(Mathematics Series Volume 1) Peetre, Jaak-New Thoughts on Besov Spaces-Duke University (1976)

NEW THOUGHTS ON

BESOV SPACES

JAAI< PEETRE Tekniska Hogskolan · Lund

DUKE UNIVERSITY MATHEMATICS SERIES I

Published by Mathematics Departinent

Duke University Durham, N.C. 27706, U.S.A.

Contents

Preface

Adverti semen t for the reader

l . General back gro�d

2. Preliminaries on interpol at ion spaces

3. De fin ition and basic propert ies o f Besov spaces

4 . Compari son o f Besov and poten tial spaces

5 . More on interpol ation

6. The Fourier tran s form

7 . Mul tipliers

8 . App roximate pseudo-i denti ties

9. S tructure of Besov spaces

1 0 . An ab stract generalizat ion o f Besov spaces

11. The case 0 < p < 1

12. Some s trange new sp aces

Appendix

References

Con ten ts

Preface .

Thi s compi lation i s ba sed in es sence on a co urse

taugh t at Duke during Apr i l 1 97 4 . I t i s to some exten t an

expanded and revi sed vers ion o f my ear l ier notes " Fun der ingar

om Besov rum" (= " Thoughts about Be sov spaces " ) from 1 967 .

Although the l a tter were written in Swedi sh they too have had

a certain ci rculat ion with in the mathematical community. I f

I am not mi s taken the original ta lk s were i n part given in

French, because of the presence of a Ruman ian mathematician .

I cannot help to try to imagine what woul d have happened i f

they had been edi ted up i n my native tongue E s thon ian! There

are a l ready several excel lent treatments of the sub ject o f

Besov space s i n book form . I am think ing of the books by

S tein, Niko lskij and Triebel. Niko lsk i j ' s i s in Rus s i an and

Triebel ' s is in German, and has not yet ( 1 974 ) appeared which

leave s on ly the rather short t reatmen t o f S tein for the

Engl i sh speak ing reader . Closest to our treatmen t comes the

one by Triebel . But thi s i s not a mere co incidence because

Triebel too takes in part hi s in sp iration from " Funderinga r " .

However our object ive i s somewhat di f feren t - we are o riented

towards appl icatio n s in appro ximation theory, Fourier analys i s

etc . , rather than partial di f feren tial equation s - there i s

not that much overlap a fter a l l s o a separate pub l ication might

be motivated . Anyhow my basic mo tive has been j u st to make

thi s type of metho d better known among mathematicians . At

happy moments I have thought they deserve i t .

Finally I take the advantage to thank my col le ague s at

the Duke Mathematics Department for their hospitality , and

in particular Pro fe s sor Morri s We isfe l d , without whose constant

enco uragement these note s might not have been written .

Durham , April 1 9 7 4

J . P .

Advertisement for the reader

The service o f the following few lines i s to provide the

reader some indications how to best read this me ss , i f he

really must .

The text is divided into twelve chapters . Each chapte r

is followed b y "note s " which contain some brie f historical

remarks . Within the main body o f the chapters there are no

bibliographical re ferences . I apologize at once to all those

whose works I have forgotten to quote (or , even worse , have

misquoted ! ) .

Now an informal account of the contents of e ach indivi-

dual chapter :

Chap . l trie s to sketch the historical deve lopment ,

starting with Riemann and Dirichlet ' s principle , which i n the

theory of partial di fferential equations leads to the introduc-

tion of Sobolev and finally potential and Besov spaces . We

al so give a heuristic argument for the particular de finition

of Besov space s (based on a Tauberian condition ) which we are

going to employ .

Chap . 2 give s a rapid survey of relevant portions of the

theory of interpolation space s . We intend to do a lot o f

interpolations !

Chap . 3 i s where we really s tart . After brie fly reviewing

tempered distributions and the Fourier trans form we give the

precise de finition of our space s , indicate al so various variants

and generalizations , and develop their basic properties .

The following six chapte rs 4 - 9 are devoted to various more

special deve lopments, and app l i cat ions.

In Chap . 4 we make a more de tailed comparison o f Be sov

and potential space s . Thi s give s us an opportunity to in tro

duce some Calde ron- Zygmund and Paley - Littlewood theory .

In Chap . 5 we prove some more deep re sul ts on interpo lation

of Be sov and potent ial space s .

In Chap . 6 (which in some sense i s preparatory for Chap.

7 ) we study the Fourier trans form .

In Chap . 7 we s tudy mul tipliers , both Fourie r multip liers

and o rdinary one s , in Be sov sp ace s and al so in potential

space s or wh at i s the s ame, at le a s t when Fourie r mul tip l ie r s

are conce rne d, Lp . In particul ar we shal l give a brie f di s

cu s s ion of the famous mul t ip l ier prob lem for the ball , a l

though t h i s re ally has n o t much t o d o with the main topi c o f

the se lec ture s, the s tudy o f Be sov space s .

In Chap . 8 we give a more gene ral (but s ti l l equivalent ! )

de fini tion o f Be sov sp ace s than the one used in the previous

discussion , i . e . the one forced upon the reade r by me diat ion

of the he uri stic arguments pre sented in Chap . l . In many

problems thi s give s a much gre ater de gree o f flexib i l ity. We

pre sent the sub j ect in such a way that we a l so ge t con tact with

some que s tions in approximation theory connected with the notion

of s aturation .

In Chap . 9 we s tudy Be sov space s from the point o f view

of topo logical ve ctor space s . Except in some e xception a l

l imiting cases , it i s possible to show that they indeed are

isomorphic to some rather simple matrix space s .

The remaining chapters are devoted to various general

izations of the previous theory .

In Chap . 1 0 we brie fly indicate a certain abstract

generalization of Besov space s .

In Chap . 1 1 we consider the generalization to the case

0 < p < 1 . I t turns out that this i s rel ated to the Fe fferman

Stein-Weiss theory o f Hardy spaces .

Finally in Chap . 1 2 there are indicated various auxiliary

more or less natural looking generalizations of Besov space s .

In the Appendix I have put some additional material which

either did not fit into the main development or I simply forgot

to include at the first writing.

As for style , the dis cussion in the e arlier chapters is

rather complete , with most detai l s written out . In the later

chapters many proofs have been le ft out so the reader probably

has to do a lot of work himsel f .

Quotation : The beginner • • • should not be discouraged if • • •

he finds that he does not have the prerequisites for reading the prerequisites .

P. Halmos (previously quoted by M. Reed and B. S imon)

Chapter l . General background .

This Chapter i s expository and there fore no proo fs will

be given in general . Our principal aim is to arrive in a

semi-heuristic way at a certain de finition of Besov space s

which is the one that our sub sequent treatment will be based

upon .

The entire subject is intimately rel ated to seve ral

other branches of analysis : partial di fferential equation s ,

calculus of variations , approximation theory , theoretical

numerical analysi s , Fourier·analysis , etc . But pre sently we

fix attention at the former two only . ( Later on however

"p . d . e . " will fade in the background • . • )

We wil l try for a while to fol low the historical road .

Let us start with Riemann and Dirichlet ' s principle . Let � n oo

be an open set in 1R with a C boundary a�. The Dirichlet

problem consists of finding a function u de fined in � satisfying

the Laplace equation

( l ) /'::, u = 0 in � and the Dirichlet boundary condition

( 2 ) u = g on a� where g is a given fun ction . Dirichlet ' s principle is now

a recipe for obtaining the solution u o f problem ( l ) - ( 2 ) :

l

2

Consider the ( Dirichlet ) integral 2 D (u ) = !� ! grad u l dx

within the class of functions u which already satisfy ( 2 ) . The solution i s the one member u o f thi s class which minimize s D ( u ) . What Riemann overlooked was the question o f existence . This became clear only after the criticism o f Weierstrass who produced a counter-example in a related s ituation . A way out of the di fficulty is to consider D (u ) as a norm in a vector space . If we al so agree to leave the realm of classical calculus and take the derivatives in a generalized ( distributional sense we get indeed a complete space , thus the norm being quadratic , a Hilbert space , denoted by Using a stand-ard result from Hilbert space theory we than get at least a generali zed longing to

( distributional , weak ) solution of ( 1 ) - ( 2 ) , bel w2 ( � ) , but we are le ft with the problem of demon-

strating that this really is a classical ( strong ) solution . This will not be considered here . Instead we look at the space w; ( � ) , and generalizations of it , per se . If we admit derivative s up to order k and take pth powers instead o f squares we

k formally obtain the space s Wp ( � ) s tudied by Sobolev in the 3 0 ' s . (Other name s that ought to be mentioned in this context are Beppo Levi , Friedrichs , Morrey , etc . ) I f p � 2 they are no longer Hilbert spaces but Banach spaces . They are then particularly useful in non-linear problems .

So much for history . Let us now write down the precise

de finition . Let l.::_p .:::_ oo k integer > 0. Then we set

which space we equip with the norm

II til _ k w- (rl) p L:

al <k

3

Here LP ( r.l) is the Lebesgue space o f measurable functions f the pth power of which jf j P i s integrable , which space again we equip with the norm

I f k = 0 o f course we simply � = � ( JRn ) .

get w0 (rl) p I f r.l = JRn we write

We stop to explain our notation . We denote the general point of JRn by x = ( x1 , ••• , xn ) and the n-dimensional volume element by dx = dx1••· dxn . Partial derivation with re spect to a multi-index a =a1 • • • av of order I a I = v is denoted by D D Cl a al· ··av

= axal ·· · always take derivatives required that

� • Needless to point out again we av in distributional sense , i . e . , it i s

f ,D rv f cp dx = ( -1 ) I a I f f D cp dx OG u. rl 0:'

4

00 fo r e ve ry " te s t f un ction " cjJ t: V (S""l ) , i . e . , cjJ i s C with compact

support conta ined in st .

We next state without proof a n umbe r of re sults about the

space s r/ (S""l ) • p

Comple tene s s theorem . r/ (ll ) i s a Banach space . p

Den s ity theorem . Ck (ll ) l'\ r/ (ll ) i s den se in Wk (ll ) . Also , p p

if ll i s bounde d , ck (TI ) C: r/ (st ) i s den se in r/ (st ) . p p k

Thus Wp ( ll ) can al so be iden tified as the ab stract com-

pletion of "nice " functions in the norm I I f I I k ( Friedrich s ) •

w- (st > Embe dding theorem ( Sobo lev 1 9 3 8 ) . We ha� an embedding

( 3 ) T : � {st ) -+ L ( st ) i f _E_ > _E_ - k k<� p P 1 P1

- p I p 0

00 More gene ral l y , st1 being any C n1-dimensional s ubman i fo ld con-

tained in st , we have an embe dding ( re s triction )

k n l n n n-n l ( 3 ' ) T : w - ( st ) -+ L ( st1 > i f -- > - - k , k < - , k > --

P P 1 pl p p p

We have al so the fo l lowin g more e lementary complement o f ( 3 ) :

( 4 ) k n n T : w- ( st ) -+ Lip ( st ) i f s 1.::_ k - -

P , k >

P- , 0 < s 1 .::_ 1 .

p sl

(Notice that the corre sponding complemen t o f ( 3 ' ) i s not o f

any intere s t . Why ? )

Exte n s ion theo rem . Eve ry f t: � ( ll ) i s the re s triction p

( to st) of some g t: � = � ( ffin ) • Moreove r ho lds

in f

ove r al l g extending f with the infimum taken .

5

Remark . Concerning the two foregoing theorems see al so

Appendix , A and B .

In other words we have the following quotient represen-

tation :

where , general ly speak ing , (�)F denote s the subspace o f �

spanned by those functions in � the support o f whi ch is con-

tained in F . (Actually one can find a mapping ( se ction o f T)

S : Wk ( Q ) -+ � such that T oS = id so that \(< ( Q ) can be identi-P p p fied with a complemented subspace of � . ) The quotient space

representation provides the pos sibility of carrying over many re

sults from the spe cial case Q = JRn to the case o f a general

Q • E . g . the Density theorem can be e stabl i shed in this fashion .

Conversely , it i s also convenient in particular in more com-

pl icated instances , to use the quotient repre sentation as a

de finition . We shall there fore in what follows mostly take

Q = JRn •

Let us however also men tion the fol lowing rather elementary

re sul t .

Invariance theorem . � ( Q) i s not changed i f we make a p 00

local C change o f coordinate s .

Thi s provides us with the pos sibility o f de fining W� (Q )

when Q i s a mani fo ld not embedded in JRn ( at least i f Q i s

compact ) .

6

We end our survey o f Sobolev space s . We now fix attention

to the problem of de fin ing space s analogous to wf when the p

intege r k i s repl ace d by any re al n umbe rs ( a kind o f Sobolev

space s of " fractional o rde r " ) . Seve ral reasons fo r why thi s

is o f importance wil l appe ar l ater on . I t turn s out howe ve r

that there i s no unique (natural ) way to achieve this . The

fo l lowing possib i l i tie s are available :

1° Potential or Liouvil le space s P� , where s re a l ,

l � p � oo . Let J ( 1 - /::, ) 1/2 where

a 2 a 2 /::, = --2 + • • • +

CJx2 i s the Lap lacian . Such a " symboli c "

axl n ope rator we alway s de fine using the Fo urie r tran s form . I . e .

denoting the Fourie r tran s form by A we require that

Jf ( O

In the s ame way fractional powe rs o f J are de fine d by the

formula

where s • de note s the space o f tempe re d di stribution s . ( Some -

time s we also nee d the " homogeneous " ope rato r I = ,;-::::-;;:-. The

powe rs I s are the gene ra lized potential o f M. Rie s z . I f

s = - 2 we get the Newton potenti al . s

The operato r s J are

occa s iona l ly cal led the Be s se l potential s . ) We then de fine

7

which space is equipped with the norm

It i s possible to show that ( use Mikhlin ' s multiplier theorem)

wk if s = k p intege r..:_ 0 , 1 < p < oo

so at least for 1 < p < oo Ps i s a true generalization of Wpk

} p 2° Besov or Lipschitz space Bsq where s real , p

1 2P1 q � oo . Be fore giving the de finition let us right away

remark that

so Bsq is a true generalization of � only i f p p p Let us also notice that

= Lip s

On the other hand

i f O < s < l .

Bl oo i s the Zygmund class o f smooth functions . 00 In general we only have

q 2 .

( 5 ) s oo + B p

8

The problem o f demons trating the eq uivalence i s in

general a non-trivial one . The si tuation is comp l i cated by

the fact that there are in the l iterature a mul titude o f

diffe rent but equivalent de finition s . Mo st o f the de fin i t ions

are goo d only in ce rtain interva l s of s . Le t u s try to make

a survey :

a . 0 < s < 1 . We set

Bsq = { f I fEL & ( J p p :rn.n

q

with 6 hf ( x ) = f ( x+h ) - f ( x ) . I f q = oo the interpre tation of

the de f in ing expre s s ion i s

sup

The no r.m i s given by

Below ( b . - j . ) we do not write down the e xpre s s ion fo r the norm

because it can be formed in exactly the s ame manner .

b . 1 < s < 2 . We set

in b .

Bpsq = { f j f�:: W1 & D.f�::Bs-l ,q (j=l, • • • ,n) } p J p

9

c . 0 < s 1- integer . Extension o f the procedure initiated

d . 0 < s < 2 . We can now set

q )

( k=integral part of s )

dh ) 1/q < 00 } I hj n

with �� f (x ) = f ( x+2h ) - 2 f (x+h ) + f ( x ) .

e . Procedure analogous to the one in b . and c . We use

kth order differences

k 2.:

\) =0 ( - l )k ( k ) f ( x+ v h) . \)

It is plain that the de finition indicated under the

headings a . -c . al l are somewhat related . Now we indicated a

somewhat different approach first deve loped systematical ly in

the thesis of Taibleson but which has its roots in the works

of Hardy-Littlewood in the 3 0 ' s .

f . 0 < s < l . Let u = u (x , t ) be the ( tempered) solution

of the boundary problem

a 2 u ;:: - b. u i f t > 0 a t2

u ;:: f i f t 0 ,

in othe r words the Po i s son inte gral o f f :

u ( x , t )

Then we have

g . o < s < 2 . Now

Bsq=

{ f \ fE L & ( f; p p

holds

l \t2

{

t

a 2 u -2

1 \L Cl t p

ts

q

q

f ( y ) dy .

dt ) 1/q < 00 } T

dt ) 1/q < 00 } T

1 0

h . 0 < s . Exten s ion o f the procedure begun in f . and g .

