RECENT DEVELOPMENTS IN THE THEORY OF FUNCTION SPACES H. TRIEBEL Sektion Mathematik, Universitiit Jena DDR-6900 Jena, Universit~its Hochhaus

i. Introduction

The word "function spaces" covers nowadays rather different bran-

ches and techniques. In our context function spaces means spaces of

functions and distributions defined on the real euclidean n-space R n

which are isotropic, non-homogeneous and unweighted. More precisely,

this survey deals with the spaces s and F s Bp,q p,q on R n which cover

Holder-Zygmund spaces, Sobolev-Slobodeckij spaces, Besov-Lipschitz

spaces, Bessel-potential spaces and spaces of Hardy type. First we try

to describe how the different approaches are interrelated, inclusively

few historical remarks. Secondly, we outline some very recent develop-

ments which, by the opinion of the author, not only unify and simplify

the theory of function spaces under consideration considerably, but

which also may serve a starting point for further studies.

2. How to Measure Smoothness?

Let R be the real euclidean n-space. The classical devises to mea- n

sure smoothness are derivatives and differences. If one wishes to ex-

press smoothness not only locally but globally, in our case on Rn, then

function spaces, e.g. of Lp-type, seem to be an appropriate tool. We m

use standard notations for the derivatives D e and the differences A h,

D = al an if x : (Xl,...,x n) 6 Rn, ~ = (~l,...,en),l~i =

8x I ...Sx n n

= Ze. j=l 3

and m m-l.l

A f(x) = f(x + h) - f(x)~ A h = A h A h

if x 6 R n, h E Rn, and m = 2,3,... Furthermore,

IPdx) I/p , lif[Lp1[ = ( f{f(x) 0 < p S ~ ,

R n

with the ssual modification if p = ~. Recall that S and S" stand for

the Schwartz space of all complex-valued infinitely differentiable

rapidly decreasing functions on R n and the space of all complex-

96

-valued tempered distributions on R , respectively. Of course, the n

spaces L with 0 < p S ~ have the usual meaning (complex-valued P

functions).

Definition i. (i) (H~lder-Zygmund spaces). Let s be a positive

number and let m be an integer with 0 < s < m. Then

C s = {flf 6 L~'YflcSllm = nflL U + sup lhl-Sl~f(x) l < ~}. (I)

xe R n 0 #he R n

(ii) (Sobolev spaces). Let 1 < p < ~ and let m be a natural

number. Then

W~ = {flf e Lp, nf~wmn~ = ~ lIDafIL II < ~}. (2)

Remark I. Let 0 < s < 1. Then

lJflCS~l= sup If(x) l + sup If(x) - f(y)l (3) 2 xeR x%y Ix-yt

n

are the familiar norms in the Holder spaces C s. If s is a positive

fractional number, i.e. 0 < s : [ s] + {s} with Is] integer and 0 <

< {s} < 1 then ( ~ can be extended by

liD~fiL~n + ~ ILD~fIc{S} il i " OSI~ISI s) I~ l= l s]

The corresponding spaces are the well-known Holder spaces (on R n) as

they had been used since the twenties. It had been discovered by A.

Zygmund [29] in 1945 that it is much more effective to use higher dif-

ferences than derivatives combined with first differences. Definition

l(i) must be understood in this sense. In particular if s is given

then all the admissible norms IIflcSH are equivalent to each other. The m

spaces ~D have been introduced by S.L. Sobolev [16] in 1936. The deri-

vatives involved must be understood in the sense of distributions.

In the fifties several attemps hade been made to extend the spaces

1 W2,... and to re- from Definitio I, to fill the gaps between Lp, Wp,

place the sup-norm in (i) by other norms. On the basis of quite diffe-

rent motivations S.M. Nikol'skij introduced in the early fifties the

spaces A s with s > 0, 1 < p < ~ (we always prefer the notations p,~ used below which are different from the original ones) and L.N.Slobo-

deckij, N.Aronszajn and E.Gagliardd defined the spaces A~,p~ with s > 0,

1 < p < ~. The next major step came around 1960. Let F and F -I be the

Fourier transform and its inverse on S', respectively. Let s

I sf = F-I[(1 + i~12)2Ff], f 6 S', -~ < s < ~ (4)

Definition 2. (i) (Besov-Lipschitz spaces). Let s > 0, i < p <

97

and 1S q S ~. Let m he an integer with m > s. Then

A s = {f{f 6 Lp, lfflA s" ~ + P,q p,qHm-~ HflLp

1

÷ ( Ilhl-squA~f(.)[L iS q ~h )q < ~} R n P lht n

(usual modification if q = ~).

