817100381710038171003817119871119901(120583)

≲1

120601 (120591119870119861)(

1

120583 (120591119870119861)int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(102)


119870119903)

119888

sub 119861(119909 119903)119888


10038161003816100381610038161003816119868120573119891

2(119909)10038161003816100381610038161003816le int

(120591119870119861)119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le int119861(119909119903)

119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

=

infin

sum

119895=0

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le

infin

sum

119895=0

(2119895

119903)120573minus119899

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲

infin

sum

119895=0

(2119895

119903)120573 1

120583 (119861 (119909 2119895+1119903))int119861(1199092119895+1

119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲ (

infin

sum

119895=0

(2119895

119903)120573

120601 (119861 (119909 2119895+1

119903)))10038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ int

infin

119903

119905120573minus1

120593 (119905) 11988911990510038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ 120595 (119903)10038171003817100381710038171198911003817100381710038171003817119871119901120601

(103)


1

Ψ (119861 (119886 119903))(

1

120583 (119861 (119886 119903))int119861(119886119903)

10038161003816100381610038161003816119868120573119891

2

10038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(104)



(120583) to 119871119902



120573is bounded

from 119871119901120601(119883) to 119871

119902Ψ(119883)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










= sup119895isin119869



119896(Ω119895)

1

120601 (119861)

(1

1003816100381610038161003816100381611986110038161003816100381610038161003816

int119861

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

= sup119895isin119869

10038171003817100381710038171003817119891

119895

10038171003817100381710038171003817119871119901120601119895119896(Ω119895)

(79)



119896and 120588(infin)

119896 119871B

119901120601119896(Ω) and 119871Q

119901120601119896(Ω) and so



BΩ



by QΩ

119896


119898120572 Clearly

120601119898120572


119901120572119896(Ω) in place of 119871Q

119901120601119898120572







119901120572119896(Ω)


(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817119871Q119901120572119896

(Ω) 119891 isin 119871

Q119901120572119896

(Ω) (80)10038171003817100381710038171198911003817100381710038171003817LQ119901120572119898

(Ω)le 11986210038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) 119891 isinL

Q119901120572119896

(Ω) (81)


Q119901120572119898

(Ω) andLQ

119901120572119896(Ω) sub LQ







of them in QΩ

119896 Thus

1

|119876|120572(1

|119876|int119876

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le1

|119876|120572sum

119895

(1

|119876|int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le sum

119895

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572(1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

int119876119895

10038161003816100381610038161198911003816100381610038161003816

119901

)

1119901

le 21198991198731003817100381710038171003817119891

1003817100381710038171003817119871Q119901120572119896

(Ω)

(82)



LQ119901120572119896(Ω) and 119876 isin QΩ

119898 QΩ

119896 Then

inf119911isinC

1

|119876|120572(1

|119876|int119876

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

=1

|119876|120572+1119901

10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

(83)

where 119871119901




1199011015840


=

1 and 1198711199011015840



119876


10038171003817100381710038171198911003817100381710038171003817119871119901(119876)C

= sup10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

120595 isin 1198711199011015840

0(119876)

100381710038171003817100381712059510038171003817100381710038171199011015840le 1

(84)


119862(119896119898 1199011015840


119898



119895isin

QΩ


119895isin 119871

1199011015840

0(119876) 1 le


119895and 120595

1198951199011015840le 119862120595

1199011015840 for

119895 = 1 119873 and 120595 = sum119873

119895=1120595

119895 Take 120595 isin 119871119901

1015840



subcubes 119876119895119873


119895119873


Then

1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816

int119876

119891120595

10038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

10038161003816100381610038161003816100381610038161003816100381610038161003816

int119876

119891(

119873

sum

119895=1

120595119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

119891120595119895

100381610038161003816100381610038161003816100381610038161003816

=1

|119876|120572+1119901

119873

sum

119895=1

100381610038161003816100381610038161003816100381610038161003816

int119876119895

(119891 minus 119891119876119895

)120595119895

100381610038161003816100381610038161003816100381610038161003816

le 119862

119873

sum

119895=1

1

10038161003816100381610038161003816119876

119895

10038161003816100381610038161003816

120572+1119901

(int119876119895

100381610038161003816100381610038161003816119891 minus 119891

119876119895

100381610038161003816100381610038161003816

119901

)