For all the se case s we have a t least s > 0 . Howeve r it

is e a sy to modi fy the above appro ach so as to cove r the case

of negative s ( and s = 0 ) too .

i . s < l . Con s ide r in pl ace o f u the solution v=v ( x , t )

o f the boundary p rob lem

Then holds

if t > 0

v = f if t = 0

{ f l f E: L I & ( -6 00 I I t �� I I .L p s t

j . s real . Analogous .

11

q dt ) 1/q < 00 } t

We are now faced with the problem o f see ing what is common

in all these case s . First let us consider a smal l variant o f

a . , the cases b . -e . being analogous :

a ' . O < s < l . One can show that

I l lite . fj I L Bsq=

00 { f I fs L & (!0 J

p p ts

q p ) dt 1/q .

t) < oo ( J=l, • • • ,n) }

where e j = ( O , • • • , l , . • • , O ) i s the j th bas i s vector of En .

I f we compare a ' . with f . say , we see that the integral s

are built up in same fashion . We have thus to confront the

integrands only , i . e . the expres sions lit f and t� e . a t

J respectively . It i s now readily seen that they both are the

e ffect of a translation invariant operator depending on t acting

on f , i . e . of the form ¢ t* f where ¢ t are " test functions "

depending on t . The dependence on t i s now particularly simple :

X <P ( t

or , expressed in terms of Fourier trans forms ,

¢ ( t� ) 1

where <jJ is a given test function . Indeed we find

and

/"-!-, f < O te . J

i tt: ' A

(e J - l ) f ( � )

t l � l e -t l � l f { � )

1 2

re spectively . We are thus lead to try the fol lowing de finition

( 6) Bsq= { f I l: { fooo ( t -s I I <Pt * f I I L ) q �t ) 1/q < oo } P finitely p

many <jJ

under suitable re strictions on <P and s . What are the restrictior to be imposed on <P and s? Let us here devise every crude necessary condition for <P • In view of ( 5 ) we have in any case the requirement Thus we must have

1 3

Restricting attention to the case p q and using Plancherel ' s

formula we get

Replace now � by t-l�

and let t -+ 0:

i � . which thus i s a nece ssary condition . I f = e J _ l ( 7)

implie s 0 < s < 1 and i f � ( � ) = I � J e- J � J , s..2_1 . Thus ( 7)

helps us to explain partly the re striction imposed on s in

these case s . Of course we cannot expect to get the complete

answer with such crude weapons . Next we observe that ( 7) , on A

the other hand , certainly is ful filled i f � vanishes in a

neighborhood of 0 and oo . Moreover we fix attention to the

case when we can do with j ust one � in (6) - obviously �t ( � )

cannot vanish for all t at some point � . We are thus lead

to impose the fol lowing condition o f Tauberian character

( analogous to Wiener ' s ) :

( 8 ) { t � I t > o } n { ¢ "� o} 'I p' for each � 'I o

14

In fact it will be enough to work with a s tronge r form o f it

( 8 I ) s upp ¢ { b - l 1 1

where we o ften for convenience choo se to work " in b ase 2 " 1

tak ing thus b = 2 . To te l l the whole truth we have al so to

add a term o f the type I 1 � * f l I L where � satis fie s p

{; t= o } ={ I t;. l < 1 } •

We have a l so ove rlooke d the re gularity conditions to be im-

posed on ¢ ( and � ) . But all th is wi l l be made more pre c i se in

due course ( Chap . 3 ) .

In concl usion we insert here two simple il lustrative

examples where the e s sence o f the technique based on the

Taube rian condition ( 8 ' ) wil l be apparent .

Let us howeve r first po int out that the re are a l s o o the r

more con structive de finit ion s of Be sov space s .

a . Approxima tion theory ( s > 0 ) . Let us con s ide r the

be s t approximation of f in L by e xponential function of type p

< r :

( 9 ) E ( t 1 f ) = inf i I f-g l l L where s upp g c { I t;. l < r } p

Then holds

Bsq= { f! :fE: L & (!000 ( rsE (r, f) ) q

p p dr ) 1/q < oo

• r

1 5

I f n = 1 and p = q = oo thi s contains the non-periodic analogue

o f the classical results o f Bern stein and Jackson for approxi-

mation by trigonometric polynomial s .

b . Interpolation ( s real ) . For real interpolation holds

Bsq with s p

e . g . by de finition ( c f . Chap . 2 )

00 { f I uo

with

(10 ) K(t , f ) = K ( t , f ;

dt ) 1/q < 00 } T

Notice the formal analog between ( 9 ) and ( 1 0 ) . Using complex

interpolation we get inste ad

s s [P 0 P 1) with s p I p 8

Now to the examples that were promised.

Example 1 . Weierstrass non-differentiable function .

Weierstrass showed in 1 872 that the function

00 ( 11 ) f ( x )

\) =1 \) \) a cos (b x ) where a < l

was not di fferentiable at any point provided

16

It i s needless to point out here the profound influence that

this counter-example has exercised in the devel opment of

analysis as a whole . In 1916 Hardy e xamined Weierstras s '

function and he demonstrated the s ame re sult under the weaker

assumption ab �1 . We shall now give a simple proo f of Hardy ' s

resul t . In place o f ( 1 1 ) we consider the more general function

( 1 2 ) f (x ) = 00 ib \!X I cv e

v=O

where { cv } i s any sequence with I I cv I < oo •

I claim that the following holds true .

Proposition . Let f be given and assume that for some

s > 0

( 1 3 ) f (x) = 0( I x I s ) , I x l > 0

soo Then holds -v s c = O (b ) and \) fE: B • 00 An analogous statement

holds with 0 replaced by o .

Proof . Let us take Fourier transforms in ( 12 ) : We get

1 7

where 8 i s the delta function . Using ( 8 ' ) it now fol lows

¢ (1) Cv 8 (� -b\!} where one takes t -\! b

With no los s of general ity we may assume that ¢ ( 1} = 1 .

There fore taking the inverse Fourier transform we end up with

\) ib X

Cv e

In particular holds thus

On the other hand , s ince

Jcp t (-y ) f (y ) dy

( Note that n = 1! ) -we obtain using ( 13)

1 !</>(- z) f (y ) dy t t

The proof o f fEBsoo is similar. 00 Having established the proposition it is easy to prove

the non-differentiabil ity o f the Weierstrass function . Take

thus c \) = av with

1 8

a < 1 and ab � 1 and assume f i s dif fe ren-tiable at some point x0 . With no loss of generality we may assume that x0 = 0 (by translation , if nece ssary ) and that f ( O ) = f ' ( O ) = 0 ( by subtracting a finite number o f terms , if necessary ) . Thus ( 1 3 ) holds with s = 1 and o in place of 0 .

We conclude that av = 0 (b- v) . But this clearly implie s ab < 1 , thus contradicting our hypothesis .

Example 2 . Riemann ' s first theorem on trigonometric serie s . In his famous memoir on trigonometric serie s from 1 8 5 9 Riemann considered functions or , better , distributions of the form

00 f ( x ) n=-oo

with em = 0 ( 1 ) as j m j �oo and ( for convenience ) c0 = 0 . In order to study the summability of the serie s he considered the ( formal ) second integral

00 F (x ) l: m=-oo

(Notice that -F ' = f ( in distributional sense , o f course ! ) . ) The " first theorem" re ferred to above now simply says in our language that F sB:oo (which is the same as the Zygmund class ) . We leave the particulars of the verification to the reader .

1 9

Notes

For a modern treatment of the variational approach to

Dirichlet ' s problem see Lions [ l ] or Lions-Magenes [ 2 ] . In

partial diffe rential equations the space w; {Q ) is al so often

denoted H1 {Q ) . One o f the classical papers by Sobolev i s

[ 3 ] . See also his book [ 4 ] . The first systematic treatment

of Bsq (Q ) of s > 0 with de fin itions of the type a . -e . using p finite diffe rence s is Be sov [ 5 ] . The spaces Bsq (Q ) , s f p intege r are o ften denoted by w; {Q ) , known as Slobode cki j space s.

s oo s The spaces Bp (Q ) are o ften denoted by Hp {Q ) , known as Niko lsk

space s . For other works o f the Soviet (= Nikolski j ) School

( Nikol ski j , S lobodecki j , I lin , Kudrj avcev, Lizorkin , Besov,

Burenkov , etc . ) see the book by Niko l ski j [ 6 ] and also the

survey articles [ 7 ] and [ 8 ] . Somewhat outdated but still read-

able are further the survey articles by Magenes-Stampachia [ 9 ]

and Magenes [ 1 0 ] where also the applications to partial dif fer-

ential equations are given . In the case p = q = 2 see Peetre

[ ll ] , Hormander [ 12 ] , Voleviv-Panej ah [ 1 3 ] . The t �eatment of

nLipschitz spaces " in Stein [ 14 ] , Chap . 5 i s based on Taibleson ' s

approach [ 1 5 ] . All of the relevant works of Hardy and Littlewood

can be found in vol . 3 of Hardy ' s collected works [ 1 6 ] . In this

context see also the relevant portions of Zygmund ' s treatise

[ 1 7 ] . These authors are concerned with the periodic 1-dimensional

case ( T1 rather than llin ) . The first systematic treatment of Beso '

space s using the definition with general ¢ was given in [ 18 ]

2 0

( c f . also [ 19 ] ) . But the special case p = q = 2 appears

alre ady in Hormander ' s book [ 1 2 ] where also the Tauberian

condition i s stated ( see notably op . cit . p . 4 6 ) . The l atter

was later , apparently independently , rediscove red by H. S .

Shapiro who made applications of it to approximation theory

( see his lecture notes [ 2 0 ] , [ 2 1 ] ) . The constructive charac

terization via approximation theory is utilized in Nikolskij ' s

book [ 7 ] . ( C f . also forthcoming book by Triebel [ 2 9 ] ) .

Concerning classical approximation theory see moreover e . g .

Akhie ser [ 2 3 ] o r Timan [ 1 4 ] . The characte rization via

interpolation originate s from Lions ( see e . g . Lions-Peetre

[ 2 5 ] ) . The tre atment o f the Weierstrass non-di ffe rentiable

function given here goe s back to a paper by Freud [ 2 6 ] ( see

also Kahane [ 2 7 ] ) . Riemann ' s theory of trigonometric series

can be found in Zygmund [ 1 7 ] , chap . 9 .

Quotation : Les auteurs ont et e soutenus par Inte rpol .

J. L . Lions and J. Peetre

Chapter 2 . Pre l i minarie s on int er po l ati on spac es .

Thi s chapter i s e ssentially a digression . We want to

give a rapid survey o f those portions of the theory of inte r

polation space s which will be used in the sequel .

First we review howeve r some notions connected with

topological vector spaces .

The most important class of topological vector space s are

the locally convex space s . In a locally convex space E there

exists a base o f neighborhoods o f 0 consisting o f symmetric ,

balanced , convex sets I , i . e . aUCU i f J a/ .2_1 . and

( 1) ( 1- T) U + T UC U i f 0 < T .2_ 1 .

A subclass of the locally convex space s are the normed

spaces . In a normed space E the topology comes from a norm ,

i . e . a realvalued functional l l x/ I de fined on E such that

( 2) // x + Y 11.2. I I x I I + I IY I I ( triangle inequality)

/ I ex I I = l c I 1 /x / I ( homogeneous )

// x / I > 0 i f x 1- 0 , //OJ / = 0 ( positive de finite )

A complete normed space i s t .ermed a Banach space .

In the type o f analysis we are heading for , howeve r , a

somewhat larger class of topological spaces i s needed , namely

2 1

2 2

the locally quasi- convex one s . Thi s me an s that we repl ace

( 1 ) by

( 1 I ) ( 1-T ) U + T UC A U i f 0 < ' 2 1

where A i s a cons tant 2_ 1 whi ch may depend on U . I n the s ame

way we arrive at the concept o f quasi-norme d space and quas i -

norm i f we rep lace ( 2 ) b y

( 2 1 ) I I x + Y l l < A ( I I x l l + I I Y l l ) ( quasi -triangle inequal ity)

Note that ( 2 1 ) ce rtainly ho lds true if

1 ( 2 " ) l l x + Y l l < < l l x l l p + I I Y I I P ) P (p-triangle inequa lity)

! -1 where A and p are re l ated by A = 2P (0 < p 2 1 ) . A complete

quasi-norme d space we call a quasi-Banach space . The quasi-

normed space s can a l so be characte r i ze d be ing local ly bounded .

The dual o f a topo logical ve cto r space E i s denote d by E 1 •

I t always carries a local ly convex topo logy which i s compatible

wi th the dua li ty , for instance the we ak topo logy or the s trong .

In particular i f E is a quasi-Banach space then E ' i s a Banach ! space in the strong topo logy .

We pause to give some examples of quasi -Banach space s .

Example l ( Lebe sgue space s ) . Let � be any me asure space

equipped with a me asure w . I f 0 < p < oo we de fine L =L W ) - p p

to be the

(with the

space i f

Also note

space o f J.l -measurable fun ctions

1 I I f l l L p

= ( !Q I f ( x ) l p d J.l) p

usual interpretation i f p = 00 ) .

l � p � oo but

that , as is

only a quasi-Banach

wel l-known , L ' :::L p p '

2 3

such that

Thi s is a Banach

space i f 0 < p < 1 .

where .!. + 1 1 p' = p

( conj ugate exponent ) , in the former case ( excluding p = 00) I

while , by a theorem by Day , L ' = 0 in the latter case {unless p ]J has atoms ) . Thus , the Hahn-Banach theorem being violated,

we see that quasi-normed space s may behave quite differently from Banach space s .

Example 2 ( Lorentz spaces ) . Let Q and JJ be as before .

I f 0 < p , q � oo we de fine L = L (Q ) to be the space o f pq pq ]J-measurable functions such that

1 1 I I fll L pq

(f; ( tp f* (t) ) q �t )q

Here f * denote s the decreasing rearrangement o f I fl Notice

the formal analogy with the de finition of Besov-space s . We

see that L = L • The space L is also known as weak pp p poo

Lebesgue or Marcinkiewicz space and i s sometimes denoted by

Lp * (or Mp ) • On 1 y i f 1 < p � oo , 1 � q � oo or p = q = 1 is Lpq a Banach space . In all other cases it i s a quasi-Banach space .

One can show that L ' :::: L , , if 1 < p < oo , l � q < oo or P = q = 1 . pq p q E xa mple 3 ( Hardy space s ) . If 0 < p __ < oo we define H =H ( D ) p p

to be the space o f func tion s holomo rphic in the un it

di sc D = { z I I z I < l} C C such that

l

s up u; lf ( re ie) l p d 8) p

0< r< l

2 4

I f l � p � oo HP

i s a Banach sp ace , o therwi se a quasi -Banach

space . By the cla s s i cal theo rem of M . Rie s z on con j ugate

functions we have H ' ::: H , i f l < p < oo • The dual o f H1 has p p

re cently been iden ti fied by Fe ffe rman-S tein . The dual o f

0 < p < 1 on the o ther hand was previou s ly dete rmined by

Duren Romberg and Shields . I t i s e s sential ly the Be s o v ( o r l

Lip s chit z ) space BE - l , oo • The theory o f Hp sp ace s has ( to

some extent ) been e xtende d to seve ra l variab le s by Stein and

We i s s . I n the non-pe riodic case , whi ch i s the one o f inte re st

to u s , they de fine the sp ace Hp ( JRn:l ) using a suitab le

gene ra l i z ation o f the Cauchy-Riemann equation s . We re turn

to the se space s late r on ( Ch ap . 11 ) .