(ii) (Bessel-Dotential sDaces). Let -~ < s < ~ and 1 < p <

Then

H s = {flf e S',HfIHSll = HI fIL II < ~] . (6) p p s p

Remark 2. The Besov-Lipschitz spaces A s have been introduced by P,q

O.V. Besov [2,3] (following the way paved by S.M.Nikol'skij). They

proved to be one of the most successful scales of function spaces. The

two sup-norms in (i) (with respect to x E R n and h E R n) are splitted

in (5) in an Lp-nOrm and an Lq-nOrm. In some sense these spaces are the

appropriate extensions of the spaces C s in the way described above

and they fill the gaps between the Sobolev spaces in a reasonable way,

although the Sobolev spaces are not special cases of the spaces A s P,q

if p ~ 2. As in the case of the spaces C s all She admissible norms

nfIA~,qn m = (with different m's) are pairwise equivalent. The spaces

H s have been introduced bv A.P.Calderon [5] and N.Aronszajn, K.T. P

Smith [ i]. First we remark that

H s = W s if s = 0,i,2,... and 1 < p < P P

s fill the gaps between the Sobolev In other words, also the spaces Hp

spaces and extend these spaces to negative values of s. But more impor-

tant, successful method, the Fourier-analytic approach, or the spec-

tral approach, which we discuss in the next section.

3. The Fourier-Analytical Approach

We return to (4) and (6). Let ~ be the Laplacian on R n and let E

be the identity. Recall that

(E - A)f : F-I[(1 + i~J2)Ff], f 6 S' .

More general, the fractoonal powers of E - A are given by s s I

(E - A)2f = F-I[(1 + J~t2)~Ff] , feS', -~<s< ~

In other words, f 6 H s if and only if (E - ~)s/2 f 6 L . This gives a P P

better feeling what is going on in (6). In particular, smoothness is

measured in the Fourier image by the weight-function g(~) = (i ,i~12) s/2,

and the growth of this weight-function at infinity represents the degree

of smoothness. Let h(~) be another positive smooth weight-function,

not necessarily of the above polynomial type. In order to provide a

better understanding of the Fourier-analytical method we dare a bold

speculation: If hl(~) and h2({) are two weight-functions with the

same behaviour at infinity then they generate the same smoothness

class in the above sense. It comes out that something of this type is

correct (via Fourier multiplier theorems), but we shall not try to

make this vague assertion more precise. But on the basis of this

speculation we try to replace the above weight-function g(~) =

= (i + I~12) s/2 by more handsome weight-functions which offer a great-

er flexibility. If I~I ~ 2 j with j = 0,i,2,... then g(~) ~ 2 js. Hence

one can try to replace g(~) by a step function g(~) with g(~) ~ 23s if

[~i ~ 2 j. This replacement is a little bit too crude, but a smooth

version of this idea is just what we want. We give a precise formula-

tion. Let ~({) E S with

1 supp 9 C {~I~ S l~i ~ 2]

and

~(2-J{) = 1 if ~ # 0 .

Functions with these properties~ exist. Let ~j(~) = ~(2-3~) if j = 1,2,

3,... and ~0(~) = 1 - ~l~j(~). Then ~0(~) has also a compact support. j s

of (I + I~i2) 2 is now given by ~ 2Jsgj({). We The desired substitute

introduce the pseudodifferential operators j=0

~9(D)f(x) = F-l[~j(~)Ff](x), x e Rn, j = 0,1,2, .... f 6 S'. (7)

This makes sense because by the Paley-Wiener-Schwartz theorem ~j(D)f(x)

is an analytic function in R n for any f E S'. Furthermore, by a theorem

of Paley-Littlewood type we have

IIf IHSll 12Js~j(D)f(.)12)i/21Lpll, -~ < s < ~, 1 < p<~ (8) II( P j=0

(in the sense of equivalent norms). This is the substitute we are look-

ing for. Now we can ask questions.Does it make sense to replace the

12-norm in (8) by an lq-norm (or quasi-norm)~ 0 < q ~ ~ ? Is it reaso-

nable to interchange the roles of Lp and 12 (or more general lq) in

(8)?