1119901

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω)

(85)

Hence

1

|119876|120572+1119901

100381710038171003817100381711989110038171003817100381710038171198711(119876)C

le 11986211987310038171003817100381710038171198911003817100381710038171003817LQ119901120572119896

(Ω) (86)


(Ω)le 119862119873119891LQ

119901120572119896(Ω)




119901120572119896(Ω)



119898



119896




1015840

0(119861)




1199011015840

0(119861) 1 le 119895 le 119873 such

that supp(120595119895) sub 119861

119895and 120595

1198951199011015840le 119862120595

1199011015840 for 119895 = 1 119873

and 120595 = sum119873

119895=1120595





lt 119877 and 120576 gt1 minus 119877

1015840

119877 there exists 1198731015840

= 1198731015840

(1198771015840


0 119877) = 119909 isin R119899


points 1199091 119909


119861 (1199090 2119877 minus 119877

1015840

) 119861 (1199090 119877

1015840

) sub cup1198731015840

119895=1119861 (119909

119895 120576119877) (87)

(if 1198771015840

= 0 then we set 119861(1199090 0) = 0)




0)

with 1199030= 119898119889

(2)

(1199090 120597Ω) Let 119877

119895= (119895119904)119903



119895= 119878(119909

0 119877




0 119877

119895+1)

119861(1199090 119877



0and 1198771015840

= 119877119895minus1=

((119895 minus 1)119904)1199030(so that 1198771015840


1015840


0 119903


119904minus1

119895=1119873

1015840

((119895 minus 1)119895 2119895 119899)


119896 take

119861(119909lowast

0 119903

lowast


0 119877



(120596 1199090) le

119889(2)

(120596 119909lowast

0) + 119889

(2)

(119909lowast

0 119909

0) and 119889(2)

(119909lowast

0 119909

0) = ((119904 minus 1)119904)119903

0=

((119904 minus 1)119904)119898119889(2)


119889(2)

(119909lowast

0 120597Ω) ge 119889

(2)

(1199090 120597Ω) (1 minus 119898

119904 minus 1

119904) (88)


119903lowast

0=2

1199041199030=2

119904119898119889

(2)

(1199090 120597Ω) lt

2

119904

1

1 minus 119898119889

(2)

(119909lowast

0 120597Ω)

le 119896119889(2)

(119909lowast

0 120597Ω)

(89)



Q11990112057212

(Ω) and sdot 119871Q119901120572

(Ω)= sdot

119871Q11990112057212

(Ω)(the


119901120572(Ω) LB




119871Q119901120572(Ω) = 119871

B119901120572(Ω) L

Q119901120572(Ω) =L

B119901120572(Ω) (90)



119876or 119876



infinle sdot



119861119876isinBΩ

radic119899119896 if119876 isin QΩ

119896 and119876

119861isin QΩ

radic119899119896 if 119861 isinBΩ

119896 Moreover

|119861119876| = 119888

1|119876| and |119876

119861| = 119888

2|119861| where 119888

1and 119888

2 depend on the


119901120572radic119899119896(Ω) For any 119861


inf119911isinC

1

|119861|120572(1

|119861|int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

le 1198881015840

2inf119911isinC

1

1003816100381610038161003816119876119861

1003816100381610038161003816

120572(1

1003816100381610038161003816119876119861

1003816100381610038161003816

int119876119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

(91)

Consequently 119891LB119901120572119896

(Ω)le 119888

1015840

2119891LQ

119901120572radic119889119896


that LQ119901120572radic119899119896



119901120572(Ω) and sdot LB

119901120572(Ω)

le

119862 sdot LQ119901120572


in an analogous way


Ω)





119870= 119870(2119870 + 1)









120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ 120601 (119861) 120583(119861)1119901

(92)


11120588 is bounded on 119871119901


119901120601120591119870120588(119883) to 119871

119901120601120588(119883)


11120588 thatis

119872120588119891 (119909) = sup

119909isin119861isinB120588

1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 (93)



splitting119891 = 1198911+119891


1= 119891120594

120591119870119861


119872120588119891

2(119909) le sup

1198611015840

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 (94)