That much for quas i -Banach space s . We are ready to

turn to inte rpo lation space s . Roughly spe aking it i s an

attempt to treat various fami l ie s o f concre te space s (potential ,

Be sov , Lebe sgue , Loren tz , Hardy , etc . ) from a common point o f

view . To be mo re precise let there be given two qua s i-Banach

space s A0 and A1 and a Hausdorff topo logical ve ctor space A

and assume that both A0 and A1 are continuously embe dde d in A •

-+ • The entity A = { A0 , A1} w1l l then be termed a quasi -Banach

couple . We sha ll now indicate seve ra l procedure s whi ch to a

25

-+ given quasi-Banach couple A associate a quasi-Banach space -+ -+ F ( A ) continuously embedded in A. The dependence o f F ( A

-+ on A wil l be o f a functorial character so we wil l say that

-+ F ( A ) i s an interpolation functor o r , by abuse of language ,

interpolation space .

1° Complex space s ( Calderon ) . Here we have to re strict

ourselve s to the Banach case only . Let o < 8 < 1 . We say

that -+ a E [ A 1 8 = [A0 ,A1J 8 i f and only i f there i s an f = f ( z ) ,

z= x + iy , such that

( a ) f ( z ) i s holomorphic and bounded i n the strip

0 < x < 1 with value s in A0 + A1 with continuous boundary values

on the boundary l ines x = 0 and x = 1 ,

( b ) f ( iy ) i s continuous and bounded with value s in A0 ,

( c ) f ( l + i y ) i s continuous and bounded with value s in A1 ,

( d ) a = f ( 8 ) . -+

We equip [ A le with the norm

I I a l l r A. 1 8

Example 1 . We have

inf max ( l l f ( iy) l l , I I f ( l+iy) I I A ) . f A o 1

1 p

1 -e + Po ( 0 < 8 < 1 ) .

This i s e ssentially just a restatement of the classical inter-

polation theorem of M. Rie s z -Thorin ( 1 9 3 9 ) . The main step

2 6

in the proof is always the construction of an "optimal" f . Here the following function wil l do :

f ( z ) = I a I

p ( 1- z ) P o

Example 2 . We have

Ps i f s p

sgn a .

This was already stated in Chap . 1 and a proof will be given in Chap . 3 .

We shall not discuss any o f the deeper propertie s of + A ] e but content ourselve s to state the following :

+ {AO , Al }

+ {BO , Bl } Interpolation theorem. Let A = and B =

+ + be two Banach couples and let T : A +B be a morphism of couples ( i . e . a linear mapping such that T : A0+s0 T : A1+B1

+ + continuously ) . Then T : [A] e+ [B ] e continuously . Moreover holds for the operator norms the convexity inequality

Here generally speaking I I T I I A , B sup I I Ta I I � I a I I a�O A + that [Al e i s an interpolation space o f exponent e . We say

This is j ust a mere restatement of the functorial character + of [A] e ( modulo the verification o f the convexity inequality ! ) .

2 7

We leave the details to the reade r .

2° Real s pace s (Lions , Gagliardo ) . Now we can consider

the general quasi-convex situation . First we introduce two

auxil iary functionals , termed the K- and the J- functional

re spectively , as fol lows : If 0 < t < oo, a E A0 + A1 ( l inear hul l ,

j oin ) we put

+ K ( t , a ;A ) K ( t , a ; A0A1 ) = inf

a=a0+a1

I f 0 < t < oo aEA0 n A1 ( intersection , meet , pullback ) we put

+ J ( t , a ;A) max ( I I a I I A , t I I a I I A ) •

0 1

They are in a sense dual to each other . Let 0 < 8 < 1 ,

0 < q ,::;, 00

or

1 )

o r

Now we can de fine

< = >

+ aE (A ) Sq

K( t , a ) ) q te

dt ) 1/q < 00 T

< = > ( Banach case only) 3 u = u ( t ) ( 0 < t < oo ) :

dt ) 1/q < oo and 2 ) t a = foo u ( t ) dt o T

2 8

\) 00 J (2 , u v)

<=> u 1 , + 2 , • • • ) : 1 ) 2:: < 00 u \) ( \)::: 0 , � v e \) =- 00 2 00

and 2 ) s = 2:: u v v=- oo

equip -+ We (A } 8q with the quasi-norm

I I a i l (A) u; (K ( t , a ) ) q dt ) 1/q 8q t8 T

::: (Banach case only) inf f; ( J (t 1 U ( t ) ) ) q dt ) 1/q u t t

\) 00 J ( 2 , a)

) q ) 1/q • ::: inf ( 2:: 2\) 8 u \) =- 00

Thus there are several de finition s : in the Banach case three , in the general quasi-Banach case only two. For re ference s we s tate this as a theorem.

Equivalence theorem. These three ( tw o) de finitions are equivalent .

That we have to exclude the middle de finition i n the Banach case is connected with the fact that there is no nice theory of integration of functions with value s in a space which is not necessarily locally convex.

Example 1 . We have

L if pq 1 p

1-8 + P o

8 pl

( 0 < 8 < 1 ) .

Notice that no condition is imposed on q0 and q1 • Thus in

particular we have

L i f p 1 p

1- 8 -- + Po

8 ( 0 < 8 < 1 ) . pl

2 9

This i s e ssentially a restatement of the clas sical inter-

polation theorem of Marcinkiewicz ( 19 39 ) . The key to those

re sults are certain explicit expre ssions for the K-functional ,

the s implest of which is the following :

K ( t , a ; Ll , � )

Example 2 . Recently Fe ffe rman , Riviere and Sagher have

extended the above results to the case of Hp space s , and even

their Lorentz analogues H pq In particular

H pq i f 1 p

8 ( 0 < 8 < 1 ) pl

holds .

Example 3 . It was mentioned in Chap . 1 that

Bsq 1. f ( 1 8 ) 8 p s = - s o+ sl ( 0 < 8 < 1 )

I t i s also possible t o show that

Bsq i f s p ( 0 < 8 < 1 ) .

30

The proofs wil l be found in Chap . 3 . Example 4 . This example i s essentially a simpler special

case of the pre ce ding one . Let c0 be the space o f continuous bounded functions in lR = ( --oo , oo ) , with the norm

I I a l l o c sup I a (x ) I

and let 1 C be the space of functions whose first derivative exists and belongs to c0 , with

I I a l l 1 c sup I a ' (x ) I

( Here we are cheating a little bit , since this become s a true norm only after identi fication of functions which differ by a constant. ) Then

0 1 ( C 1 c ) eoo ( o < e < 1 )

holds , where Lip6 is the space of functions satis fying a Lipschitz (Holder) condition of exponent e , with the norm

sup l a ( x ) - a (y) I I x - Y l 6

This follows at once from the following expression for the K- functional :

31

K ( t , a ; c0c1 ) :::: w ( t , a) = sup I a ( x+h ) -a (x ) l (modulus of continuity) I h l 2t

Because it is so s imple and because the argument i s typical for

seve ral of the fol lowing proofs we will for the reader's

bene fit display a detailed

Proof : Easy side . We have the fo llowing two obvious

e stimate s

w ( t , a ) < 2 sup I a ( x ) I = 2 I I a I I 0 c

w ( t , a) 2 t sup I a ' ( x ) I < 2 t I I a I I 1 c

I f a = a0 + a1 then follows

Taking the inf we thus get

w ( t , a ) 2 2 K ( t , a) .

Hard (er) s ide . We must find a decomposition a = a0+a1

which is , i f not optimal , at least "approximately" optimal .

We choose

a0 (x) • l(X) ... .!_ Jt a (x+y ) dy , t 0

We then have

� f: ( a (x ) - a (x + y) ) dy

a ' (x ) 1 a ( x + t ) - a (x )

t

which clearly yields

I I a0 I I 0 < w < t , a > c

I I I I < W( t , a) al 1 t c

Thus we end up with

K ( t , a ) < I I a0 I I 0 + t I I a1 I I 1 .2. 2 -

c c

The proo f i s complete .

w ( t , a) .

Remark . The above can be generalized to the more general situation

A0 = E = any Banach space ,

3 2

fol lowing

A1 = D ( A) = the domain of the infinitesimal generator

3 3

of a semigroup o f uniformly bounded operator s G (t ) i n E;

i . e . we have

G ( t+a ) = G ( t ) G ( s ) , G ( t ) -rid s trongly as t-rO ,

! I G ( t ) I I < C , Aa = l im G ( t ) a-a i f a € D (A ) . t+O t

Now we can prove

( 3 ) K ( t , a) ::: sup i! G ( s ) a-a l l . 0 < s �t

If we also impose the fol lowing additional requirement

t i l G ( t ) Aa l l � C ,

which in particular implie s that G ( t ) i s a holomorphic semi-

group , we can also prove that

K ( t ,a ) ::: sup s I IG ( s ) A a I I 0 < s � t

The details are le ft fo� the reader . Thi s perhaps helps the

reader to understand why the various de finitions of Besov

space s in Chap . 1 , under the headings a . - j . , are equivalent .

In Chap . 8 we shall howeve r , give a different ( equivalence )

3 4

proof. We no·w list some auxiliary properti.t! s of the spaces

+ (Atq • First we have an Interpolation theorem. Analogous to the interpolation

theorem in the complex case . We al so compare the real and the complex space s . Comparison theorem. (Banach case only) • We have

+ + + CA) e 1 C [Al e c (A) eoo C O < e < 1 ) .

A particular instance of it is the relation ( see Chap . 1 ) :

+ Proof (out line ) : 1 ) Let a E (A) 8 1 • Then a admits !00 dt the representation a = 0 u ( t ) T . with a suitable u . We

obtain a representation a = f ( e ) with a holomorphic f simply by taking

+ whi ch is a kind of Mel lin transform. It follows that a E [Al e •

+ 2 ) Let a E [Al e • Then a admits a representation

a = f ( e ) with f holomorphic . Obviously we have

K ( t , f ( iy) ) < l lf ( iy) I I A ,:s c 0

K ( t , f ( l+iy) ) � t iif ( l+iy) IIA � C t 1

3 5

Thus the three l ine theorem, usually named after Doetsch ,

but real ly due to Lindelo f , I have been told , yields

K ( t , a) 8 K ( t , f ( 8 ) ) � c t

-+ and we have aE (A) 800 The proof i s complete .

We also mention anothe r

Comparison theorem. We have

-+ -+ ( ) C (A) i f ql =< q2 • A 8q 8q 1 2

In particular we see that

The mos t important re sult of real interpolation is how-

ever the following :

Reiteration theorem. Let -+ X { X0 , x1 } be any quasi-

Banach couple such that

-+ -+ (A) e . q .cxi C (A) e . oo ( i=O , l)

l l l

for some q0 and q1 > 0 but

that

Then it follows

3 6

+ (A) e q i f e = (l- n) e0 + n e1 ( o < n < 1 ) .

This explains why in examples 1-3 we need not impose any conditions on q 0 and q1 .

Final ly we mention the fol lowing powerful Duality theorem. ( Banach case only) • Assume that

A0 (') A1 is den se in both A0 and A1 • Then

hold s . This contains in particular the result concerning the

dual o f Lorentz space . We also get

( B sq ) I � B-s , q I i f 1 � p < oo 1 � q < oo p p i

Indeed we use the duality theorem along with the fact that

s -s ( P ) 1 :::: p 1 i f 1 �p < 00 p p

( One can also determine the dual when 0 < q < 1 . In the Be sov case one finds :

( B sq ) I � B-s , oo i f 1 � p < oo , 0 < q < 1 • ) p � p i

3 7

We conclude this chapter b y giving some applications

which are intended to display the power of the technique o f

inte rpolation space s . -a

We consider the potentials I where we res trict our-

selve s to the case 0 < a < n . In Chap . 1 they were de fined

using the Fourier tran s form. Howeve r , it is al so possible

to de fine them as certain convolutions . Indeed

-a I f ( x ) = c I I a-n x-y f (y ) dy

where c i s a constant expressible in r factors . The

following holds true .

Thea rem 1 . We have

It implies the following

-a I L +L p q i f 1 = q 1 p

a n n' 1 < p< a ·

Coro llary . We have � +L i f .!_ = .!_ - � 1 < p < !!,k P q q P n '

Indeed this i s preci sely the Sobolev Embedding theorem

( Chap . 1 ) , except that we cannot capture the case p = 1 in

this way .

Proo f o f corollary : For simplicity let us take k 1 .

We start with the following Fourie r transform identity

f n l:

j=l

Taking the inve rse Fourier transform we get

f n l: a . * D . f

j =l J J

3 8

where -i � j 1 1 � 1 2 • The important point i s that one has C l x l 1 -n h f < T ere ore one gets

l f l < C I-l l grad f l = c l x l l-n * l grad f l

1 Let now f e: Wp • Then j grad f I e: L by de finition and p f e:Lq by th . 1 where � 1 1 = p - n . The proof i s complete .

Proof o f th . 1 : The proof goes via O ' Neil ' s inequality stated be low .

a-n lx I E:L poo i f 1 q

1 l - + - -p p l

Indeed it (n- a) p= n . o r , a fter

There remains

is readily There fore

e limination

checked that lx I a-n

o f P ,

* f EL q if 1

q

i f 1 p

O ' Neil ' s inequality . a e:L poo' f E:Lp => a * f ELq i f 1 1 l = - + - -1 1 a* fE:L i f p p p .!:.. + .!:.. - 1 l a * f where a e:L poo Then we have

T : L1 -+ L poo (by Minkowsky ' s inequality ) ,

T : L p' l -+L oo (by the fact that L ' � L ) • p 1 poo By interpolation we then get

But

T : ( L1 , L , l ) -+ ( L , L ) 8 r • p 8 r poo 00

(Ll , Lp ' l ) 8 r

( L ' poo L ) 8 oo r

L

= L

i f 1 -pr p

if 1 qr q

Elimination o f 8 give s pre cisely 1

Thus we have

T : L -+ L pr qr

q

3 9

1 -8 8 -1- + PT ,

1- 8 + 8 -p 00

1 + 1 l . -p p

Finally we take p r and notice that Lpp = Lp ' L CL which qp q yields

We are through .

Next we conside r , takin g n = 1 , the Hilbert transform

Hf (x ) = 1 * f ( x ) X = p . v . � f (�-y) dy = l im s -+o

f f (x-y) dy IY 12:E: y

where the integral thus i s a principal value (p . v. ) in the

sense o f Cauchy. Notice that

4 0

"'

i sgn t;, f ( t;,)

In the case o f T1 ( the periodic case ) this is the operation

which to a function , given by the boundary values of a harmonic

function in the unit disc D , assigns the conj ugate fun ction .

The fo llowing classical re sul t holds true .

Theorem 2 . We have H : Lip -+ Lip , s s O < s < l .

Later on we shall prove much more general resul ts { for

arbitrary r , general Be sov space s and general convolution

operators ) . 00

Proof of th. 2 : We expre s s H as a sum H = L: Hv where V =-oo

H f { x ) \) J f {x-y ) d y , IV y

( This amounts to about the s ame as taking principal value s ! )

We write now

Hvf (x ) =

DHvf (x )

+

!I \)

= -

f ( x-y) dy !I y \)

!I f (x-y) dy -2 \) y

f (x+2 v ) = 2

This give s the e stimate s

f ( x-y) -f (x ) dy y

f ( x-2v +l ) + \) f (x-2 )

2 v+l

Df ( x-y) dy . y

2\) +

or in terms o f the J-functional

or c 2v

l i t I I 1 c

or c l it I I 1 c

J ( 2 v

1 Hv f ) < c I I f I I 0 c or c 2

vI I t I I 1

c

4 1

I f we apply this to an arbitrary decomposition f = £0 + £1 we can al so write this in terms of the K-functional :

\) \) J ( 2 1 Hv f ) < C K ( 2 , f )

0 1 \) v s I f f ELips , because Lips = (C , c ) e s ' we have K ( 2 , f ) � c 2

and thus J ( 2v , Hvf ) � C 2v s . Now recall that Hf = IHv f .