Definition 3. (i) Let -~ < s < ~, 0 < p S ~ and 0 < q S ~. Then

B sp,q= [fif 6 S',llfJB~,qll = (3~02Jsqu~j(D)f(.)ILpllq)i/q.= < ~] ~ (9)

(usual modification if q = ~) .

(ii) Let -~ < s < ~, 0 < p < ~ and 0 < q ~ ~. Then

F s = {flf e S',IIflF s n = ~( ~ 2Jsql~j(D)f(.)lq)I/qlL U<~] (i0) P'q P'q ~ j=0 P

99

(usual modification if q = ~ ).

s and Remark 3. For all admissiSle values s,p,q the spaces Bp,q

F s are quasi-Banach spaces (Banach spaces if p > 1 and q > i)~ and p,q - _

they are independent of the chosen function ~ (in the sense of equiv-

alent quasi-norms). Maybe this fact is not so astonishing if p and q

are restricted by 1 < p < ~ and I < q < ~, because in those cases the

Fourier multiplier theory for Lp with 1 < p < ~ and its vector-valued

counterparts can be taken as hints that something of this type may be

valid. But it was a big surprise, also for the creators of this theory,

that these definitions make sense even if 0 < p s 1 (and 0 < q ~ i).

The only exception is p = ~ in the case of the spaces F s (but even P,q

in this case one can do something after appropriate modifications).

The above definition of the spaces B s is due to J.Peetre [11,12]. P,q

The spaces F s have been introduced by the author [19], P.I.Lizorkin P,q

[I0] and J.Peetre [13]. Fro the greater part of the theory of these

spaces a restriction to p Z i, q ~ i would be artifical. But from a

technical point of view such a restriction often simplifies the proofs

because one has the elaborated technique of Banach space theory at

hand (and one avoids a lot of pitfalls which are so abundant if p<l).

Systematic treatments of the theory of the spaces B s and F s have P,q P,q

been given in [14] (mostly restricted to B s with 1 ~ p S ~ ) and P,

[23] (with [21,22] as forerunners, cf. also ~20]). Again one can ask

questions. What is the use of these spaces? What is the connection of

these spaces and those ones introduced in Section 2? As far as the

latter question is concerned one has the following answer.

Theorem i. (i) Let s > 0. Then

C s = B s

(ii) Let i < p < ~ and -~ < s < ~. Then

H s = F s P P,q '

(in particular, ~ = F s if m = 0,1,2, and 1 < p < ~ ) p,2 . . . .

(iii) Let s > 0, i < p < ~ and i ~ q ~ ~. Then s = B s Ap,q p,q

(iv) Let 0 < p < ~. Then F 0 p,2

Hardy type.

(11)

(12)

(1 3)

is a (non-homogeneous) space of

Remark 4. Proofs may be found in [23], cf. also Sections 6 and 7.

4. Points Left Open

The Fourier~analytical approach proved to be very useful in con-

lO0

nection with applications to linear and non-linear partial differen-

tial equations, cf. [20,23] as far as linear equations are concerned.

In the recently developed method of para-multiplications by J.M.Bony

and Y.Meyer (in order to obtain local and microlocal smoothness asser-

tions for non-linear partial differential equations) characterizations

of type (II) play a crucial role. An extension of these methods to the

full scales B s and F s has been given by T.Runst [15] (there one P,q P,q

can also find the necessary references to the papers by Bony, Meyer).

There is no claim that this paper gives a systematic description

of the history of those function spaces which are treated here. We o-

mitted few important developments. But we wish to mention at least few

key-words and some milestone-papers. Interpolation theory plays a cru-

cial role in the theory of function spaces since the sixties. The out-

standing papers are those ones of J.-L.Lions, J.Peetre [9] and A.P.

Calderon [6]. A systematic approach to the theory of function spaces

from the standpoint of interpolation theory has been given in [20].

Another important approach to the theory of function spaces is the

real variable method in the theory of Hardy spaces and the elaboration

of the technique of maximal functions. The milestone-paper in this

field is C.Fefferman, E.M.Stein [7].

