Research ArticleEstimates for Parameter Littlewood-Paley 119892lowast
120581Functions on
Nonhomogeneous Metric Measure Spaces
Guanghui Lu and Shuangping Tao
College of Mathematics and Statistics Northwest Normal University Lanzhou 730070 China
Correspondence should be addressed to Shuangping Tao taospnwnueducn
Received 18 January 2016 Accepted 17 March 2016
Academic Editor Yoshihiro Sawano
Copyright copy 2016 G Lu and S Tao This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Let (X 119889 120583) be a metric measure space which satisfies the geometrically doubling measure and the upper doubling measureconditions In this paper the authors prove that under the assumption that the kernel of Mlowast
120581satisfies a certain Hormander-type
condition Mlowast120588
120581is bounded from Lebesgue spaces 119871119901
(120583) to Lebesgue spaces 119871119901
(120583) for 119901 ge 2 and is bounded from 1198711
(120583) into1198711infin
(120583) As a corollary Mlowast120588
120581is bounded on 119871
119901
(120583) for 1 lt 119901 lt 2 In addition the authors also obtain that Mlowast120588
120581is bounded from
the atomic Hardy space1198671
(120583) into the Lebesgue space 1198711
(120583)
1 Introduction
In 1958 Stein in [1] firstly introduced the Littlewood-Paleyoperators of the higher-dimensional case meanwhile theauthor also obtained the boundedness of the Marcinkiewiczintegrals and area integrals In 1970 Fefferman in [2] provedthat the Littlewood-Paley 119892lowast
120581function is weak type (119901 119901) for
119901 isin (1 2) and 120581 = 2119901 With further research about Little-wood-Paley operators some authors turn their attentions tostudy the parameter Littlewood-Paley operators For exam-ple in 1999 Sakamoto and Yabuta in [3] considered theparameter 119892lowast
120581function Since then many papers focus on the
behaviours of the operators among them we refer readers tosee [4ndash6]
In the past ten years or so most authors mainly study theclassical theory of harmonic analysis on R119899 under nondou-bling measures which only satisfy the polynomial growthcondition see [7ndash12] Exactly we assume that 120583 which is apositive Radon measure on R119899 satisfies the following growthconditions namely for all 119909 isin R119899 and 119903 isin (0infin) there existconstant 119862 and 0 lt 119889 le 119899 such that
120583 (119861 (119909 119903)) le 119862119903119889
(1)
where 119861(119909 119903) fl 119910 isin R119899
|119909 minus 119910| lt 119903 The analysisassociated with nondoubling measures 120583 as in (1) hasimportant applications in solving long-standing open Pain-leversquos problem and Vitushkinrsquos conjecture (see [13 14])Besides Coifman andWeiss have showed that the measure 120583is a key assumption in harmonic analysis on homogeneous-type spaces (see [15 16])
HoweverHytonen in [17] pointed that themeasure120583 as in(1) may not contain the doubling measure as special cases Tosolve the problem in 2010 Hytonen in [17] introduced a newclass of metric measure spaces satisfying the so-called upperdoubling conditions and the geometrically doubling (respsee Definitions 1 and 2 below) which are now claimed non-homogeneousmetricmeasure spacesTherefore if we replacethe underlying spaceswith nonhomogeneousmetricmeasurespaces many known-consequences have been proved stilltrue for example see [18ndash22]
In this paper we always assume that (X 119889 120583) is a non-homogeneous metric measure space In this setting we willestablish the boundedness of the parameter Littlewood-Paley119892lowast
120581functions on (X 119889 120583)In order to state our main results we firstly recall some
necessary notions and notation Hytonen in [17] gave out thedefinition of upper doubling metric spaces as follows
Hindawi Publishing CorporationJournal of Function SpacesVolume 2016 Article ID 9091478 12 pageshttpdxdoiorg10115520169091478
2 Journal of Function Spaces
Definition 1 (see [17]) A metric measure space (X 119889 120583) issaid to be upper doubling if 120583 is Borel measure on X andthere exist a dominating function 120582 X times (0infin) rarr (0infin)
and a positive constant 119862120582such that for each 119909 isin X 119903 rarr