Using again Lips = (C0 , c1 ) 8s and the second of the e quivalent

de finitions using the J-functional , we thus conclude

The proo f i s complete .

4 2

Notes

Concerning topological vector space s see the book by

Kothe [ 2 8 ] which also contains a brief treatment of locally

bounded space s . Concerning the dual o f L in the quasipq Banach case see Haaker [ 2 9 ] where a short proof of Day ' s

theorem also is indicated. See al so Cwikel-Sagher [ 3 0 ] ,

Cwikel [ 31 ] . For Lorentz space in general there i s the ex-

cel lent survey article by Hunt [ 3 2 ] . The classical theory of

Hp spaces can be found in Duren ' s book [ 3 3 ] . The dual of

Hp ( D ) i f 0 < p < 1 was determined b y Duren-Romberg-Shields n+l [ 34 ] . Their re sult was extende d to the case o f Hp ( lR + ) by

Wal sh [ 35 ] . The dual o f H1 ( lR �+l ) was determined by Fefferman

and Stein in their fundamental work [ 36 ] . ( The case o f H1 ( D)

i s of course implicitly contained there in . ) For an introduction

to H spaces of seve ral variables see Ste in-Weiss [ 3 7 ] , Chap . 3 p or Stein [ 14 ] , Chap . 7 .

For a more detailed treatment o f interpolation space s

we re fer to Chap . 3 of the book by Butzer-Berens [ 3 8 ] .

Seve ral other books deal ing with interpolation spaces are now

in preparation , by Bergh-Lofstrom [ 3 9 ] , by Krein-Petunin-Semenov

[ 4 0 ] , by Triebel [ 2 2 ] etc . Then we shall content ourselves with

j ust a sketch of the historical development of the theory . F irst -

of a l l , a discussion of the classical interpolation theorems of

(M . Riesz - ) Thorin and Marcinkiewicz can be found in Chap . 12

of Zygmund ' s treati se [ 1 7 ] . The abstract theory of interpolation

spaces was created around 1 9 6 0 by Lions , Gagliardo , Calderon ,

Krein and others . The complex spaces are studied in

4 3

Calde ron [ 4 1 ] . The real space s are studied in Lions-Peetre

[ 2 5 ] and in the pre sent form - with explicit mention of

K ( t ,a ) and J ( t ,a ) - in Peetre [ 4 2 ] . The extension to the

quasi-Banach case come s later . See Kree [ 4 3 ] , Holmstedt [ 4 4 ] ,

Sagher [ 4 5 ] , Peetre -Sparr [ 4 6 ] . In the latte r work it is not

even assumed that the space s are vector space s , i . e . the

additive structure alone enters . Concerning integration in

quasi-Banach spaces, see Peetre [ 4 7 ] and the works quoted

there . Concerning interpolation of H spaces see Fe ffermanp Stein [ 36 ] , Riviere-Sagher [ 4 8 ] , Fe fferman-Riviere-Sagher [ 4 9 ] .

Compare with the classical treatment in [ 1 7 ] , Chap . 1 2 . More

pre cise results conce rning the comparison o f the complex and

the real space s can be found in Peetre [ S O ] , [ 5 1 ] . In the

latter paper there is also mentioned a third type o f inter-

polation method which somehow lie s in between the real and the

complex. Concerning the dual o f Bsq when 0 < q < 1 (or q =00 ) p see Peetre [ 5 2 ] . See also Flett [ 5 3 ] . The present treatment

of O ' Ne i l ' s inequality [ 5 4 ] can be found in Peetre [ 5 5 ] . See

al so Peetre [ 5 6 ] where the same type of technique is applied

to general integral operators which need not be translation

invariant. For Young ' s inequality (via Thorin ' s theorem) see

[ 1 7 ] , Chap . 1 2 . The result on Rie sz potential s ( th . 1 ) i s

due to Sobolev [ 3 ] but was later independently redis covered

by Thorin [ 5 7 ] . In the case of T1 i t stems from Hardy and

Littlewood ( see [ 1 7 ] , Chap . 12 and Hardy [ 1 6 ] ) . The treatment

of the Hilbert trans form is likewise taken from [ 5 5 ] .

4 4

Quotation : The sphere i s the mos t uniform o f solid bodie s • • •

Origen, one o f the Fathers of the Church , taught that the blessed would come back to l i fe in the form o f sphere s and would enter rolling into heaven •

J . L . Borge s "The Book o f Imaginary Beings"

Chapter 3 . De finition and basic propertie s o f Be sov space s .

Now we are ready to embark on a more systematic study of Besov spaces .

First we collect for re ference some basic facts concerning tempered distributions and Fourier transforms which we have already freely made use o f in the preceding. Let S be the space o f rapidly for all multi-indice s

decreasing s a, S , x D a

functions , f ( x ) = 0 ( 1 )

I f we equip i t with the family of semi-norms

sup X

i . e . f E S as x -+ oo

<=>

I I '

S becomes a Frechet space . Obviously S i s stable for derivation and multiplication with coordinates : for all a , S f E S =>x13o f E S and these are , moreove r , continuous a operations . The dual space S ' = S ' ( JRn) is called the space o f tempered distributions . By abuse o f notation the duality between s • and S i s generally written as an integral :

< f , g> f f ( X ) g ( X ) dX i f f E S 1 1 g E S •

JRn

e . g . i f 8 i s the "de lta function " then we have

4 5

<o , g > = g ( O ) = JlRo ( x ) g (x ) dx i f g E: s

By duality D and v. extend to S ' .

In dealing with the Fourier trans form it is often convenient

to have in mind two spaces lRn , one " latin " space lRn = lRn with X the general element x = (x1 , . . . , xn ) and the dual " greek " space

R� with the general element � = ( � 1 , • • • , � n ) , the duality

+ xn E;, n . Thi s is also natural

from the point of view of physics where x often is " time " ( sec ) - 1 and t;, " frequency" ( sec ) so that x E;, i s " dimensionles s " . I f

f E: s = S its Fourier trans form is an e lement X given by

A f ( t;, Ff ( t;, ) -ixE;, J e f ( x ) dx .

::IRn X

A

That f E: St;, can be seen from the basic formulas

( 1 )

( 2 )

( D f ) a

(x sf )

More generally

( 1 ' ) F (a * f )

( 2 ' ) F (bf )

( i t;, ) f . a

( - iD ) S Ff . E;,

Fa F f ,

1 Fb * F f ,

A

f = Ff of

4 6

under suitable assumption on a and b . We wil l also need the formula

"'

( 3 ) f ( t t;, ) where

The inverse Fourier trans form is given by

This Fourier inversion formula has as a simple consequence Plancherel ' s formula ( for S ) :

J n I f ( x ) I 2 dx JR. X

1

Since F and F-l are continuous operations -1 F : Sx + St;, , F : St;, + Sx , they extend the duality to tempered

distributions S ' . Formulas ( 1 ) and ( 2 ) (or ( 1 ' ) and ( 2 ' ) re -main valid for tempered distributions .

Remark . Using instead duali ty and Plancherel ' s formula , F has an extension to an (essentially) isometric mapping F : L2 -+ L2 • This is the classical Plancherel ' s theorem in

modern language . We have also F : L1 + L oo or even r : L1 + c 0 ( the space o f continuouiil functions tending to 0 at oo ) which is Riemann-Lebesgue ' s lemma . By interpolation we get F : Lp+ Lp ' or even F : L + L if 1 < p < 2 . p p ' p Hausdorff-Young and Paley .

These are the theorems of

4 7

I f f i s a function o r a ( tempered) distribution we

denote its support by supp f , i . e . the smal le st closed set

such that f vanishes in the complement . Appealing once A

more to a physical language supp f consists of those

frequencie s which are needed to build up f from linear com-

binations of characters ix s e •

that

Now let { v(S) 'I 0 iff E;,s int 1\ where 1\= {2 � s �2

( Tauberian condition )

( 6 ) l;v(O 1 _:_cs>o i f s;s�s={ (2-s)-1 2v

.::_lsiS2-€)2v

( 7 ) I o6¢v ( 01 � c6 2-v I S I for eve ry S

Sometime s we shal l also require that

00 ( 8 ) 1 ( or

V=L: -oo <Pv (X) = o ( x ) )

Let also cil be an auxiliary "test function " such that

( 9 ) cp € s

(10) �(0 'I 0 i f E;, s int K , whe re K ={lsi �1}

( Tauberian condition )

4 8

( 1 1 ) I � <t,; > I > c s > o if t,; s K €

Example 1 . I f ¢ i s any function in s with supp = RO ' A I <t> <t,; > I .:. c s > o i f t,; s Ro I € then we can de fine

A = � (f,; /2v <l>v (E,;) ) . I f we in addition assume

then we get ( 8 ) , upon replacing <l>v (f,; ) by

i f neces sary . In this case we can take

-1 A l: </> v by A

that </> (E,; ) > 0

This special type o f te st function , we encountered already in Chap . 1 , except that there we used a discrete parameter t ( roughly - v t ::: 2 ) •

We are now in a position to formulate our basic de finitions De finition 1 . Let s be real , 1 � p � oo , 0 < q � oo • Then

we set

{ f l f s S '

& < 00 }

(Besov space )

This space we equip with the (quasi- ) norm

4 9

Some words o f explanations are in orde r here . Compared with Chap . 1 two changes have been made . Firstly , the para-meter q has 0 < q < 1 included in the range . Thi s means Bsq is not always a Banach space . Secondly , as already noted , p we used the di screte parameter V ( V= 0 , � 1 , � 2 , • • • ) instead o f the continuous one t ( O <t < oo). That we neve rtheless ob-tain the same space s at least i f 1 � q � oo wil l be clear l ater on . It wil l be al so proven in due course that the definition is independent o f the particular test functions {¢v}�=-oo

and l .

Definition 2 . Let s real , 1 �P � oo Then we set (with J �)

= { f If s S ' & I I Js f I I L < oo } p

(potential space )

This space we equip with the norm

I I f I I s = I I Js f I I L pp p

This is exactly as in Chap . 1 . " Example 2 . I f f :: o ( de lta function ) , so that f

then we have n p' 00

(.!. + f s B p p

and this is the bes t result s > - � or - n q < s :;: p' I p ' <P * f :;: <P\) * 0 :;: <P\) and it is \)

( Use ( 2 ) and ( 7 ) to e stimate

1 p' 1 )

in the sense that f � B-sq p 00 In easy to

n \) p 2

fact we have see that

We have also

5 0

:;: 1 ,

i f

i f s < - n p' but it is not possible to make as strong a con-

elusion as in the Be sov case . Example 3 . More general ly , i f f ( � ) :: l � l -0 in a neighbor-

00 hood of oo and i s C e lsewhere then 0-n/p ' oo f s B ' and this p i s again best possible in an analogous sense .

Example 4 . More generally i f in a neighborhood o f oo and C00 elsewhere then f s B0-n/p' ,q if q > .!. • p T

It i s often convenient and sometime s even nece ssary to work with "homogeneous " . ( quasi- ) norms ( i . e . homogeneous with respect to dilations , or in D ) . We there fore also de fine the following modified space s .

Definition 1 . We set

This space we equip with the ( quasi - ) norm

00 2: V =-co

De finition 2 . We set (with I r-IS:)

{ f/f E S' & / /Is f//L p

This space we equip with the norm

5 1

< 00 }

Here arises however a certain complication . Namely they

are not true ( quasi - ) norms since they are not positive de finite

( indeed //f//.sq= O<r=> f is a polynomial ) . The same phenomena Bp

we encountered already in Chap . 2 in connection with the

example with Lips . Al so Isf cannot be defined for all f E S.

Indeed we would l ike to have I sf ([;) =/E; / s 1 ([;) as in the s case of J • But the fact that [;= 0 i s a s ingularity i f s < 0

i s an obstacle .

The remedy for all this is to do the calculus modulo poly-

nomials , of degree < d , where d is a suitable number . Let us

5 2

give a complete analysis of the situation . For simplicity and with no e ssential loss of generality

we may assume that ( 8 ) i s val id. Let us consider the doubly infinite serie s

00 V =�oo <f>v * f .

It i s easy to see that one hal f o f i t , namely the serie s

00 L: <j> * f V=O V

converge s weakly in S ' for any f s S ' , for the Fourier trans-formed serie s

does so . Indeed we have an e stimate of the type _,

I f f ( � ) g ( O d� l < c L: ! l � l l o8g (O i d� if gs S l a l � m , I B I � m

A Applying this to <f>v ( � ) g ( � ) and forming the sum we readily obtain the convergence o f

-VA Indeed it turns out that each term i s 0 (2 ) with A > 0 . The

hal f of our serie s

-1 E O and = 0 i f s < O i t is not conve rgent . I t i s how-

ever true that the derived series

00 \) =--00

converge s in S ' i f I a I i s sufficiently large 1 say 1 i f

I a I � d . To see thi s we use the above e stimate in the case o f

the series

( The term A A

alone gives 0 ( 2 VB) with B > 0 but the extra

factor kills B i f d > B!) The convergence o f the derived

serie s is however equivalent to the existence of sequence

{ PN }�=l o f polynomials of degree < d such that

00 E

v=-N <P * f -\)

converge s in S ' as N + oo • In other words the serie s con-

verge s modulo polynomial s . I t i s clear that the limit di ffers A

from f by a distribution with supp f = {O } 1 in other words

5 4

a polynomial . To summarize we have thus shown that each f E: S ' has the representation

00 f L: v =- oo ¢v * f (modulo polynomial )

+ polynomial.

We want however to say a l ittle bit more about how big the number d must be .

First we state the following lemma that will do us great service in what fol lows too .

holds

holds

Lemma l . Let f E: s• with supp f

( 1 2 )

( 1 3 )

Assume that

( 1 3 I )

supp f

l n (p

R (r )

K ( r ) ={ I � I � r } • Then

r l we can as well substitute J for I . Remark . I f f E: S '. to say that supp f is compact i s

by the Paley-Wiener theorem the same as to say that f is an entire function of exponential type . We see therefore that ( 1 3 ) is nothing but Bernstein ' s famous inequality ( first stated

for

5 5

1 T and trigonometric polynomials ) .

Proof : S ince eve rything i s homogeneous in r we may as

wel l take r = 1 . (Also it would have been sufficient to

prove ( 12 ) for p1 = oo, for in view of Holder ' s inequality we

have

A

1-8 � - + . ) p 00

Now let ¢ be any function in S with supp ¢ compact and

� ( � ) = 1 i f � E:K ( l ) ( =K0 ) . Then the identity f = ¢* f

holds . Using Young ' s inequality ( see Chap . 2 ) we now get

= c I I f i l L p 1 1 p + 'P" -1 .

This finishe s the proof o f ( 12 ) . To prove ( 13 ) we use the

identity Da f = Da ¢* f . Minkowsky ' s inequality then yields

and we are through . The proof of ( 1 3 ' ) goes along similar

l ine s .

Let now f b e a di stribution which fulfils the condition -vs in def . 1 . Then holds in particular I l ¢v* f l I L = 0 ( 2 ) or

v ( n -�) P by ( 12 ) (with pl oo ) i l ¢v * f i i Loo= 0(2 P ;:> ) . We there fore

see that i f s < � our series conve rges i n L and s o i n S' . p 00 A similar argument shows that thi s is true al so i f

56

n s = p , q ;;; 1 . With the help of ( 1 3 ) (with p = oo ) we can

extend this argument to the derived serie s . We find that it

converges in S ' i f l a l ,2: d and s < d + n or s = d + !!. p p

q �1 . Thus to summarize the situation I l f l I becomes a true Bsq p

(quasi- ) norm i f , with d as above , we agree to do the calcula-

tions modulo polynomials of degree < d , excluding at the s ame

time polynomials of degree > d .