119909 isin 1198611015840 and 119903(1198611015840


1

120601 (119861)

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 le

1

120593 (119903 (119861))(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le1

120601 (1198611015840)(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

(95)

Consequently

1

120601 (119861)119872

120588119891

2(119909) le

10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119909 isin 119861 (96)


119872120588is bounded on 119871119901

(119883 120583) give

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891)

119901

119889120583)

1119901

le sum

119894=12

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891

119894)119901

119889120583)

1119901

le 119888119901

1

120601 (119861) 120583(119861)1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

le 1198881015840

119901

1

120601 (120591119870119861) 120583(120591

119870119861)

1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

le 11988810158401015840

119901

10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

(97)


119871119901120601120588

≲ 119891119871119901120601120591119870120588


119901120582(R119899



(R119899


120593(119903) = 119903120572 120572 le 0 so that 120601 = 120601




119901120601119903120572

into itself


120573by letting

119868120573119891 (119909) = int

119883

119891 (119910)

119889(119909 119910)119899minus120573

119889120583 (119910) 119909 isin 119883 (98)


120593 (119903)


int

infin

119903

119905120573minus1

120593 (119905) 119889119905 ≲ 120595 (119903) 119903 gt 0

(99)


120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ Ψ (119861) 120583(119861)1119902

(100)



(119883 120583) to119871119902


119901120601(119883) to 119871

119902Ψ120588(119883)


le sdot 119871119902Ψ


1

Ψ (119861)(1

120583 (119861)int119861

1003816100381610038161003816100381611986812057311989110038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(101)



1+ 119891

2

1198911= 119891120594

120591119870119861 and 120591





119868120573and (100) we write

1

Ψ (119861)(1

120583 (119861)int119861

10038161003816100381610038161003816119868120573119891

1

10038161003816100381610038161003816

119902

119889120583)

1119902

le1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171003817119868120573119891

1

10038171003817100381710038171003817119871119902(120583)

≲1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171198911

1003817100381710038171003817119871119901(120583)

≲1

120601 (120591119870119861)(

1

120583 (120591119870119861)int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(102)


119870119903)

119888

sub 119861(119909 119903)119888


10038161003816100381610038161003816119868120573119891

2(119909)10038161003816100381610038161003816le int

(120591119870119861)119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le int119861(119909119903)

119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

=

infin

sum

119895=0

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le

infin

sum

119895=0

(2119895

119903)120573minus119899

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲

infin

sum

119895=0

(2119895

119903)120573 1

120583 (119861 (119909 2119895+1119903))int119861(1199092119895+1

119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲ (

infin

sum

119895=0

(2119895

119903)120573

120601 (119861 (119909 2119895+1

119903)))10038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ int

infin

119903

119905120573minus1

120593 (119905) 11988911990510038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ 120595 (119903)10038171003817100381710038171198911003817100381710038171003817119871119901120601

(103)


1

Ψ (119861 (119886 119903))(

1

120583 (119861 (119886 119903))int119861(119886119903)

10038161003816100381610038161003816119868120573119891

2

10038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(104)



(120583) to 119871119902



120573is bounded

from 119871119901120601(119883) to 119871

119902Ψ(119883)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119898



119896




1015840

0(119861)




1199011015840

0(119861) 1 le 119895 le 119873 such

that supp(120595119895) sub 119861

119895and 120595

1198951199011015840le 119862120595

1199011015840 for 119895 = 1 119873

and 120595 = sum119873

119895=1120595





lt 119877 and 120576 gt1 minus 119877

1015840

119877 there exists 1198731015840

= 1198731015840

(1198771015840


0 119877) = 119909 isin R119899


points 1199091 119909


119861 (1199090 2119877 minus 119877

1015840

) 119861 (1199090 119877

1015840

) sub cup1198731015840

119895=1119861 (119909

119895 120576119877) (87)

(if 1198771015840

= 0 then we set 119861(1199090 0) = 0)




0)

with 1199030= 119898119889

(2)

(1199090 120597Ω) Let 119877

119895= (119895119904)119903



119895= 119878(119909

0 119877




0 119877

119895+1)