120582(119909 119903) is nondecreasing and for all 119909 isin X and 119903 isin (0infin)
120583 (119861 (119909 119903)) le 120582 (119909 119903) le 119862120582120582 (119909
119903
2
) (2)
Htyonen et al in [18] proved that there exists anotherdominating function
120582 such that 120582 le 120582 119862120582le 119862
120582and
120582 (119909 119910) le 119862
120582
120582 (119910 119903) (3)
where 119909 119910 isin X and 119889(119909 119910) le 119903 Based on this from now onlet the dominating function in (2) also satisfy (3)
Now we recall the notion of geometrically doubling con-ditions given in [17]
Definition 2 (see [17]) A metric space (X 119889) is said to begeometrically doubling if there exists some 119873
0isin N such
that for any ball 119861(119909 119903) sub X there exists a finite ball cover-ing 119861(119909
119894 1199032)
119894of 119861(119909 119903) such that the cardinality of this
covering is at most1198730
Remark 3 (see [17]) Let (X 119889) be a metric space Hytonen in[17] showed that the following statements aremutually equiv-alent
(1) (X 119889) is geometrically doubling
(2) For any 120598 isin (0 1) and ball 119861(119909 119903) sub X there exists afinite ball covering 119861(119909
119894 120598119903)
119894of 119861(119909 119903) such that the
cardinality of this covering is at most1198730120598minus119899 Here and
in what follows1198730is as Definition 2 and 119899 = log
2119873
0
(3) For every 120598 isin (0 1) any ball 119861(119909 119903) sub X can containat most119873
0120598minus119899 centers of disjoint balls 119861(119909
119894 120598119903)
119894
(4) There exists 119872 isin N such that any ball 119861(119909 119903) sub X
can contain at most 119872 centers 119909119894119894of disjoint balls
119861(119909119894 1199034)
119872
119894=1
Hytonen in [17] introduced the following coefficients119870119861119878
analogous to Tolsarsquos number 119870119876119877
in [7]Given any two balls 119861 sub 119878 set
119870119861119878
fl 1 + int
2119878119861
1
120582 (119888119861 119889 (119909 119888
119861))
d120583 (119909) (4)
where 119888119861represents the center of the ball 119861
Remark 4 Bui and Duong in [21] firstly introduced the fol-lowing discrete version
119870119861119878
of 119870119861119878
as in (4) on (X 119889 120583)
which is very similar to the number119870119876119877
introduced in [7] byTolsa For any two balls 119861 sub 119878 119870
119861119878is defined by
119870
119861119878= 1 +
119873119861119878
sum
119894=1
120583 (6119894
119861)
120582 (119888119861 6
119894119903119861)
(5)
where the radii of the balls 119861 and 119878 are denoted by 119903119861and
119903119878 respectively and 119873
119861119878is the smallest integer satisfying
6119873119861119878
119903119861ge 119903
119904 It is easy to obtain
119870119861119878
le 119862119870119861119878 Bui and Duong
in [21] also pointed out that it is incorrect that119870119861119878
sim119870
119861119878
Now we recall the following notion of (120572 120573)-doublingproperty (see [17])
Definition 5 (see [17]) Let 120572 120573 isin (1infin) A ball 119861 sub X isclaimed to be (120572 120573)-doubling if 120583(120572119861) le 120573120583(119861)
It was stated in [17] that there exist many balls whichhave the above (120572 120573)-doubling property In the latter part ofthe paper if 120572 and 120573
120572are not specified (120572 120573
120572)-doubling ball
always stands for (6 1205736)-doubling ball with a fixed number
1205736gt max1198623 log
26
120582 6
119899
where 119899 fl log2119873
0is considered as
a geometric dimension of the space Moreover the smallest(6 120573
6)-doubling ball of the form 6
119895
119861 with 119895 isin N is denotedby
119861
6 and sometimes 1198616 can be simply denoted by 119861
Now we give the definition of the parameter Littlewood-Paley 119892lowast
120581functions on (X 119889 120583)
Definition 6 (see [22]) Let 119870(119909 119910) be a locally integrablefunction on (X times X) (119909 119910) 119909 = 119910 Assume that thereexists a positive constant 119862 such that for all 119909 119910 isin X with119909 = 119910