We can now also give a pre cise de finition of I s . I f

f s S ' we de fine Is f by the formula

(modulo polynomials )

Each term i s here uniquely defined (by the requirement that

its Fourier transform should be l � l s ¢v f) but the sum is

determined only up to a polynomial . To s ay that fs Ps is thus

interpreted so that there exists

00 v=-oo

D Is ( cp * f ) a v -+ D g a

g s L such that p

as N -+ oo •

we agree to adopt the same identification convention for

:Ps as for :Bsq . p p

Bsq and :Bsq s and f>; ) The connection between ( or P p p p

al so apparent now. Namely i f f s Bsq or f Bsq then we p € p

is

have

5 7

Now by the above remark <P * f i s an entire function of 00

e xponential type ( � 1 ) , thus in particular c In other

words the di stributions in Bsq and p Bsq have the same local p

regularity properties .

In what follows we shall mos tly work with Bsq but many p of the proofs are valid for ( Readers should check

thi s point each time ! ) In the applications we will often en-

counter Bsq and not p

We also indicate two more generalizations of B;q and

Ps First we notice that in the definitions the underlying p •

space Lp = Lp ( ffin ) could be replaced by any translation in-

variant Banach space of functions or distributions X. ( Such

spaces are sometime s termed homogeneous . ) For the new spaces

we sugges t the fol lowing notation : Bsqx , PsX . We may also

introduce analogous Sobolev spaces WkX . In the same way we

use Bsqx , PsX , �X .

Example 5 . I f X

we have :

Example 6 . If X

Lp we are back in the old case .

L ( Lorentz space ) we also write pr

= Ps L pr

Thus

5 8

The se spaces we may call Lorentz Besov, Lorentz potential , Lorentz-Sovolev spaces .

( i . e . Example 7 . Another important case i s

f E: F-l L <=> f E: L ) p p • The space s are related to certain space s Ksq 1' nt d d b B · p ro uce y eurl1ng and Herz . The precise relation is

F-1 Ksq p or

Secondly we notice that if � is any quasi-Banach space o f sequence s then we may replace the de fining condition by

We then obtain space which might be denoted by � B X ( and analogously B � in the homogeneous case ) . Clearly we get B � = Bsqx if � = £sq where a = { a v } � =O E: � iff

� ( 2\! s Ia) ) q ) l/q < oo. Such space s were introduced by v=O Calderon . We shall not consider this generalization here . on the other hand the spaces Bsqx even in a more general , abstract form will be di scussed in Chap . 10 .

Now , all de finitions being made , we can start our study of Be sov spaces . We begin with a completeness result .

Theorem l . I f l,�:q � oo B�q is a Banach space , i f

0 < q < l Bsq i s a quasi-Banach space . p

5 9

First we prove a use ful technical lemma .

Lemma 2. Let g E V' F;, (not tempered ) in lRn

F;, ) •

F = g . As sume that

( the space o f all di stributions

( 14 ) 00

+ ( l: v = O

A

Define fv and F by fv = ¢v g and

A

Then there e xists fs Bsq such that p f = g . ( In particular this

holds for g E S F;, ) •

Proof : We may assume that ( 8 ) i s valid . It i s clear 00

that F E S ' . We shal l prove that the serie s

converges in S ' . I f the sum i s denoted by f ' and i f we put

f = F + f ' 1 then it fol lows that <Pv * f = fv 1 *f = F so that A

f E Bsq • p It is also clear that f = g 1 because o f condition

( 5 ) • To e stablish the convergence of 00 l: f

V=l \! in

other hand it suffice s to prove the convergence of

S ' on the

Z I-0f \) =1

in L for cr sufficiently large . To do thi s we use (12) and 00 ( 13 ' ) o f Lemma 1 to conclude that

< c 2 n - cr ) p

I I f I l L � \) p c 2

<

( n s ) v P - cr -

It is now clear that the series converges in L i f 00 n cr > P - s .

We al so take this opportunity to mention the following

use£ul characteri zation of

6 0

Le1Tli\1a 2 ' • Let { fv } � =l be a sequence and F a member o f s • such that supp fvC Rv , and supp F C K and assume that ( 1 4 )

00 holds true . Then the series L: fv converge s in S • . Let \) =l

f ' be its sum and de fine f = F + f ' . Then f E: Bsq p •

Conversely every ft.: Bsq can be obtained in such a manner . p Proof : The direct part can be proved along lines similar

to the proof of Lemma 2 (or using it ) . For the converse it suffices to take fv = <Pv * f , F = iP * f .

Let u s also make some general remarks concerning complete-ness . Let E be any quasi-normed space . By the Aoki-Rolewicz lemma every quasi-normed space can be P-normed , for p ( 0 < p � l ) sufficiently small . In other words we may assume that the quasi-norm of E is a P-norm , i . e .

Then E i s complete , thus quasi-Banach , i ff every serie s 00 L: x . i=l 1 such that is convergent in E . The

proof i s the same as for the normed space , in which case we o f

Let

course can al low ourselve s to take p = l . Now finally to the Proof 00 L: f .

i=l 1

o f be

Th . l . a series

00 L: i=l

Bsq p is p -normable in Bsq p such that

j jf . j j p < oo 1 Bsq p

i f P = min (q , l ) .

We shall show that it converges in S • to some element

f E: Bs q and that p

(15 ) II f I I sq � < 'I I If. I I P ) 11P •

Bp i=l 1 B�q

()()

6 1

Applying the same estimate to the " tai l " E f; and letting i=M -'-

M -+ oo we see that the serie s in fact conve rge s to f in Bsq •

p To establish ( 1 5 ) we first observe that for each v the series

()() 2:

i=l �v * fi converges in Lp and so does ()() E <P * f . •

i=l 1 Denote A

the sums by f and F re spectively . Then we have supp fv CR v' "'

supp FC K . Also it is easy to see that

()() + ( 2:

V =Q

Using Lemma 2 we see that fE: Bsq and ( 15 ) follows . p Next we consider various comparison (embedding) theorems .

First we compare Bsq with S and S ' which is a rather trivial p matter .

Theorem 2. We have a ( continuous ) embedding S -+ B�q ·

Al so S i s dense in Bsq i f p , q < oo. p Proof : Let f E: S Consider fv = �v* f . Then by ( 7 )

for any o, IDS f v ( .;) 12.c i � I - I SI-o holds . Using (2) we find

or , for any k ,

i f ( x ) i .:S. C 2v (n-o ) / ( 1 + ( 2v l x l ) k v -

It follows that

1 I I f I I C 2v (n ( l- p-) -O ) v L � p

6 2

Taking o sufficiently large we see that fE: B=q for any s , p , q . The continuity of the embedding follows readily from the above estimate s . To prove the density of S i t suffice s to remark that if q < co the subspace o f those f in Bsq such p

A that supp f i s compact certainly i s dense in Bsq , i . e . p the exponential functions . I f suffice s now to invoke the classical fact that if p < co the exponential functions are dense in L (non-periodic analogue of the Weierstrass approxip mation theorem) . The proof i s complete .

Theorem 3 . We have an embedding Bsq -r S ' . p Proo f : Only the continuity has to be verified . To this

end it suffice s to remark that if f EBsq then p is sufficiently large (more precise result will be given in a moment ! ) and that thi s corre spondence i s a continuous one . For the embedding L -r S ' i s apparently a continuous one . co

Next we compare Be sov spaces with the same p . Theorem 4 . We have the embedding

i f s1 < s o r

Al so Bsl -+ps -+ p p

sao B . p I f

6 3

s = k integer > 0 then

Bk l -+ � -+ Bkoo p p p Moreover � = Pk i f 1 < p < oo (or k = 0 ) . p p

Proof : As was already stated in Chap . 2 , this can be

proved using interpolation and the theorem below. However a nSq C nslql direct proof re sults easily i f we notice that N N

under the said conditions relating the parameters . The proof

of the s tatement involving P s is left to the reader . The p proo f o f the last statement concerning Wk will be postponed p to Chap . 4 . I t i s based on the Mihklin mul tiplier theorem.

Much more interesting i s the fol lowing

Theorem 5 (Besov embedding theorem) . We have the em-s q

bedding Bsq -+ B 1 provided p pl Proof : After al l the se preparations , the proof can

almost be reduced to a triviality . Let Then by

( 12 ) of Lemma 1

in the said conditions on the parameters . Since clearly s lq

<P * f E L for any p1� p we see that f�::B • The proof i s pl pl

complete .

For comparison we write down the corre sponding result

for potential space s .

Theorem 6 (potential sl s lp

embedding P5 -+ P B p pl pl

embedding theorem) . We have the

provided � - s = n - s , pl 1 p

6 4

p1 > p , s1 < s and 1 < p < oo . It admits the fol lowing immediate Corollary ( Sobolev embedding theorem) . We have � + L p pl

provided n p - k , p1 2:, p , k integer 2:, 0 and l < p < oo .

Remark . As we know the corollary remains true for p = 1 too but this calls for a special proof ( c . f . Chap . 1 ) .

Be fore proving Thm . 6 we first settle the que stion of real interpolating Besov and potential spaces , for the proof requires interpolation . The result i s already known to us from Chap . 2 .

Theorem 7 . We have

Bsq i f p s =

It has several important corollaries Corol lary 1 . We have

Proo f : use the reiteration theorem ( Chap . 2 ) . sq oo Corollary 2 . Bp does not depend on { </Jv }v =l and 

( satis fying ( 4 ) - ( 7 ) and ( 9 ) - ( 11 ) respectively) . Proo f : Corollary 3 .

does not depend ko kl (W W ) -P , P e q -

s = ( l-8 ) k0 + ek1 ( o < e < 1 ) .

on { <Pv } and •

Bsq if p

6 5

00 Corollary 4 . Bsq

p i s invariant for a local c change o f

coordinates ( in JRn ) • X Here come s the

Proof o f th . 7 : Let By ( 1 31) o f Lemma 1 (with

J in place o f I ) and Minkowsky ' s inequality we have

( 1 6 ) I I <Pv* I l L � C 2 -V s iiJs ( <Pv * f ) I l L p p

C 2 -V s I I <Pv * Js ti l L � C 2 -V s I I <P ) I L I I Jsf iiL p 1 p

( The e stimate ! I <Pv l I L � C re sults from ( 7 ) ! ) Let now 1 so sl f E:( PP , PP ) eq and consider any de composition f = £0+ £1 . By

the triangle inequality we get

I I <Pv * f i i L � I I <Pv * f0 1 1 L + I I <Pv * f1 1 1 L � p p p

< c 2 -v so l l f o l l ps o + 2v ( s 0-sl )

I I £ I I sl p 1 pp

Taking the inf over al l such de compositions we get

In a s imilar way we obtain

Hence

+ (

Since , generally speaking ,

K ( t , a ) < max

00 l:

\) =1

t ( 1 , - ) K ( s , a) , s

6 6

because K (t , a ) i s a concave function of a , we see that the last expression e ffective ly can be estimated by

00 e C ( f (t- K ( t , f ) ) q 0

dt ) 1/q = T c l lf l l s ( P 0 p

s s Thus we have proven ( P 0 P 1 ) C Bsq . p ' p 8 q p

For the converse inclusion let With no loss of generality W@ may assume that ( 7 ) holds true . Using again ( 16 ) we obtain

We have likewise the estimate

Since s

f E: ( P 0, p

so s l J ( l I cp * f ; p I p ) _:s_ c I I cp * f I I p p - Lp 00

+ E ¢ * f the equivalence theorem yie lds V=l V thus proving ::l • The proo f is complete .

6 7

We still have to prove th . 6 . I t i s however convenient

to insert here the following re sult which has been more or

less implicit in the pre ceding discussion .

Theorem 8 . For any n we have an i somorphism

Jn : Bsq -+ Bs- n ,q and an i somorphism Jn· Ps -+ Ps-n p p . p p 0

Thi s means also that in what follows we can often take s = 0 .

Proof : The s tatement for potential space s results at

once from de f . 2 . For Be sov space s it fol lows by interpol ation

( th . 7 ) or by a s imple dire ct argument .

Corollary l . The isomorphism class of B;q depends only

on q ,p . The i somorphism class of Ps depends only on p . p

i f

Proo f : Obvious .

Corollary 2 . 1 1- e + � P Po P1

Proo f : Use

( Ps p I

0 ( o < e <

Ps ) p 8p l l )

0

For completene ss ' sake let us also mention the following

elementary re sul t , the proo f of which i s le ft to the reade r .

Theorem 9 . For

D : Bsq ->-Bs-1 a � q and a p p

any multi-index a we have continuous maps

0 . Ps __,. Ps- I a I a · P P •

Now final ly to the

Proo f o f th . 6 : For the sake of simplicity let us take

s1 0 which by th . 8 is no re striction . By th . 5 and th . 4

where n By interpolation - s . p

6 8

( th . 7 ) ' keeping p fixe d but varying s , p1 we get

PP + s; +

now with

Lp oo • 1

A new interpol ation ( using cor . 2 of th . 8 ) s fixed and varying

Passing back to a general s we have

. s g1ve s P + L + Ip . s p plp 1

Ps + p 1 A final p pl interpolation ( th . 7 and cor . 2 of th . 8 ) , this time with s , p1 s p fixed and p 1 s1 varying , leads to Ps + B 1 The proof i s p pl complete .

Remark . The above proof actually yields more than the

Ps sl ,... slp theorem says , namely + P 1 • B • ( Here we used the p plp plp Lorentz-potential and Lorentz-Besov space s . )

The real interpolation o f potential space s was settled

in th . 7 . As we said , one gets Besov space s as the result

of the interpolation . Now we write down the corre sponding

resul t for complex interpolation .

Theorem 1 0 . We have

s s [ p 0 p 1 ] P ' P e P; i f s = ( l-8 ) s0+ e s1 ( o < e < 1 ) and l < p < oo

Proof (outline ) : Let s s 1 fo: [P P

o ' P P l e • Then there exists

a vector valued holomorphic function h ( z ) such that

h < e ) = f , sup 1 1 h < i Y ) 1 1 s 0 � c p -p

sup l l h ( l+iy) l l ps l � c p

( and sup l l h ( z ) l l s 0 s 1 < oo ) . O < Re z < l pp + pp

Consider the function

6 9

so ( l- z ) + slz J h ( z )

with value s in L • I t can be proven ( e . g . using the Mikhlin p multiplier theorem , cf . Chap . 4 ) that the operators Jiy ( y real )

are bounded in Lp and that

( 1 7 )

for some A > o . Using ( 1 7 ) we see that

hl ( 8 ) Jsf , sup l l hl ( iy ) I l L ,;S c ( 1 + I Y I ) A ,

p

sup I I h 1 ( 1 + i y ) I I L ,;S C ( l + I Y I )A .

I t i s possible to show that the

Chap . 2) is s ti l l applicab le . s s

f sP; . This prove s [ Ppo

' Pp1 ]

p

three l ine theorem ( see

Thus we conclude Jsf s Lp and

Ps For the converse we have p · to find an at least approximately optimal repre sentation

f = h ( 8 ) . A natural choice is

h ( z )

Using ( 1 7 ) one gets then the estimates

70

so it i s not pre ci sely what one wants to but the difficulty

can be easily overcome by replacing h ( z ) by m ( z ) h ( z ) where

m ( z ) i s a scalar valued holomorphic function , with m ( 6 ) = 1 and which behave s as 0 ( I Y I -A> as z -+ oo in 0 < Re z < 1 . We

won ' t enter into the details since we shal l later on give

another proo f ( Chap . 5 ) . k

Corollary 1 . [Wp 0 ,

l < p < oo .

s

Corol lary 2 . s ()() The space s P are invariant for a local C p change of coordinates i f 1 < p < oo .