119861(1199090 119877



0and 1198771015840

= 119877119895minus1=

((119895 minus 1)119904)1199030(so that 1198771015840


1015840


0 119903


119904minus1

119895=1119873

1015840

((119895 minus 1)119895 2119895 119899)


119896 take

119861(119909lowast

0 119903

lowast


0 119877



(120596 1199090) le

119889(2)

(120596 119909lowast

0) + 119889

(2)

(119909lowast

0 119909

0) and 119889(2)

(119909lowast

0 119909

0) = ((119904 minus 1)119904)119903

0=

((119904 minus 1)119904)119898119889(2)


119889(2)

(119909lowast

0 120597Ω) ge 119889

(2)

(1199090 120597Ω) (1 minus 119898

119904 minus 1

119904) (88)


119903lowast

0=2

1199041199030=2

119904119898119889

(2)

(1199090 120597Ω) lt

2

119904

1

1 minus 119898119889

(2)

(119909lowast

0 120597Ω)

le 119896119889(2)

(119909lowast

0 120597Ω)

(89)



Q11990112057212

(Ω) and sdot 119871Q119901120572

(Ω)= sdot

119871Q11990112057212

(Ω)(the


119901120572(Ω) LB




119871Q119901120572(Ω) = 119871

B119901120572(Ω) L

Q119901120572(Ω) =L

B119901120572(Ω) (90)



119876or 119876



infinle sdot



119861119876isinBΩ

radic119899119896 if119876 isin QΩ

119896 and119876

119861isin QΩ

radic119899119896 if 119861 isinBΩ

119896 Moreover

|119861119876| = 119888

1|119876| and |119876

119861| = 119888

2|119861| where 119888

1and 119888

2 depend on the


119901120572radic119899119896(Ω) For any 119861


inf119911isinC

1

|119861|120572(1

|119861|int119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

le 1198881015840

2inf119911isinC

1

1003816100381610038161003816119876119861

1003816100381610038161003816

120572(1

1003816100381610038161003816119876119861

1003816100381610038161003816

int119876119861

1003816100381610038161003816119891 minus 1199111003816100381610038161003816

119901

)

1119901

(91)

Consequently 119891LB119901120572119896

(Ω)le 119888

1015840

2119891LQ

119901120572radic119889119896


that LQ119901120572radic119899119896



119901120572(Ω) and sdot LB

119901120572(Ω)

le

119862 sdot LQ119901120572


in an analogous way


Ω)





119870= 119870(2119870 + 1)









120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ 120601 (119861) 120583(119861)1119901

(92)


11120588 is bounded on 119871119901


119901120601120591119870120588(119883) to 119871

119901120601120588(119883)


11120588 thatis

119872120588119891 (119909) = sup

119909isin119861isinB120588

1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 (93)



splitting119891 = 1198911+119891


1= 119891120594

120591119870119861


119872120588119891

2(119909) le sup

1198611015840

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 (94)



119909 isin 1198611015840 and 119903(1198611015840


1

120601 (119861)

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 le

1

120593 (119903 (119861))(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le1

120601 (1198611015840)(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

(95)

Consequently

1

120601 (119861)119872

120588119891

2(119909) le

10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119909 isin 119861 (96)


119872120588is bounded on 119871119901

(119883 120583) give

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891)

119901

119889120583)

1119901

le sum

119894=12

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891

119894)119901

119889120583)

1119901

le 119888119901

1

120601 (119861) 120583(119861)1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

le 1198881015840

119901

1

120601 (120591119870119861) 120583(120591

119870119861)

1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

le 11988810158401015840

119901

10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

(97)


119871119901120601120588

≲ 119891119871119901120601120591119870120588


119901120582(R119899



(R119899


120593(119903) = 119903120572 120572 le 0 so that 120601 = 120601




119901120601119903120572

into itself


120573by letting

119868120573119891 (119909) = int

119883

119891 (119910)

119889(119909 119910)119899minus120573

119889120583 (119910) 119909 isin 119883 (98)


120593 (119903)


int

infin

119903

119905120573minus1

120593 (119905) 119889119905 ≲ 120595 (119903) 119903 gt 0

(99)


120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ Ψ (119861) 120583(119861)1119902

(100)