5. Harmonic and Thermic Extensions

The interest in Hardy spaces has its origin in complex function

theory: traces of holomorphic functions in the unit disc or the upper

half-plane on the respective boundaries. A generalization of this idea

yields a characterization of functions and distributions of the spaces

B s and F s on R as traces of harmonic functions or temperaturs in p,q p,q n +

Rn+ 1 = [(x,t)Ix • Rn, t > 0} cn the hyperplane t = 0, which is identi-

n ~2 fied with R n. We reformulate this problem as follows. Let A =

j=l ~x~ 3

be the Laplacian in R and let f 6 B s or f E F s . What can be said n p,q p,q

(in the sense of characterizing properties) about the solutions u(x,t)

and v(x,t) of the problems

( ~zu + AU)(x,t) = 0 if (x,t) • + u(x,0) f(x) if x E R (14) ~t2 Rn+l} = n

(harmonic extension) and

(Sv Av)(x,t) = 0 if (x,t) • + (15) - Rn+ll v(x,0) = f(x) if x • R n

(thermic extension)? At least in a formal way the solutions u(x,t) and

101

V(x,t) are known,

t u(x,t) = P(t)f(x) = c ~ n+l f(y)dy, x e Rn, t>0 (16)

R n (|x_yl2÷ t 2) 2

(Cauchy-Poisson semigroup) and 2

n _Ix-~l - m

v(x,t) = W(t)f(x) = ct 2 fe 4t f(y)dy, x e Rn, t > 0 (i 7) R n

(Gauss-Weierstrass semigroup). If f E S' is given, then (17) makes sen-

se. Furthermore, (16) must be understood in the following theorem via

limiting procedures. If a is a real number we put a+ -- max (0,a).

Theorem 2. Let 9 0 E S with 90(0) # 0.

(i) Let - ~ < s < ~, 0 < p < ~, and 0 < q < ~. Let k and m be non-

negative integers with k > n( 1 - i)+ + max (s,n( 1 - i) ) and 2m > s. Then P P + 1

1 U~0(D)flLp[I + (.0 f t(k-s)q" ~t ~0kp(t)f Lpliq ~-dt )q (18)

and s 1 i (m-~)qll0mw(t)flL ii q at )q ll~0(D)flL II + ( f t (19)

P 0 3t m P ~-

(modification if q = ~ ) are equivalent quasi-norms in B s . If s > P'qil - [i in (i 8) (i 9) can be replaced by fIL II. n( 1 I)+ then II~0(D)flL p ~ P

(ii) Let -~ < s < ~, 0 < p < ~ and 0 < q < ~. Let k and m be non-

n i~ i)+) and 2m > s. -negative integers with k > min(p,q) + max (s,n(~ -

Then ]

H~o(D)flLp~ + ll(0~it(k-s)q ~t~.)lq~kp(t)f~ ~)qILpll (20)

1 and I (m-~)qi~mw(t)f(.)lq ~- )qILpll (21) II~0(D)flLpII + I{( ~ t ~ dt --

0 @ t m

(modification if q = ~ ) are equivalent quasi-norms in F s . If s > P,q

n(p - i)+ then ll~0(D)flLpll in (20), (21) can be replaced by ]IfILplI.

Remark 5. Characterizations of the above type have a long history.

As far as the classical Besov-Lipschitz spaces A s and the Bessel- P,q

-potential spaces H s are concerned the first comprehensive treatment P

in the sense of the above theorem has been given by MoH.Taibleson [ 18],

cf. also T.M.Flett [ 8]. In this context we mention also the books by

P.L.Butzler, H.Berens [ 4] and E.M. Stein [ 17] where one can find

many informations about characterizations of the above type (for the

classical space) and the semigroups from (16) and (17), cf. also [20,

I02

2.5.2, 2.5.3]. More recent results (characterizations of the spaces

s and s in the sense of the above theorem) have been obtained Bp,q Fp,q

by G.A.Kaljabin, B.-H.Qui and the author. The above formulation has

been taken over from [25] (cf. also [23, 2.12.2] where we also gave re-

ferences to the papers by B.-H.Qui and G.A.Kaljabin).

6. Unified Approach

Up to this moment we said nothing how to understand that the

apparently rather different approaches via derivatives, differences,

Fourier-analytical decompositions, harmonic and thermic extensions,

always yield the same spaces B s and F s In [23] we proved equiva- p,q p,q"

lence assertions of the above type mostly by rather specific arguments,

cf. also [14,22]. But recently it became clear that there exists a

unified approach which covers all these methods, at least in principle,

and which sheds some light on the just-mentioned problem. We follow

[25] where [24] may be considered as a first step in this direction.