Theorem 11 Let 119870(119909 119910) satisfy (6) and (11) and let Mlowast120588
120581be
as in (9) with 120588 gt 12 and 120581 gt 1 Suppose thatMlowast120588
120581is bounded
on 1198712
(120583) ThenMlowast120588
120581is bounded from119867
1
(120583) into 1198711
(120583)
Applying the Marcinkiewicz interpolation theorem andTheorems 9 and 10 it is easy to get the following result
Corollary 12 Under the assumption of Theorem 10 Mlowast120588
120581is
bounded on 119871119901
(120583) for 119901 isin (1 2)
The organization of this paper is as follows In Section 2wewill give somepreliminary lemmasTheproofs of themaintheorems will be given in Section 3 Throughout this paper119862 stands for a positive constant which is independent of themain parameters but it may be different from line to line Forany 119864 sub X we use 120594
119864to denote its characteristic function
2 Preliminary Lemmas
In this section we make some preliminary lemmas which areused in the proof of the main results Firstly we recall someproperties of119870
119861119878as in (4) (see [17])
Lemma 13 (see [17]) (1) For all balls 119861 sub 119877 sub 119878 it holds truethat 119870
119861119877le 119870
119861119878
(2) For any 120585 isin [1infin) there exists a positive constant 119862120585
such that for all balls 119861 sub 119878 with 119903119878le 120585119903
119861 119870
119861119878le 119862
120585
(3) For any 984858 isin (1infin) there exists a positive constant 119862984858
depending on 984858 such that for all balls 119861119870119861
119861
984858 le 119862984858
(4) There exists a positive constant 119888 such that for all balls119861 sub 119877 sub 119878 119870
119861119878le 119870
119861119877+ 119888119870
119877119878 In particular if 119861 and 119877 are
concentric then 119888 = 1(5) There exists a positive constant such that for all balls
119861 sub 119877 sub 119878 119870119861119877
le 119870119861119878 moreover if 119861 and 119877 are concentric
then 119870119877119878
le 119870119861119878
4 Journal of Function Spaces
To state the following lemmas let us give a known-result(see [19]) For 120578 isin (0infin) the maximal operator is defined bysetting that for all 119891 isin 119871
1
loc(120583) and 119909 isin X
119872(120578)119891 (119909) fl sup
119876ni119909119876doubling
1
120583 (120578119876)
int
119876
1003816100381610038161003816119891 (119910)
1003816100381610038161003816d120583 (119910) (13)
is bounded on 119871119901
(120583) provided that 119901 isin (1infin) and alsobounded from 119871
1
(120583) into 1198711infin
(120583)The following lemma is slightly changed from [8]
Lemma 14 Let 119870(119909 119910) satisfy (6) and (7) and 120578 isin (0infin)Assume that M120588 is as in (8) and Mlowast120588
120581is as in (9) with
120588 isin (0infin) and 120581 isin (1infin) Then for any nonnegative function120601 there exists a positive constant 119862 such that for all 119891 isin 119871
1003816100381610038161003816le 120574119905 119891119900119903 120583-119886119897119898119900119904119905 119890V119890119903119910 119909 isin X
(23)
where 120574 is some positive constant depending only on (X 120583)and there exists a positive constant 119862 independent of 119891 119905 and119894 such that if 119901 = 1 then
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
Definition 1 (see [17]) A metric measure space (X 119889 120583) issaid to be upper doubling if 120583 is Borel measure on X andthere exist a dominating function 120582 X times (0infin) rarr (0infin)
and a positive constant 119862120582such that for each 119909 isin X 119903 rarr
120582(119909 119903) is nondecreasing and for all 119909 isin X and 119903 isin (0infin)
120583 (119861 (119909 119903)) le 120582 (119909 119903) le 119862120582120582 (119909
119903
2
) (2)
Htyonen et al in [18] proved that there exists anotherdominating function
120582 such that 120582 le 120582 119862120582le 119862
120582and
120582 (119909 119910) le 119862
120582
120582 (119910 119903) (3)
where 119909 119910 isin X and 119889(119909 119910) le 119903 Based on this from now onlet the dominating function in (2) also satisfy (3)
Now we recall the notion of geometrically doubling con-ditions given in [17]