(119883 120583) to119871119902


119901120601(119883) to 119871

119902Ψ120588(119883)


le sdot 119871119902Ψ


1

Ψ (119861)(1

120583 (119861)int119861

1003816100381610038161003816100381611986812057311989110038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(101)



1+ 119891

2

1198911= 119891120594

120591119870119861 and 120591





119868120573and (100) we write

1

Ψ (119861)(1

120583 (119861)int119861

10038161003816100381610038161003816119868120573119891

1

10038161003816100381610038161003816

119902

119889120583)

1119902

le1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171003817119868120573119891

1

10038171003817100381710038171003817119871119902(120583)

≲1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171198911

1003817100381710038171003817119871119901(120583)

≲1

120601 (120591119870119861)(

1

120583 (120591119870119861)int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(102)


119870119903)

119888

sub 119861(119909 119903)119888


10038161003816100381610038161003816119868120573119891

2(119909)10038161003816100381610038161003816le int

(120591119870119861)119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le int119861(119909119903)

119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

=

infin

sum

119895=0

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le

infin

sum

119895=0

(2119895

119903)120573minus119899

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲

infin

sum

119895=0

(2119895

119903)120573 1

120583 (119861 (119909 2119895+1119903))int119861(1199092119895+1

119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲ (

infin

sum

119895=0

(2119895

119903)120573

120601 (119861 (119909 2119895+1

119903)))10038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ int

infin

119903

119905120573minus1

120593 (119905) 11988911990510038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ 120595 (119903)10038171003817100381710038171198911003817100381710038171003817119871119901120601

(103)


1

Ψ (119861 (119886 119903))(

1

120583 (119861 (119886 119903))int119861(119886119903)

10038161003816100381610038161003816119868120573119891

2

10038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(104)



(120583) to 119871119902



120573is bounded

from 119871119901120601(119883) to 119871

119902Ψ(119883)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119870= 119870(2119870 + 1)









120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ 120601 (119861) 120583(119861)1119901

(92)


11120588 is bounded on 119871119901


119901120601120591119870120588(119883) to 119871

119901120601120588(119883)


11120588 thatis

119872120588119891 (119909) = sup

119909isin119861isinB120588

1

120583 (119861)int119861

10038161003816100381610038161198911003816100381610038161003816 119889120583 (93)



splitting119891 = 1198911+119891


1= 119891120594

120591119870119861


119872120588119891

2(119909) le sup

1198611015840

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 (94)



119909 isin 1198611015840 and 119903(1198611015840


1

120601 (119861)

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816 119889120583 le

1

120593 (119903 (119861))(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le1

120601 (1198611015840)(

1

120583 (1198611015840)int1198611015840

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

le10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

(95)

Consequently

1

120601 (119861)119872

120588119891

2(119909) le

10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

119909 isin 119861 (96)


119872120588is bounded on 119871119901

(119883 120583) give

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891)

119901

119889120583)

1119901

le sum

119894=12

1

120601 (119861)(1

120583 (119861)int119861

(119872120588119891

119894)119901

119889120583)

1119901

le 119888119901

1

120601 (119861) 120583(119861)1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120588

le 1198881015840

119901

1

120601 (120591119870119861) 120583(120591

119870119861)

1119901

(int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

+10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

le 11988810158401015840

119901

10038171003817100381710038171198911003817100381710038171003817119871119901120601120591119870120588

(97)


119871119901120601120588

≲ 119891119871119901120601120591119870120588


119901120582(R119899



(R119899


120593(119903) = 119903120572 120572 le 0 so that 120601 = 120601




119901120601119903120572

into itself


120573by letting

119868120573119891 (119909) = int

119883

119891 (119910)

119889(119909 119910)119899minus120573

119889120583 (119910) 119909 isin 119883 (98)


120593 (119903)


int

infin

119903

119905120573minus1

120593 (119905) 119889119905 ≲ 120595 (119903) 119903 gt 0

(99)


120601 (120591119870119861) 120583(120591

119870119861)

1119901

≲ Ψ (119861) 120583(119861)1119902

(100)