The basic idea is to extend the admissible functions ~ and ~j in (7)

and (9), (i0), Such that corresponding (quasi-)norms in the sense of

(9), (i0) cover automatically characterizations of type (18), (19) and

(5). We recall that

~(tD)f(x) = F-l[~(t.)Ff](x) = ct k 8kp(t)f(x) if ~(~) =

= j~ike_l~ I ~t k (22)

and

~(~t D)f(x) ct m 8mW(t)f(x> if ~(~) ~ J~12me -I~2 : (2 3) 8 t m

F u r t h e r m o r e we r e m a r k t h a t t h e d i s c r e t e q u a s i - n o r m s i n ( 9 ) and ( i 0 )

h a v e a l w a y s c o n t i n u o u s c o u n t e r p a r t s , i . e .

! i Spl lq d t )q ( 2 4 ) II + ( I t - s q l l ~ ( t d ) f ( . ) l ~--- II~0(D)flLp 0

is the continuous substitute of the quasi-norm in (9) and

I i [I~0(D)flLpll + If(0 / t-sqI~(tD)f(" )lq ~--dt )qILpll (25)

is the continuous substitute of the quasi-norm in (I0). This replace-

ment of "discrete" quasi-norms by "continuous" ones is a technical mat-

ter and has nothing to do with the extension of the class of admissible

~'s which we have in mind. If one puts (22),(2 3) in (24)~(25) then

one obtains (18)-(21)° Of course one has to clarify under what condi-

tions for the parameters involved this procedure is correct. However

before giving some details we ask how to incorporate derivatives and

103

differences in this Fourier-analytical concept. We have

~(D)f(x) = cD~A~f(x) if ~(~) = ~(e i~h- i) m, (26)

n with ~ = (el,...,en#, m natural number, and Sh = ~ {jhj, ~e= ~ii...

...$~n. The three functions ~ in (29),(2 3)~(26) h~ve in common that n

they tend to tero if ISt + 0 (even if e = 0 in (28)). In addition the

functions ~ from (22),(2 3) have the same property if I~I + =. If one

compares these functions ~ with the function ~ from Section 3 used

in Definition 3 then it seems to be at least plausible that one can

substitute ~ in (9)~(10) by the functions ~ from (22)~(2 3) if k and

m are chosen sufficiently large. As for the function ~ form (26)

this question is more delicate. First one has no decay if ~ tends to

infinity and secondly one has not only to handle an isolated function

but a family of functions parametrized by h E R (and, maybe, by e). n

We return to these questions later on and formulate a result which co-

vers in principle all cases of interest.

Let h(x) C S and H(x) E S with supp h C {yl l yl ~ 2}, supp H C {yI 1 1

K lyl ~ 4}, h(x) = i if ]xl ~ i, and H(x) = i if ~ K Ixl S 2.

Theorem 3. Let 0 < p ~ ~, 0 < q ~ ~ and -~ < s < ~. Let s 0 and s 1

be two real numbers with 1

- i)+ < s < s I and s I > n(~ - i)~ . (27) So+ n(~

Let ~0(~) and ~(~) be two infinitely differentiable complex-valued

functions on R n and R n- {0}, respectively, which satisfy the Tauberi~

conditions 1

I~0(~)I > 0 if I~t ~ 2 and I~(~)I > 0 if ~ ~ l~I ~ 2. (28)

let p = rain (1,p) and

~I(F-I ~(z)h(z) )(y)IPdy < ~ ~ (29) R sl n Izl

-ms0P sup 2 fl(F-l~(2m.)H(.))(y)IPdy < ~ , (~)

m= i, 2, .. R n

and (~) with ~0 instead of ~. Then

ll~0(D)flLpll + ( I t-sqll~(tD)f(.)ILpllq ~ )q (3ql 0

(modification if q = ~ ) is an equivalent quasi-norm in B s . P,q

Remark 6. This formulation coincides essentially with Theorem 3 in

[25] Of course, ~(tD)f F-l[ • = ~(t.)Ff](x) and (31) coincides with (24)

This theorem has a direct counterpart for the spaces F s .Furthermore s s P'q .

there are some modifications (both for Bp,q and Fp,q) where not onzy a

104

single function ~ but families of these functions are involved, cf.

the considerations in front of the above theorem. Maybe the crucial

conditions (29) and (30) look somewhat complicated and seem to be hard

to check. But this is not the case, in particular for functions of type

(26) the formulations (29)~(30) are well adapted. Furthermore, if one

uses

IIF-I~ILvn ~ 1 S cUllH ~, 0 < v ~ i, 6 > n(~ - ~), (~)

then one can replace (29)~(30) by more handsome-looking conditions, 6

even Sobolev spaces W~) are where only Bessel-potential spaces H 2 (or

involved.