Definition 2 (see [17]) A metric space (X 119889) is said to begeometrically doubling if there exists some 119873
0isin N such
that for any ball 119861(119909 119903) sub X there exists a finite ball cover-ing 119861(119909
119894 1199032)
119894of 119861(119909 119903) such that the cardinality of this
covering is at most1198730
Remark 3 (see [17]) Let (X 119889) be a metric space Hytonen in[17] showed that the following statements aremutually equiv-alent
(1) (X 119889) is geometrically doubling
(2) For any 120598 isin (0 1) and ball 119861(119909 119903) sub X there exists afinite ball covering 119861(119909
119894 120598119903)
119894of 119861(119909 119903) such that the
cardinality of this covering is at most1198730120598minus119899 Here and
in what follows1198730is as Definition 2 and 119899 = log
2119873
0
(3) For every 120598 isin (0 1) any ball 119861(119909 119903) sub X can containat most119873
0120598minus119899 centers of disjoint balls 119861(119909
119894 120598119903)
119894
(4) There exists 119872 isin N such that any ball 119861(119909 119903) sub X
can contain at most 119872 centers 119909119894119894of disjoint balls
119861(119909119894 1199034)
119872
119894=1
Hytonen in [17] introduced the following coefficients119870119861119878
analogous to Tolsarsquos number 119870119876119877
in [7]Given any two balls 119861 sub 119878 set
119870119861119878
fl 1 + int
2119878119861
1
120582 (119888119861 119889 (119909 119888
119861))
d120583 (119909) (4)
where 119888119861represents the center of the ball 119861
Remark 4 Bui and Duong in [21] firstly introduced the fol-lowing discrete version
119870119861119878
of 119870119861119878
as in (4) on (X 119889 120583)
which is very similar to the number119870119876119877
introduced in [7] byTolsa For any two balls 119861 sub 119878 119870
119861119878is defined by
119870
119861119878= 1 +
119873119861119878
sum
119894=1
120583 (6119894
119861)
120582 (119888119861 6
119894119903119861)
(5)
where the radii of the balls 119861 and 119878 are denoted by 119903119861and
119903119878 respectively and 119873
119861119878is the smallest integer satisfying
6119873119861119878
119903119861ge 119903
119904 It is easy to obtain
119870119861119878
le 119862119870119861119878 Bui and Duong
in [21] also pointed out that it is incorrect that119870119861119878
sim119870
119861119878
Now we recall the following notion of (120572 120573)-doublingproperty (see [17])
Definition 5 (see [17]) Let 120572 120573 isin (1infin) A ball 119861 sub X isclaimed to be (120572 120573)-doubling if 120583(120572119861) le 120573120583(119861)
It was stated in [17] that there exist many balls whichhave the above (120572 120573)-doubling property In the latter part ofthe paper if 120572 and 120573
120572are not specified (120572 120573
120572)-doubling ball
always stands for (6 1205736)-doubling ball with a fixed number
1205736gt max1198623 log
26
120582 6
119899
where 119899 fl log2119873
0is considered as
a geometric dimension of the space Moreover the smallest(6 120573
6)-doubling ball of the form 6
119895
119861 with 119895 isin N is denotedby
119861
6 and sometimes 1198616 can be simply denoted by 119861
Now we give the definition of the parameter Littlewood-Paley 119892lowast
120581functions on (X 119889 120583)
Definition 6 (see [22]) Let 119870(119909 119910) be a locally integrablefunction on (X times X) (119909 119910) 119909 = 119910 Assume that thereexists a positive constant 119862 such that for all 119909 119910 isin X with119909 = 119910
Theorem 11 Let 119870(119909 119910) satisfy (6) and (11) and let Mlowast120588
120581be
as in (9) with 120588 gt 12 and 120581 gt 1 Suppose thatMlowast120588
120581is bounded
on 1198712
(120583) ThenMlowast120588
120581is bounded from119867
1
(120583) into 1198711
(120583)
Applying the Marcinkiewicz interpolation theorem andTheorems 9 and 10 it is easy to get the following result
Corollary 12 Under the assumption of Theorem 10 Mlowast120588
120581is
bounded on 119871119901
(120583) for 119901 isin (1 2)