(119883 120583) to119871119902


119901120601(119883) to 119871

119902Ψ120588(119883)


le sdot 119871119902Ψ


1

Ψ (119861)(1

120583 (119861)int119861

1003816100381610038161003816100381611986812057311989110038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(101)



1+ 119891

2

1198911= 119891120594

120591119870119861 and 120591





119868120573and (100) we write

1

Ψ (119861)(1

120583 (119861)int119861

10038161003816100381610038161003816119868120573119891

1

10038161003816100381610038161003816

119902

119889120583)

1119902

le1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171003817119868120573119891

1

10038171003817100381710038171003817119871119902(120583)

≲1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171198911

1003817100381710038171003817119871119901(120583)

≲1

120601 (120591119870119861)(

1

120583 (120591119870119861)int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(102)


119870119903)

119888

sub 119861(119909 119903)119888


10038161003816100381610038161003816119868120573119891

2(119909)10038161003816100381610038161003816le int

(120591119870119861)119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le int119861(119909119903)

119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

=

infin

sum

119895=0

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le

infin

sum

119895=0

(2119895

119903)120573minus119899

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲

infin

sum

119895=0

(2119895

119903)120573 1

120583 (119861 (119909 2119895+1119903))int119861(1199092119895+1

119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲ (

infin

sum

119895=0

(2119895

119903)120573

120601 (119861 (119909 2119895+1

119903)))10038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ int

infin

119903

119905120573minus1

120593 (119905) 11988911990510038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ 120595 (119903)10038171003817100381710038171198911003817100381710038171003817119871119901120601

(103)


1

Ψ (119861 (119886 119903))(

1

120583 (119861 (119886 119903))int119861(119886119903)

10038161003816100381610038161003816119868120573119891

2

10038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(104)



(120583) to 119871119902



120573is bounded

from 119871119901120601(119883) to 119871

119902Ψ(119883)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119883 120583) to119871119902


119901120601(119883) to 119871

119902Ψ120588(119883)


le sdot 119871119902Ψ


1

Ψ (119861)(1

120583 (119861)int119861

1003816100381610038161003816100381611986812057311989110038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(101)



1+ 119891

2

1198911= 119891120594

120591119870119861 and 120591





119868120573and (100) we write

1

Ψ (119861)(1

120583 (119861)int119861

10038161003816100381610038161003816119868120573119891

1

10038161003816100381610038161003816

119902

119889120583)

1119902

le1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171003817119868120573119891

1

10038171003817100381710038171003817119871119902(120583)

≲1

Ψ (119861) 120583(119861)1119902

10038171003817100381710038171198911

1003817100381710038171003817119871119901(120583)

≲1

120601 (120591119870119861)(

1

120583 (120591119870119861)int120591119870119861

10038161003816100381610038161198911003816100381610038161003816

119901

119889120583)

1119901

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(102)


119870119903)

119888

sub 119861(119909 119903)119888


10038161003816100381610038161003816119868120573119891

2(119909)10038161003816100381610038161003816le int

(120591119870119861)119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le int119861(119909119903)

119888

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

=

infin

sum

119895=0

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816

119889(119909 119910)119899minus120573

119889120583 (119910)

le

infin

sum

119895=0

(2119895

119903)120573minus119899

int1199102119895119903le119889(119909119910)lt2

119895+1119903

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲

infin

sum

119895=0

(2119895

119903)120573 1

120583 (119861 (119909 2119895+1119903))int119861(1199092119895+1

119903)

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889120583 (119910)

≲ (

infin

sum

119895=0

(2119895

119903)120573

120601 (119861 (119909 2119895+1

119903)))10038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ int

infin

119903

119905120573minus1

120593 (119905) 11988911990510038171003817100381710038171198911003817100381710038171003817119871119901120601

≲ 120595 (119903)10038171003817100381710038171198911003817100381710038171003817119871119901120601

(103)


1

Ψ (119861 (119886 119903))(

1

120583 (119861 (119886 119903))int119861(119886119903)

10038161003816100381610038161003816119868120573119891

2

10038161003816100381610038161003816

119902

119889120583)

1119902

≲10038171003817100381710038171198911003817100381710038171003817119871119901120601

(104)



(120583) to 119871119902



120573is bounded

from 119871119901120601(119883) to 119871

119902Ψ(119883)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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