Remark 7. Theorem 2 follows from Theorem 3 and its F s -counte[- P,q

part. One has to use the functions ~ from (22)~(2 3).

7. Characterizations via Differences

In principle one can put ~ from (26) in Theorem 3 and its F s p,q- counterpart. One can calculate under what conditions for the pamaterers

(29)~(30) are satisfied. However as we pointed out in front of Theorem

3 one has to modify Theorem 3, because one needs now theorems with fa-

milies of functions ~ instead of a single function ~. This can be

done, details may be found in [25]. We formulate a result what can be

obtained on this way.

1 Theorem 4. (i) Let 0 < p S ~, 0 < q S ~ and n(~ - i)+< s < m,

where m is a natural number. Then 1

llflLpll + ( lhl~f lhl-sqnAmflLpnqh dhlhl ~ )q (33)

(modification if q = ~ ) is an equivalent quasi-norm in B s n P'q

(ii) Let 0 < p < =, 0 < q S ~ and min(p,q) < s < m, where m is a

natural number. Then 1

llflLpll + 0( f lhl-sqj(~f)(.)lq dh )qlLpU (~) lhlSl lhl n

(modification if q = ~ ) is an equivalent quasi-norm in F s P,q

Remark 8. We refer for details to [25] where we proved many other

theorems of this type via Fourier-analytical approach from Section 6

and few additional considerations. However the theorem itself is not

new, it may be found in [23, 2.5.10, 2.5.12]. But the proof in [23] is

more complicated and not so clearly based on Fourier-analytical results

in the sense of Theorem 3. On the basis of Theorem 4 one has now also a

105

better understanding of (ii) and (i 3). We prefered in the above theo-

rem a formulation via differences only. But one can replace some

differences by derivatives, as it is also suggested by (26).

8. The Local Approcah

The original Fourier-analytical approach as described in Section 3

does not reflect the local nature of the spaces B s and F s o If P,q P,q

x E R n is given then one needs a knowledge of f on the whole R n in

order to calculate ~j(D)f(x) in (7). This stands in sharp contrast to

the derivatives D~f(x) and the differences A~f(x) with lh[ ~ 1 as they

have been used above. However the extended Fourier-analytical method

as described in Section 6 gives the possibility to combine the advan-

tages of the original Fourier-analytical approach and of a strictly

local procedure. We give a description. Let k 0 6 S, and k E S with

supp k 0 C [y[ [y[ ~ i], supp k C [y[ [yl ~ I],

(Fk0)(0) • 0 and (Fk)(0) # 0. n ~2

Let k N = ( ~ 2 )Nk, where N is a natural number. We introduce the j:l 8xj

means

K(kN,t)f(x) : fkN(Y)f(x + ty)dy, x 6 Rn, t > 0, R n

where now N = 0,1,2,... This makes sense for any f E S'.

(~

Theorem 5. (i) Let -~ < s < ~, 0 < p ~ ~ and 0 < q S ~. Let

0 < c < ~, 0 < r < ~ and 2N>max (s,n(~ - i)+). Then

1 r lIK(k0,e)flLpl[ + ( f t-sqllK(kN,t)flLpll q ~ )q (

0

(modification if q = ~ ) is an equivalent quasi-norm in B s p,q"

(ii) Let -~ < s < ~, 0 < p < ~ and 0 < q s ~. Let 0 < e < ~,

0 < r < ~ and 2N > max (s,n(~ - i)+). Then P i

nK(k0,E)fILpa + ]l(0frt-sqJK(kN,t)f(.)lq t--dt )qlLpt[ (37

(modification if q : ~ ) is an equivalent quasi-norm in F s . P,q

Remark 9. It comes out that the above theorem can be obtained

from Theorem 3 and its F s -counterpart. On the other hand it is clear P,q

that (3) describes a local procedure.

Remark I0. With the help of Theorem 5 one can simplify and unify

several proofs in [23], cf. e.g. [26]. But it is also an appropriate

106

tool to handle psudodifferential operators, cf. [28], and to introduoe

spaces of B s and F s type on complete Riemannian manifolds (which P,q P,q

are not necessarily compact), cf. [27].

References

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