PROBLEM . To extend th . 1 0 and its coro llarie s to the

case p = 1 or oo

Remark . The proof o f th . 1 0 breaks down for p = 1 or

oo since ( 1 7 ) is not val id in this case . It is not even known

h h s . k t at t e space s P1 ln Wl proce s s .

are s table under any interpolation

Remark . In Chap . 5 we wil l al so discuss the more

re fined results involving the interpol ation of Besov and

potential spaces when al l parameters vary s imultaneously .

The characteriz at ion of Besov spaces as inte rpolation

space s obtained in th . 7 is in some sense a constructive

characterizat ion . Another constructive characterization can

be obtained via approximation theory . ( It i s in fact related

to Lemma 2 ' above . ) Be fore stating the exact result let us

say something about approximation theory in general .

The theory o f approximation can be s tudied from many

71

aspe cts . Among the theorems proved here are density theorems .

A typical such resul t - in fact the prototype - is the

Weierstrass approximation theorem ( 1 8 75 ) . In the case of llin

it says that. the e xponential functions are dense in Lp i f

l < p < oo. On T1 we have to take the trigonometric polynomial s .

O f course Weierstrass himse l f did not have L but C , the space p u o f bounded uni formly continuous functions . By the way both

uni form continuity and uniform convergence make their appear-

ance in thi s context . On a compact topological group it is

the Peter-Weyl theorem.

Returning to Weierstrass and :JRn let us set

E ( r , f ) = in� jjf - g j j supp gCK ( r ) Lp

which quantity is called the be st approximation o f f in Lp by functions of exponential type � r . Then the Weierstrass

theorem can be rephrased as

E ( r , f ) ..., ( l ) , r -+ oo for any f s L • p

Another type o f theorems are now conce rned with the degree of

accuracy of the approximation . What ( smoothnes s ) propertie s

have to be imposed on f in order to assure that

E ( r , f ) = O ( r-5 ) , r -+ oo

72

whe re s i s a given number > 0 ? 1 In the case of T and

p = oo the problem was completely solved by Bernstein and

Jackson in the 1 9 1 0 ' s , except for the case s = inte ger which

was filled in much late r on ( 19 4 5 ) by Zygmund . The answer i s

that f must be in soo B • The case o f several variables 00 was probably first considered by Nikolski j ( around 1 9 5 0 ) . Now

we give a general treatment of thi s problem within our general

framework .

Theorem 1 1 . Let s > 0 , 0 < q .::_ oo Then we have

dr l/q - ) < 00 r

Proof : For simplicity we take 1 2_ q :::._ oo . The case

0 < q < 1 require s some slight changes in the argument . Let

f s Bsq De fine p •

Then we get

Hence

g <P* f +

<

oo s q dr ) 1/q ( fl ( r E ( r , f ) ) r

00 2v+l q dr ) 1/q � L: !2 ( rs E ( r , f ) ) --

V=O r

00 00 ( 2 V S 1/q < C ( L: L: I I <PV+A * f I I ) q ) L V=O

00 � c L: A =0

00 L:

v=O

A=O p

( 2 \) s I I v* f doe s not change i f f i s replaced by f-g .

Thus we have

I I<Pv* f I lL � C E ( r , f ) i f r� 2v -l . p

It now fol lows readily

dr -r

and thus upon summation

00 ( L:

v = O

00 c ( L: v = O

74

The proof i s complete .

Final ly we determine also the dual o f our space s .

Theorem 12 . Let s real , 1 � p < oo 1 � q < oo Then

holds

Proo f : The statement for potential space s follows

readily from th . 8 and the fact that L ' � L , i f 1 � p < oo p p Indeed let SE ( Ps ) ' and de fine T by putting T ( f ) = S (J- sf ) . p Then TEL ' p • Hence , by the above there exi sts hEL , such p that T ( f ) = ffh There fore we find

s ( f ) J f g

where we have set g = Jsh . It i s plain that gEP�� · I t i s

now easily seen that the mapping S + g : ( Ps ) ' + P-� actually p p is an i somorphism.

The statement for Be sov space s fol lows from the one for

potential space s , which we j ust prove d , using interpolation

( th . 7 ) . We prefer however to give a direct proof . If E i s

any Banach space we denote by £ sq ( E ) the space o f sequences

a = {a }00 0 with \) \)=

00 q 1/q 2: ( 2 vs I I av I I E ) ) < oo • v=O

75

We use the fact that (Q, sq (E ) ) ' � Q, -sq ' ( E ' ) i f 1 � q < oo . Also we recal l that ( E E9 E ) ' �E ' E9E ' for any Banach space s 1 2 1 2

E1, E2• Let now s t:: ( Bsq ) ' We de fine T by setting p •

S ( f ) i f F cp * f v

and extend it by Hahn-Banach ' s theorem to the who le of the

space Lp ® Q,sq ( Lp ) . By the above there e xist

{g }00 _0 t:: Q,-sq ( LP ) such that v v-

00

G EL , and p

f F G dx + Z:: f fv gv dx v =O

It i s no e s sential res triction to assume that A

supp G C 2 I< , supp g vc 2 Rv De fine now g = G +

By Lemma 1' ( conveniently modi fied ) we see that

00 z:: gv • v=O -sq ' g E B p '

Also we can see that S ( f ) = J f g . We leave i t to the reader

to fil l in the re st o f the detail s .

76

Notes

Distributions ( or generalized functions ) have a long

history . They can be traced back to Sobolev ( 19 3 4 ) but his

work went unnoticed for a long time . They were rediscovered

and popularized by L . Schwartz ( 19 4 5 ) . An introduction to

distributions can be found in many modern texts on functional

analysis ( e . g . Yoshida [ 5 8 ] ) or partial dif ferential equations

(e . g . Hormander [ 1 2 ] ) .

Concerning the " classical " Fourier transform the be st

source is perhaps Stein-Weiss [ 3 7 ] .

As was said already in Chap . 1 our treatment o f Besov

space s is the one o f Peetre [ 1 8 ] . In the special case

p = q = 2 this can be found already in HBrmander [ 1 2 ] . The

basic underlying ideas can also be said to be in the work o f

Hardy-Littlewood ( and Paley-Littlewood) i n the 3 0 ' s .

The pre sent approach to Be sov spaces has been used in

seve ral papers . Let us mention Lofstrom ' s thesis [ 5 9 ] where

problems pertaining to theoretical numerical analysis are

treated. The point of view of Shapiro [ 2 0 ] , [ 2 1 ] and in

particular that of Boman-Shapiro [ 6 0 ] come s close to ours .

( These are directed towards questions o f approximation theory

and the Besov spaces do not enter explicitly , c f . Chap . 8 . ) Concerning Bernste in ' s inequality ( Lemma 1 ) see books in

approximation theory (e . g . [ 2 3 ] , [ 2 4 ] , or [ 6 ] ) . Generalizations

of the type B5qx or more generally EP x have been considered

7 7

by quite a few authors : Golovkin [ 6 1 ] , Calderon [ 4 1 ] ,

Torchinsky [ 6 2 ] to mention j ust a few . The functorial point

of view appears e . g . in Grisvard [ 6 3 ] and in Donaldson [ 6 4 ] .

The Lorentz-Besov space s are mentioned in Peetre [ 1 9 ] . The

space s Ksq make their appearance in the early work o f Beurling ' s p on spectral synthesis in the middle 4 0 ' s ( see e . g . [ 6 5 ] ) but

in ful l generality they were introduced only in Herz ' s paper

[ 6 6 ] . The connection with interpolation space s was clarified

by Peetre [ 6 7 ] and Gilbert [ 6 8 ] . See al so Johnson [ 6 9 ] , [ 70 ] .

Concerning the "homogeneous "

i s of a more dubious nature .

space s Bsq and Ps , their origin p p Anyhow they are in Peetre [ 5 5 ]

and also , perhaps in a more clear form , in Herz [ 6 6 ] . The

work of Shamir [ 71 ] , [ 7 2 ] ought to be mentioned here too . In

some problems they simply are a must . Concerning the Aoki-

Rolewicz lemma see Pee tre-Sparr [ 4 6 ] or Sagher [ 4 5 ] . As was

mentioned in the notes to Chap . 1 the characterization of

Besov spaces using interpol ation space s goes back to the work

o f Lions ( see e . g . [ 2 5 ] ) . Th . 10 on the other hand probably

comes from Calderon ( see [ 4 1 ] ) . From the hi storical point

of view it is inte re sting to note that Lions himsel f was lead

to introduce interpolation spaces in proving a special case

(p = q = 2 ) o f Cor . 4 of th . 6 . Regarding the characterization

using the best approximation see Nikolski j [ 6 ] . Regarding

Lemma 1 ' see likewise Nikolskij [ 7 ] and Triebel [ 2 2 ] •

The proof of the duality theorem given here is similar to the

one of Triebel [ 73 ] . It is not clear where Besov space s with

7 8

s � 0 really appear. Regarding the case s = 0 see Lions-

Lizorkin-Niko l skij [ 74 ] . In the special case p = q = 2

space s with "negative norms " were systematically used in

partial differential equations already in the middle S O ' s

( c f . e . g . Hormander [ 1 2 ] 1 Peetre [ 1 1 ] ) . Concerning the case

0 < q < 1 see Peetre [ 5 2 ] 1 Flett [ 5 3 ] . The result i s that

( Bsq ) ' � B-�oo if 1 � p < oo 1 0 < q < 1 . I f generally speaking p p X being any homogeneous Banach space and x 0 standing for the

closure of S in X one can show that ( ( B; oo ) 0 ) ' � B��l 1 a fact

belonging to folklore , that has been often rediscovered ( see

[ 5 �) . The proof is the same as for th . 12 .

Quotation : Exe rcise for the reade r .

Chapter 4 . Comparison of Be sov and potential spaces.

In this chapter we will present a more detailed com-

parison of Besov and potential space s . This wil l provide an

occasion to introduce some basic Calderon-Zygmund and Paley-

Littlewood theory , which wil l be needed in what fol lows . We

know already that ( Chap . 3 , th . 4 )

i f 1� p < 00

With the aid o f Planchere l ' s formula it i s also e asy to show

that

We now show that the l atter re sult can be extended at least

to the range 1 < p < oo Theorem 1 . We have

Bsp -+ Ps p p

Bs2 -+ P s p p

-+ Bs2 p

-+ Bsp p

i f 1 < p� 2

i f 2 � p < 00

Proof : In view o f th . 8 o f Chap . 3 it suffices to do

the proof in the special case s = 0 . In other words we

are going to prove that

79

80

if 1 < p � 2

i f 2 � p < ()()

To achieve this we shal l invoke the following

Theorem 2 . ( o f Paley-Littlewood type ) . Let { ¢v } and �

be as in Chap . 3 , so that ( 4 ) - ( 7 ) and ( 8 ) - ( 9 ) of that

chapter are fulfilled. Then

( 3 )

holds .

()() I I £ I l L * I I � * f I l L + I I ( E

p p v=O

I t is also true that

1/2 I ¢ * £ 1

2l I I v L p

( 4 ) I I f I IL "' I I p

()() 2 1/2 E I ¢v* f I ) I I L v =-oo p

( The latter fact i s needed for the analogue of th . 1 for

Bsq and Ps , of course ! ) p p We postpone the discussion of th . 2 for a moment and

complete the proof o f th . 1 . We prove ( 1 ) ; ( 2 ) i s proved in

exactly the same manner ( or using duality) •

Let fsB�p (where 1 < p < 2 ) . Then

()() E

v = 0

holds . Hence , using the fact that Q, p + Q, 2 i f p < 2 (where

81

we have written t P = t 0P ) we get

Thus

00 J ( L: \) = 0

fE: L by th . 2 . p

2 P/ 2 I * fl ) \) dx < oo

This proves the left hand part of ( 1 ) .

Conversely let fE:Lp . Then by th . 2 and using Jessen ' s

inequality we obtain

� I I <1> * f I 12 ) 112 v L V=O p

< 00

Thus f E:B02 and we have e stablished the right hand side too . p The proof is complete (modulo the proof o f th . 2 ) .

Remark . The above sugge sts the introduction of certain

new space s Fsq de fined as follows : p

f E:F;�=}f E: S' & I I * f I lL p

We wil l return to them l ater (Chap . 1 2 ) •

Now back to th . 2 . The proof o f i t will be based on the

fol lowing

Theorem 3 . ( of Calderon-Zygmund type ) . Let E1 and E2

be two Hilbert space s and consider the convolution operator

Tf ( x ) a * f ( x ) J a ( x-y) f (y ) dy

where a i s an operator val ued function and f a vector

82

valued one . Assume that

( 5 ) l x l f2: 2t l l ( a ( x-y ) -a (x ) ) e i i E dx � c l l e i i E , l y l � t - 2 1

for eve ry es E1 and every t > 0 ,

( i .e . , J J � ( O I I E E � C ) 1 , 2

Then

holds .

I f ( 5 ) holds with indice s 1 and 2 inte rchanged then ( 8 ) holds

al so with 1 < p � 2 replaced by 2 � p < oo . Thi s is , in

particular , the case i f

( 9 ) I I D a aU) I I E E � c 1 �; 1 -J a l i f l a l � n

1 , 2 -

(Mikhlin condition )

Since there are many excellent treatments in the l iterature

we are not going to give the full proof o f th. 3 . Several

8 3

comments are , however, in order .

I f E i s any given Banach space we denote by L ( E) the p space of measurable fun ctions ( on a given measure space �

with measure � ) with value s in E such that

In an analogous way we de fine the space L (�) or all other pq space s we need . I f E1 and E2 are two Banach space and

A : E1 -+ E2 is a continuous linear operator we denote its

norm by

sup I I Ae I I E I I I e I I E . efO 2 1

I f E i s a Hilbert space , Planchere l ' s formula is still

valid in the space L2 ( E ) and ( 8 ) for p = 2 i s a dire ct

consequence o f it . For 1 < p < 2 ( 8 ) results from ( 7 ) using

interpolation ( ve ctor valued analogue o f Marcinkiewicz ) . To

e xtend it to 2 < p <oo under the conditions s tated we simply

have to use duali ty . ( I t i s known that (L ( E ) ) ' ::: L , (E ' ) i f p p 1 ,;;;p,;;; oo and E is a reflexive Banach space . ) The hard part

is thus ( 7 ) . It depends on a certain decomposition of L1 func

�ns (the Calder6n-Zygmund decomposition) which again is obtained

from one covering lemma or the other . We shall not enter into

the detail s .

84

We shall have to use the general ve ctor value d form of

th . 3 . Vector valued functions will also have to be conside red

later on . For the bene fit of the reade r let us write down

conditions ( 5 ) , ( 6 ) , and ( 9 ) in the s calar valued case

( E1= E2= C ) • They read as follows :

( 5 ' ) l x l { 2r

l a ( x-y) -a ( x ) I dx � C , l y l � r for eve ry r>O .

The proo f that ( 7 ) entail s ( 5 ) ( and ( 6 ) ) wil l be given later

on ( see Chap . 6 where it belongs logically ) in this special case ,

i . e . , the s tatement ( 9' ) = > ( 7' ) , in fact , even a stronger form

o f i t . ( The statement ( 9' )=> T : Lp + LP i f 1 < p < oo i s the

Mikhlin or Marcinkiewicz multiplie r theorem . ) Here we shall

restrict ourselve s to some s impler comments . First o f al l

( 5 ' ) i s ful fil led i f we have

( 10 ) n+l I grad a ( x ) I � C/ I x I

Thi s i s , in particular , the case i f a i s homogeneous o f

degree -n (and c1 on the unit sphere S ) . I f moreove r

!8 a (x ) ds = 0 (6 ' ) i s true too . These are the operators

conside red original ly by Calde ron and Zygmund ( 19 5 2 ) . They

I

85

are characterized by being not only translation invariant but

also dilation invariant . I f n = 1 there i s , up to a constant

multiple , only one such operator - the Hilbert transform 1 ( a (x ) = x ) . Th . 3 special ized to thi s case i s nothing but

the classical theorem of M. Ries z on con j ugate fun ctions ( 19 2 7 ) .