The organization of this paper is as follows In Section 2wewill give somepreliminary lemmasTheproofs of themaintheorems will be given in Section 3 Throughout this paper119862 stands for a positive constant which is independent of themain parameters but it may be different from line to line Forany 119864 sub X we use 120594
119864to denote its characteristic function
2 Preliminary Lemmas
In this section we make some preliminary lemmas which areused in the proof of the main results Firstly we recall someproperties of119870
119861119878as in (4) (see [17])
Lemma 13 (see [17]) (1) For all balls 119861 sub 119877 sub 119878 it holds truethat 119870
119861119877le 119870
119861119878
(2) For any 120585 isin [1infin) there exists a positive constant 119862120585
such that for all balls 119861 sub 119878 with 119903119878le 120585119903
119861 119870
119861119878le 119862
120585
(3) For any 984858 isin (1infin) there exists a positive constant 119862984858
depending on 984858 such that for all balls 119861119870119861
119861
984858 le 119862984858
(4) There exists a positive constant 119888 such that for all balls119861 sub 119877 sub 119878 119870
119861119878le 119870
119861119877+ 119888119870
119877119878 In particular if 119861 and 119877 are
concentric then 119888 = 1(5) There exists a positive constant such that for all balls
119861 sub 119877 sub 119878 119870119861119877
le 119870119861119878 moreover if 119861 and 119877 are concentric
then 119870119877119878
le 119870119861119878
4 Journal of Function Spaces
To state the following lemmas let us give a known-result(see [19]) For 120578 isin (0infin) the maximal operator is defined bysetting that for all 119891 isin 119871
1
loc(120583) and 119909 isin X
119872(120578)119891 (119909) fl sup
119876ni119909119876doubling
1
120583 (120578119876)
int
119876
1003816100381610038161003816119891 (119910)
1003816100381610038161003816d120583 (119910) (13)
is bounded on 119871119901
(120583) provided that 119901 isin (1infin) and alsobounded from 119871
1
(120583) into 1198711infin
(120583)The following lemma is slightly changed from [8]
Lemma 14 Let 119870(119909 119910) satisfy (6) and (7) and 120578 isin (0infin)Assume that M120588 is as in (8) and Mlowast120588
120581is as in (9) with
120588 isin (0infin) and 120581 isin (1infin) Then for any nonnegative function120601 there exists a positive constant 119862 such that for all 119891 isin 119871
1003816100381610038161003816le 120574119905 119891119900119903 120583-119886119897119898119900119904119905 119890V119890119903119910 119909 isin X
(23)
where 120574 is some positive constant depending only on (X 120583)and there exists a positive constant 119862 independent of 119891 119905 and119894 such that if 119901 = 1 then
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
Theorem 11 Let 119870(119909 119910) satisfy (6) and (11) and let Mlowast120588
120581be
as in (9) with 120588 gt 12 and 120581 gt 1 Suppose thatMlowast120588
120581is bounded
on 1198712
(120583) ThenMlowast120588
120581is bounded from119867
1
(120583) into 1198711
(120583)
Applying the Marcinkiewicz interpolation theorem andTheorems 9 and 10 it is easy to get the following result
Corollary 12 Under the assumption of Theorem 10 Mlowast120588
120581is
bounded on 119871119901
(120583) for 119901 isin (1 2)
The organization of this paper is as follows In Section 2wewill give somepreliminary lemmasTheproofs of themaintheorems will be given in Section 3 Throughout this paper119862 stands for a positive constant which is independent of themain parameters but it may be different from line to line Forany 119864 sub X we use 120594
119864to denote its characteristic function
2 Preliminary Lemmas
In this section we make some preliminary lemmas which areused in the proof of the main results Firstly we recall someproperties of119870
119861119878as in (4) (see [17])
Lemma 13 (see [17]) (1) For all balls 119861 sub 119877 sub 119878 it holds truethat 119870
119861119877le 119870
119861119878
(2) For any 120585 isin [1infin) there exists a positive constant 119862120585
such that for all balls 119861 sub 119878 with 119903119878le 120585119903
119861 119870
119861119878le 119862
120585
(3) For any 984858 isin (1infin) there exists a positive constant 119862984858