We shal l al so - be fore return ing to the main road - amuse

ourselve s by s tating a condition emanating from Cotlar which

guarantee s ( 6 ' ) , i . e . ( 8 ) with p = 2. The nice thing is that

we then have conditions on Lp boundedness where the Fourier

tran sform doe s not enter explicitly ( i f this i s so nice ) .

We state it as

Theorem 4 . Let a satis fy ( 5 ' ) and

( 1 1) K { r) I x I I a ( x ) I dx < C r

( 1 2 ) J K ( r ) a ( x ) dx 0

( Here K ( r ) = { l i; l � r } i s the bal l with radius r and cente r 0 . ) • 0 00 "'

Then we may conclude that a E.: B1

and in particular a E.: L2 •

Note that both ( 11) and ( 1 2 ) are ful fil led in the Calderon-

Zygmund case .

The proof again will be postponed to Chap . 8 where it

belongs logical ly .

However , we can now complete a point left open in Chap . 3 .

Proo f o f th . 4 o f Chap . 3 ( completed) : Let fE.:P�

(k integer >o, l < p < oo ) . If l a l � k we can write

8 6

D f = a * J f with � (!;) = ( il;: ) / ( 1 + j �;; j2 ) s/2• It i s

a readily verified that the Mikhlin condition ( 9 ' ) i s fulfilled

since i t follows that D f s L • a p Thus f E J< p Conve rsely , let k f s Wp . Then we may wri te k - L: J f - i a i � k with suitable aa to which ( 9 ' ) again is appl icable . We

Jkf s L and g::J< . We have shown � Pk conclude that p p p p " After this detour , having gained sufficient trust for

th . 3 , let us give the

Proof o f th . 2 : We begin with the following general

observation . In some o f the previous work we have imposed on

{ rl'v } , c} conditions ( 8 ) o f Chap . 3 . We let { lj;v } , 1J' be another

set ful filling ( 4 ) - ( 7 ) and ( 9 -1 0 ) of Chap . 3 . We shall instead

require that

It is not hard to see that thi s can be achieved : s imply take

It can even be arranged that lj;v = cpv ' IJ' = <P •

and

Conside r now the mappings S and T formal ly de fined by

T : f -+ ( <P * f , { ¢v * f } �= O )

00 S : ( F , { � , } � =-00 ) + ' * F + E � * f v v v =0 v v

In view of ( 1 3 ) we have

S o T id ( identity)

so that s i s a pro jection ( retraction of T ) •

If we now apply th . 3 with E1 = c, E2 = _Q,2

( 1 4 ) T : L + Lp EB Lp ( £2) i f 1 < p� 2 . p

Indeed (9) is ful fil led for we certainly have

we

( I D <P ( c-) 12 + z a " v=O

1/2 I D � ( E;,) 12 ) < C I E;. �- I a I a v

In the same way taking E1 C we find

( 1 5 )

8 7

find

Thi s completes the proof for the range l < p � 2. The case

2� p< oo now fol lows at once by duality .

Remark 1 . A simple proof o f one hal f o f th . 2 using only

the s calar value d form of Mikhlin ' s theorem and a standard

probabilisti c argument runs as fol lows . For a random sequence

w = { + 1 , + 1 , + 1 , • • } set

T w

Again we find

with a bound uniform in w

< C I If I I L p

00 l: + * f + v�O + <Pv * f l p ) dx ) l/p p

and , as is well known , for any number A and any numerical

00 1/p 2 'f 2 1/2 E ( I A + v � 0 + a v I

p ) :::: ( I A I + V'=' 0 I a v I ) , 0 < p < oo ( Khintchine ' s inequality )

holds . ( The more "mathematical ly oriented" reader could here

89

have used Rademacher series instead in which case we use Littlewood instead o f Khintchine ! ) Inse rting this in ( 16 )

we get ( 1 4 ) anew. It would be desirab le to have a s imilar proof o f ( 1 5 ) too .

Remark 2. The above proof of th . 2 shows real ly more : That Lp is a retract of Lp ED Lp ( l) (or 1 what is the same 1

2 Lp ( £ ) - the two space s are clearly isomorphic ) . We can expre ss this in terms of a commutative diagram.

id

Since Lp and P; are isomorphic we have l ikewise

id

It is al so easy to

id

Ps T p

t ·�

� p; see that

Bsq �· s p

L ED Lp ( £2 ) p

90

Notice that the operators T and S are independent o f s and

q . Thi s wil l be use ful in Chap . 5 .

We now return to the space s B�q· We want to show that

th . 1 is in a sense the best possible . To fix the ideas again

let s = 0 . Then we shal l prove

( 1 7 ) B�q + Lp => q < min (p , 2 )

( 18 ) Lp + B��>q > max (p , 2 )

By a variation o f the arguments below one can al so show that

for the limiting case s p = 1 and p = oo th . 6 of Chap . 3 i s

already the best possible .

To fix the ide as let us take 1 < p � 2 . The case

2 � p < oo can be handled in analogous manner (or most simply

by duality ) .

The proof o f ( 1 8 ) i s then particularly simple so we

start with it . Let g be a fixed function in S such that

supp g is con tained in a sufficiently smal l neighborhood of

( 1 , 0 , • • • , 0 ) and let

us choose f as follows

A 00 f (U \)§;,0

o r

f ( x )

00 a = { a,

) v=O be any sequence .

00 L:

\) =0

00 L: a v =O v

Let

91

About our sequence of

that ¢v(� ) = 1 near "2\)

test functions {</> }00 0 we may assume \) \) =

1 x1 <l>v * f = av e g •

\) 2 e1 . There fore we have

It fo llows that

I I f I I Oq ::: I I a I I •

B Q,q p

On the other hand using basic facts on lacunary Fourier series

we readily see that

There fore i f L � BOq we must have £2 � Q,q which implies p p q > 2 = max (p , 2 ) •

For the proof o f ( 1 7 ) we con sider an f such that

n f ( � ) = I � � p' ( log i � I >- T in a neighborhood o f oo ,

()()

C e lsewhere

I t is possib le to demonstrate the asymptotic development

with

n - p' f (x ) - C l xl 1 -- T ( log jxl )

a suitab le C . There fore f E: L p

, x �o

i f f T > 1/p. On the

other hand by th. 3 o f Chap . 3 (with o = n/p ' )

T > 1/q. There fore i f s0q� L we must have p p q,;S p 0

Remark. Similar technique s can be used to show that

92

th . 5 o f Chap . 3 cannot be improved upon .

9 3

Note s .

Th . 1 i s e xplicitly s tated in Be sov [ 5 ] and Taibleson

[ 1 5 ] but its roots lie much deeper ( I f n = 1 c f . [ 1 7 ] , Chap .

14 . ) Th . 3 goe s back to Calderon-Zygmund [ 7 5 ] ( scalar valued

case , dilation invariant operators ) . They thereby e xtended

M . Rie s z theorem - which was first proved by complex variable

techniques - to the case o f several variables . Their result

has important applications to el liptic partial di ffe rential

equations ( c f . e . g . Arkeryd [ 76 ] ) . A considerable simplifi

cation and clarification of the proof in [ 75 ] was obt.ained by

Hormander [ 7 7 ] who also expl icitly stated condition ( 5 ' ) .

The vector valued case was first clearly conce ived by J . Schwartz [ 7 8 ] who used it precise ly for proving theorems

of the Paley-Littlewood type . Let us further mention Benedek

Calde r6n-Panzone [ 79 ] , Littman-McCarthy-Riviere [ 80 ] ,

Riviere [ 81 ] and for a general introduction Ste in [ 14 ] . The

Paley-Littlewood theory arose from the work o f these authors

in the 30 ' s . Again original ly complex variable technique s ,

notoriously complicated by the way , were used . See [ 1 7 ] ,

Chap . 1 3 . For the Paley and Littlewood theory in a rather

general abstract situation ( diffusion semi-group s ) see Stein

[ 82 ] . It i s intere sting to note that the Mikhlin or

Marcinkiewicz theorem historically was proved using Paley

Littlewood theory . First by Marcinkiewicz ( 1 9 3 9 ) ( see [ 8 3 ] )

for T l and then , using his resul t , by Mikhlin [ 8 4 ] ( 1 9 5 7 )

94

for :JRn . Th . 4 goe s back to the work of Cotlar [ 85 ] ( c f .

[ 14 ] ) . Concerning lacunary Fourier series see [ 1 7 ] .

Quotation : S ame as for Chap . 2 .

Chapter 5 . More on interpolation .

We know already several re sults on interpolation o f

Be sov and potential space s ( see Chap . 2 and Chap , 3 , in

particular th . 7 and th . 10 of the latter) . But in these

results the exponent p was fixed all the time (except in the

cor . to th . 8 where p varied but the other parameter s was

kept fixed ) . Now we wish to see what happens i f al l para-

meters are varied at the same time .

Taking into account remark 2 in Chap . 4 we see that the

interpolation of Besov and potential space s can be reduced

to the interpolation of the spaces t sq (A ) and Lp (A ) , i . e . ,

vector valued sequence and function space s . We there fore

begin by reviewing what i s known to be true about this .

Let us recal l the definitions o f the above space s . Let

A be any quasi-Banach space . We denote by t sq (A) , where s

real o < q .::_ oo , the space of sequences a = { av Soo=O with

values in A such that

S] being any measure space carrying the positive measure ll ,

we denote by Lp (A) , where 0 < p � oo , the space of l-1-measurable

fun ctions a = a (x ) ( x ES] ) with value s in A such that

9 5

I I a I I L (A ) p ( J� < I I a ( x ) l l i ) d ]J (x ) ) l/p < oo .

9 6

In an analogous way we introduce the Lorentz space L (A) pr where 0 < p , r � oo • More generally , w being a positive ].l -measurable function in � , we define the space o f ].l -measurable functions a = a (x) such that

The space £ sq (A) i s really a special instance of L (A ,w ) . p Indeed take :

� = { o , l , 2 , • . • } J.l ( {v } ) = 1 w (v ) = 2 v s

q = p

( discrete measure )

We there fore start with Lp (A ,w) . The following results are wel l-known and completely understood . For the proofs we re fer to the literature . We separate the complex and the real case .

Theorem 1 . ( vector valued analogue o f Thorin ) . Let A = { A0 ,A1 } be any Banach couple . Let 1 � p0 , . p1� oo holds

Then

provided 1 p

-+

l w )

1 W ( 0 < 8 < 1 ) .

Theorem 2 . Let A be a quasi-Banach couples .

9 7

( i ) vector

valued analogue o f M. Rie s z ) . Let 0 < p0 1 p1 � oo • Then holds

-+ = Lp ( (A ) 8 p 1 w)

provided 1 p I W wo

1- 8 wl 8 ( 0 < 8 < 1 )

More generally we have

and the reve rsed embedding i f r � P· ( i i ) ( ve ctor valued

analogue of Marcinkiewicz ) . Let A be any Ban ach space . Let

again 0 < Po 1 p1 � oo • Then holds

(L Pa ra (A , w) ,

provided 1 1- 8 + _j_ = p Po pl

Remark . Notice that in

L (A , w) ) 8 plrl r

( 0 < 8 < 1 ) •

part ( i i ) o f

L (A 1w ) pr

th. 2 we take

A0 = A1 = A and w0 = w1 = w . Thus we do not have a ful l

analogue o f the Marcinkiewicz theorem in the s calar case .

9 8

Let us now turn our attention to the space s £5q (A) . +

Theorem 3 . Let A = { A0 , AJ! be any Banach couple . Le t

Then holds

provided , s = ( 1- 8) s 0 + es1 ( o < e < 1 )

Proof : In view o f the above observation that £ sq (A)

i s but a special case of Lp (A ,w ) thi s is j ust a recast of

th . 1 .

Theorem 4 . ( i ) Let A = { A0 1A1} be any quasi-Banach

couple . Let 0 < q0 1 q1 � oo then holds :

( 1 )

provided

5 q £ o o (Ao ) I

1 q =

More generally holds :

( 2 )

5 q £ 1 1 (A ) ) 1 e q

I 5 < 1- 8) s 0 + e s1 < o < e < 1 ) •

The exponents min (q , r ) and max (q 1 r ) are the bes t possible .

( ii ) Le t A be any quasi-Banach space . Let

Then holds

99

( 3 ) s q Jl. 0 0 (A ) ,

s q Jl. 1 1 (A ) ) = Jl. sr (A) 8 r

provided s = ( 1- 8 ) s 0 + 8s 1 ( O < 8 < 1 ) .

( ii i ) Let A be any quasi-Banach space . Let 0 < q0 , q1 � oo Take further s 0 = s 1 = s . Then holds

provided 1 q

Proo f : ( i ) Again Jl. sq (A) being a special instan ce of

Lp (A ,w ) , ( 1 ) i s a straight forward consequence of part ( i )

o f th . 2. Let us next fix attention to the first -+ in (2) . I f r � q we can again make appeal to part ( i ) o f th. 2 . Let

us there fore assume r � q . For any sequence a = {av � =O let us write

00 a = L: a E where E =

\) =0 \) \) \) ( 0 , • • • , 0 , 1 , 0 , • • • ) (with the 1 in the v -th

position )

ain that for each v

\)S . I I s . q . < 2 ]. v Jl. J. J. (A . ) ].

From this we obtain by interpolation

i 0 , 1 ) •

whe re we have written

T = s q

Q, 1 1 (A ) ) •

1 e r

1 0 0

Assume now that r is so small that T i s r-normable . We

know that thi s i s po ssible in view o f the Aoki-Rolewicz lemma .

There fore taking r-th powers and forming the sum we get

I I I a E: I I r ) l/r � oo v =0 v v T -

00 1/r < C ( I

=0 l la.v I I (A.) ) r ) = c I I a I I sr + 8 r £ (A ) er

We contend that the first + in the following two case s :

2° r sufficiently smal l . To obtain the same con-

tention for general r ( :;; g ) we have to use interpolation .

Generally speaking , let us write

and assume that

s q £ o o (Ao ) ,

s q Q, 1 1 (A ) ) ( i = 0 I 1 ) l 8 r .

sr . + Q, 1 (A ) + T . sri 1

1

i 0 ' l ) •

Interpolating this give s

( 5 ) sr sr

n O (A+ ) , n 1 (A+ ) ) , + ( T T ) h �v �v 1\ 0 , 1 , r w ere ero er 1 r 1\

1 A + ( 0 < A < 1 ) r1

-r

By ( 1 ) , which we have already proven , we have

sro + srl 9, (A e ) , 9, (i\ > > r0 er1 A r

1 0 1

However , b y a certain complement to the reiteration theorem

we have

Thus the left hand side of ( 5 ) i s e ffe ctively 9, sr (A ) 8r •

by the same token

� right hand

T .

side is T . Altogether we have shown

sr (A+ ) 9, er + T ,

Also

under the said assumptions . Thi s finishe s the proof for the

1 0 2

first -+ in ( 2 ) . Let us turn to the second -+ in ( 2 ) . Now

the hard case is r ,2, q . For e ach v we have the inequality

\) s . 2 2 I I a) I A . � c I Ia I I s . q . (A )

( i= e, 1 ) l. � l. l. i

By interpolation we obtain

2vs l l a l l -+ < c l la. I I T A er

with T having the same meaning as above , i . e . , the middle

space appearing in ( 2 ) . I f r = oo this finishe s the proof .

The general case is obtained b y interpolating between the

cases r = q and r = oo •

Having thus e stablished ( 2 ) let us indicate a counter-

example which shows that the exponents min (q , r ) and

max (q , r ) in ( 2 ) cannot be improved on , in general . To be

precise let us assume that P is a number such that

s q s q -+ ( � o o (Ao ) ' � 1 1 (Al ) ) e r

We want to prove that p � min ( r , q ) . Taking

s1 = 0 we obtain to the right by part ( i i )

o f th . 2 a Lorentz sequence space . It i s easy to see that this

entail s by nece ssity p < q or p = q , q < r . Taking again

Ao = A = A but requiring now so =I sl part ( ii ) (which we have 1

not yet proven ) give s the space � sr (A) . This clearly shows

1 0 3

p � r . Thus min (q , r ) i s best possible . The proof that

max (q , r ) is best possib le is similar .