depending on 984858 such that for all balls 119861119870119861
119861
984858 le 119862984858
(4) There exists a positive constant 119888 such that for all balls119861 sub 119877 sub 119878 119870
119861119878le 119870
119861119877+ 119888119870
119877119878 In particular if 119861 and 119877 are
concentric then 119888 = 1(5) There exists a positive constant such that for all balls
119861 sub 119877 sub 119878 119870119861119877
le 119870119861119878 moreover if 119861 and 119877 are concentric
then 119870119877119878
le 119870119861119878
4 Journal of Function Spaces
To state the following lemmas let us give a known-result(see [19]) For 120578 isin (0infin) the maximal operator is defined bysetting that for all 119891 isin 119871
1
loc(120583) and 119909 isin X
119872(120578)119891 (119909) fl sup
119876ni119909119876doubling
1
120583 (120578119876)
int
119876
1003816100381610038161003816119891 (119910)
1003816100381610038161003816d120583 (119910) (13)
is bounded on 119871119901
(120583) provided that 119901 isin (1infin) and alsobounded from 119871
1
(120583) into 1198711infin
(120583)The following lemma is slightly changed from [8]
Lemma 14 Let 119870(119909 119910) satisfy (6) and (7) and 120578 isin (0infin)Assume that M120588 is as in (8) and Mlowast120588
120581is as in (9) with
120588 isin (0infin) and 120581 isin (1infin) Then for any nonnegative function120601 there exists a positive constant 119862 such that for all 119891 isin 119871
1003816100381610038161003816le 120574119905 119891119900119903 120583-119886119897119898119900119904119905 119890V119890119903119910 119909 isin X
(23)
where 120574 is some positive constant depending only on (X 120583)and there exists a positive constant 119862 independent of 119891 119905 and119894 such that if 119901 = 1 then
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
To state the following lemmas let us give a known-result(see [19]) For 120578 isin (0infin) the maximal operator is defined bysetting that for all 119891 isin 119871
1
loc(120583) and 119909 isin X
119872(120578)119891 (119909) fl sup
119876ni119909119876doubling
1
120583 (120578119876)
int
119876
1003816100381610038161003816119891 (119910)
1003816100381610038161003816d120583 (119910) (13)
is bounded on 119871119901
(120583) provided that 119901 isin (1infin) and alsobounded from 119871
1
(120583) into 1198711infin
(120583)The following lemma is slightly changed from [8]
Lemma 14 Let 119870(119909 119910) satisfy (6) and (7) and 120578 isin (0infin)Assume that M120588 is as in (8) and Mlowast120588
120581is as in (9) with
120588 isin (0infin) and 120581 isin (1infin) Then for any nonnegative function120601 there exists a positive constant 119862 such that for all 119891 isin 119871
1003816100381610038161003816le 120574119905 119891119900119903 120583-119886119897119898119900119904119905 119890V119890119903119910 119909 isin X
(23)
where 120574 is some positive constant depending only on (X 120583)and there exists a positive constant 119862 independent of 119891 119905 and119894 such that if 119901 = 1 then
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
1003816100381610038161003816le 120574119905 119891119900119903 120583-119886119897119898119900119904119905 119890V119890119903119910 119909 isin X
(23)
where 120574 is some positive constant depending only on (X 120583)and there exists a positive constant 119862 independent of 119891 119905 and119894 such that if 119901 = 1 then
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
![Introduction - Joint Mathematics Meetings · By Coifman’s ideas, David, Journ´e and Semmes in [3] provided the Littlewood-Paley theory for spaces of homogeneous type introduced](https://static.documents.pub/doc/80x56/60300a1e64e88a070613dff1/introduction-joint-mathematics-by-coifmanas-ideas-david-journe-and-semmes.jpg)
![GLOBAL WELL-POSEDNESS IN SPATIALLY CRITICAL BESOV …rjduan/Preprint for Homepage... · Chemin-Lerner type space (cf. [6]) with those at the uid level. Speci cally, using the Littlewood-Paley](https://static.documents.pub/doc/80x56/60300b95578cd93462553c25/global-well-posedness-in-spatially-critical-besov-rjduanpreprint-for-homepage.jpg)