( i i ) We s tart from the estimate

\) s . 2 1 I I av I I A � I Ia I I s . q .

R. � 1. (A) (i = 0 , 1 )

Applied to a general decomposition a = a0 + a1 this leads

to

s q s q R. 0 O (A) , £ 1 l (A ) )

Taking rth powers and forming the sum we get

dt l/q T) < c I I a I I s q s q

( R. 0 0 (A) 1 £ 1 1 (A) )

by the de finition of the interpolation spaces via K ( see

8 r

Chap . 2 ) . This prove s one half of ( 3 ) . For the converse let

us write

We have

00 L: u

\) =- 00 \) with u = a E: \) \) \)

s . q . £ l. 1. (A)

if \)� 0 ,

-v s . 2 1 l la 1 1 A

Thi s give s

Thus we get

I I a l l < s q s q (£ 0 0 (A ) , £ l l (A ) ) 8 r

< C I I a l l s r £ (A)

C ( oo v ( s -s0 ) v ( s 0-s1 ) z:: ( 2 J ( 2 \) =0

1 04

by the ( discrete ) de finition of the interpolation space s via

J ( see Chap . 2 ) . This completes the proof of ( 3 ) .

( i i i ) Immediate consequence of part ( i i ) of th . 2 .

PROBLEM. To find a precise de scription of s q s q

( 2 0 O (AO ) ' £ l l (A ) ) i f r � q . 1 er

We are now ready to proceed to the applications to Besov

and potential space s . Contrary to our habit we shall start

with Ps , because thi s is here that much simple r . p Theorem 5 . ( i ) Let l < p0 , p1 < oo • Then holds

s [ P 0

Po i f s =

( ii ) We also have ( in the same conditions )

l ti S

Proof : ( i ) By remark 2 in Chap . 4 we have the commuta-

ti ve diagram

s . p l p .

� l s . 2 id

1 L Ell L (Q, l ) ( i 0 , 1 ) p . p . l l

s . � p l p . l

By interpolation we obtain ( recall the functorial character

o f our interpolation " space s " ! )

s s l [ P 0 ' p

T P o pl

id 1 � L Ell L (Q, s 2 ) p p s s l �� [ P 0 p p ' pl 0

Here we have used th. 1 . From this diagram, can now be read

o f f :

s s 2 1/2 f E [ P o

, P 1 ] 8 <=> I I * f i l L + I I ( I ( 2v s I ¢ * f l ) I l L < uc P 0 P1 P v =O v P

But the latter condition means pre ci sely that Thi s

f ' i shes the proof o f part ( i ) .

Remark . I f p0 = p1 = p , part ( i ) i s j ust a re statement

of th. 9 of Chap . 3 . We thus obtain a new proof o f the latter

resul t . I f s 0 = s 1 both parts are contained i n cor . 2 of

th . 8 of Chap . 3 .

To state the full result for Bsq we need also the p

1 06

Lorentz-Besov spaces Bsq = Bsq L ( see Chap . 3 . ) . Notice pr pr that

Theorem 6 . holds

soqo s lql ]8 Bsq [B B

Po pl p

i f 1 1- 8 8 1 1- 8 8 ( l- 8 ) s0+ 8s1 ( O < -- + + I s p Po pl q qo ql

( i i ) Let 1 ,;;;, p0 1 p1 ,;;;, co I 0 < q0 1 q1 ,;;;, co . Then holds

( 6 ) soqo s1q1 Bsq (B B

P )8 q Po 1 pq

Then

8 < 1 )

1 1- 8 8 1 1- 8 8 ( l- 8) s0+ 8 s1 ( 0 < 8 < 1 } i f + P I + - I S p Po 1 q qo ql

( 7 )

More generally we have

( 8 )

if 1 p

Bs , rnin (q , r) pr

1- 8 -- +

Po 1 q

1- 8 -- + qo

I S

Bs 1rnax (q 1 r ) pr

In particular holds ( r p )

( 9 ) · ( ) s oqo slql ( ) Bs ,m1n q ,p -+ ( B ) � Bs ,max q , p

, Bp 8 p � P Po 1 P

1 0 7

Al so the exponents min (q , r ) and max (p , r) i n ( 8 ) are best

possible , at least if l � Po , p1 < oo , s 0 1- s 1 and

( 1 0 ) s o

- s l n = n n < l - - -Po pl

( i i i ) Then holds

( iv ) Let l � p < oo 0 < q0 , q 1 � oo . Then holds

1 q

8 ( 0 < 8 < l ) • ql

Remark . Part ( i i i ) i s of course j ust a re statement of

th . 7 o f Chap . 3 o f which we thus get a new proo f .

Proof : ( i ) By Remark 2 in Chap . 4 we now have the

cornm tative diagram :

s . q . B 1 1

T p . 1 �

s . q . id l L E9 .Q, 1 1 ( L ) ( i 0 , 1 ) p . p .

�

1 1 s . q .

B 1 1 P · 1

1 0 8

By interpo lation thi s yie lds

soq o s lq l [ B ' B

P l e T p

i d l � L tB Q, sq ( L ) p p

s oq o s lql � [ B ' BP J e p

s oq o He re we have used th . 3 . That f s [B , P o

fo l lows exactly as in the proo f o f th . 5 .

s q sq B l l ] <=> ft:B

P l e P

( i i ) The proof o f th i s part goe s along s imilar l ine s s tarting

with the same diagram . Fo r (6) we use ( 1 ) of th . 4 . ( 7 ) o f

course i s obtained j ust b e spe cial i z ation from (6) . For ( 8 )

we use ( 2 ) o f th . 4 . Again ( 9 ) fol lows from i t j us t be

specia l i z at ion . The only thing that remain s - and that i s the

hard point in fact - i s to see that the e xponents min ( q , r )

and max ( q , r ) are best po s s ib le . By duality ( th . 1 0 o f

Chap . 3 ) we can re strict attention to the l atter case . Assume

thus that we have for some P

( 1 1 )

As in Chap . 4 let g be a fixed funct ion in such that

supp g i s contained in a smal l ne ighborhood o f e 1= ( 1 , 0 , • • • , 0 )

and l e t a : { av }

00V =O be any scalar sequence . We de f ine f

by

109

A 00 00 f ( s ) = \) g ( s-2 e1 ) or f ( x ) = I: av gv ( x )

v=O

where thus

g v ( x ) . 2 \)

e �x g ( x ) .

Since I I g I l L does not depend on v pr

obviously , it i s readily seen that

We thus get an embedding

A and since supp gv Rv

f E: Bsq i f f a E: Q.sq . pr From ( l l ) follows

where we have used th . 4 , part ( i i ) , incidentally . But the

inverse mapping ( de fined for all functions of the form

I: av gv ) i s continuous . Thus we get Q. sr -+ Q. s p which

entails p > r . For the proof of p � q we take inste ad

A 00 00 f (s ) = I:

\) =0

wher thi s time

g ( x ) \)

-v s -vn n ix2v - vn 2 2 e g ( 2 x ) .

We require that 1° supp . gv C. Rv and 20 I 19v I I L c 2- vs pr pr

with c independent of v , with pr p and s re l ated as in ( 8) .

l l O

0 Th i s leave s us with the condi ti ons 1 n � 1 and

2 ° t;+ n !!. pi s . ( i = 0 , 1 ) . Upon e l iminat ing , we find pre ci sely l

condi tion ( 1 0 ) • I f th i s i s so we see that f E: Bsq i ff pr

a E: Q,q = £ Oq . In view o f ( 1 1 ) again this le ads to

qo ql p ( £ , £ )6 r -+ £ • By part ( i ii ) o f th . 4 we must the re fore

ne ce s s arily have P2: q .

( i i i ) Use part ( i i ) of th . 4 .

( iv) Use part ( i i i ) o f th . 4 .

PROBLEM . To find a pre ci se de scription of s oq o s lq l l' f ( BP 1 B

P )6 r r � q . ( I t i s thus not a Be sov space . )

0 1 A fte r thus having te rminated our di scus s ion o f inte rpola-

tion o f potential and Be sov sp ace s le t us indi cate a few

app l i cations of the re sul ts obtaine d . ( Other app li cation s

wil l be given l ate r . )

We be gin wi th the fo llowing impo rtant coro l l arie s .

Coro l l ary

5 = ( 1- 6 ) s o +

( 12 )

( 1 3 )

1 . Let 1 < P o 1 pl < ()()

6s l ' 1 1 - 6 6

-- + - ( 0 < p P o pl

I I£ I I s < c I I £ 1 1 1 - 6 p s o

p p P o

I I f I I < c I I f 1 1 1�06 Bsp p

p pl

and let

6 < 1 ) • Then ho lds

I I f I I 6 5 1 p pl

I I f I I 6 s l p p l

Proo f : We make use of the fol lowing gene ral re sult for

interpolation sp ace s : ( * ) Let A = { A0 1A1 } be any quasi -Banach

111

couple and let A be a space such that 1i. 8q + A for some

q > 0 . Then holds :

I f one applies ( * ) , ( 12 ) and ( 1 3 ) readily fol low , making also

appeal to part ( i ) and ( i i ) o f th . 5 re spectively .

Corol lary 2 . Let 1 < p I pl � 00 I o < q0 , ql � = 0

s = ( 1 -8 ) so + 8 s1 , 1 1- 8 p Po

Then holds :

In particula r ( p0 = q0 , p1

I I f I IBsp � c p

Proo f : Use again ( * ) in th . 6 .

+ 8 1 I pl q 1-e e -- + -qo ql

c 1 1£ 1 1 1� 8q I I£ 1 1 8s q • B 0 0 B 1 1 Po P1

q1 ) holds

I I£ 1 11- 8 sapo B Po

but now in

1 1£ 1 1 8 s p •

B 1 1 pl

con j unction

oo and

( 0 < 8

with

let

< 1) .

( 6 )

P OBLEM. To extend Cor . 1 to the case Po or p1 = 1 .

We give also an appl ication where inte rpolation i s used

in a m re e s sential way . ( Several simil ar applications will

come/later on . ) Let n = 1 and assume that 1 � p � oo • Then

VP is the space o f functions of boun ded pth variation in the

sense of Wiener , i . e . , f s V i f f for every family of disjoint p intervals Ik = [ ak , bk ] c: JR holds

1 1 2

( 1 4 )

with C depending on f on l y . I t i s re adi ly seen that thi s i s

a Banach space , taking I � � l v = in f C , a t least i f we count p

modulo polynomia l s o f de gree 0 , i . e . , mo dulo con stants .

( 1 4 )

whe re

( 15 )

Theorem 7 . We have

1 p

1 1 •p ' B + p

1 - 8 + P o

(V V ) + V P o, P l 8 P P

( 0 < 8 < 1 } .

1

1 - 00 • p I

+ B p

Al so in the s ame condi tions

- 00

(v V ) + s" P P o

I pl 8 oo p oo

Remark •

. He re we use for the first time the homogeneous

space s B. We alway s le ave to the reade r to ve r i fy in e ach

case that a re sul t prove d for B i s va lid also for B . Proo f : 1 ) We first prove the middle + in ( 1 4 ) .

Con s ide r a fixe d fami ly o f di s j oint inte rva l s Lk = [ ak , bk ] C � .

Wi th i t we a s s ociate the mapping :

It i s c lear that

P · U : V + £ 1 ( i 0 , 1 )

pl

( 1 6 ) p p -+ ( £ 0 1 £ 1 ) = Q, P

e p

11 3

It follows that i f f E (V , V ) then ( 1 6 ) holds true , i . e . , P o P1 f E VP . 2 ) For the first embedding we noti ce the obvious

embeddings

L -+ V 00 00


o 3 11 1 I • ool B oo ) e P

-+ <v 1 , voo ) e P

But by ( 9 ) o f th . taking 1 1- e 6 -1-p

1 I 1 -

o 3i1 , oo l • p Bp -+ B 00

Also y what just was proven

e + -00

) e P

so it suffice s to invoke the re iteration theorem. We get

indeed

11 4

+

3 ) In the p roo f o f the third + see Chap . 8 . (We have inc luded

it for the sake o f completene s s ) .

4 ) ( 15 ) fol lows from ( 14 ) using aga in ( 9 ) o f th . 6 .

Remark . One can also consider s l i ght ly more gene ral

space V a de fined by rep l acing ( 14 ) by p

( 14 I ) ( I I b -a l -a p I f (b ) - f ( a ) I p ) l/p < C < oo k k k k =

Obviou s ly One can also show that

l- ! v P = w1

i f 1 < p < oo . The re ader i s in vi te d to try to p p

extend th. 7 to the space s

1 1 5

Note s

Concerning th . 1 see Calderon [ 4 1 ] and concerning parts

of th . 2 (which goe s back to Gagl iardo ) Lions-Peetre [ 25 ] 1

the extension to the ful l range ( 0 100 ] i s of later origin

( see Holmstedt [ 44 ] 1 Peetre [ 86 ] ) . A counter-example for a

general vector valued analogue o f l'-1arcinkiewicz theorem was

found by Kree [ 8 7 ] . I t should al so be mentioned that the

interpolation in the case of weights first was considered

by Stein-Weiss [ 8 8 ] . Th . 4 was first s tated in Peetre [ 1 9 ]

( c f . Triebel [ 7 3 ] 1 [ 2 2 ] ) . Part ( i ) of th . 5 and al l of

th. 6 except for ( 8 ) i s due to Grisvard [ 6 3 ] . The res t comes

from Peetre [ 5 5 ] 1 [ 1 9 ] . In [ 55 ] there i s also an application

to the interpolation theorem of Stampacchia [ 89 ] . The spaces

Vp eminate from Wiener [ 9 2 ] . The l ast + in ( 14 ) goe s back

to Marcinkiewicz ( see [ 8 3 ] ) . Regarding inte rpolation see

Kruger-Solomj ak [ 9 3 ] . Inequalities of the type appearing in

Cor . 1 and Cor . 2 to th . 6 were first obtained by Nirenberg

[ 9 0 ] and Gagliardo [ 9 1 ] .

Chapte r 6 . The Fo urie r tran s form .

Thi s shorte r chapte r i s re ally an introduc tion or

preparation for Chapter 7 .

1 1 6

Le t u s begin b y re s tat ing what we a l re ady know re garding

the action of the Fo urie r trans form in various classes of

functions ( see Chap . 3) . Th i s is e s sential ly the Hausdorff-

Young theorem :

( 1 ) 1 )

and i t s re finement usual l y as sociated with Pale y .

( 2 ) F : L + L I p p p i f 1 < P __ < 2 c!. + 1 p p' 1 )

Now we turn to various vari ant s and general i z ation o f the se

re s ul t s .

We re cal l that ( 1 ) and ( 2 ) we re obtained by inte rpo lation

from the endpo int re s ul t s p = 1 and p = 2 , i . e . , by the use

of the (quantitative ve rsion o f ) Riemann-Lebe sgue lemma and

Planche re l ' s formula re spe ct ive ly . We now obse rve that

( 3 )

which i s a k ind o f improvement o f the Riemann-Lebe sgue lemma .

Inde e d i f • Q oo

f s B1 then fo r each v we have

the re fore by the said lemma A A "' fsL '1'\) 00 It fo l lows that

A f i s

bounded in the sets R V E = ( 2 -E ) -l 2v ..::_ I t;. I .::_ ( 2 - E ) 2

v 1

11 7

uni formly in v i f E > 0 i s fixed . But these sets obviously

overlap i f E i s sufficiently small . There fore f E L oo • �ve

now interpol ate between ( 3 ) and Planchere l . Noticing that • 0 2 B2 = L2 1 thi s yields :

F :

But by Thm. 6 1 ( 8) of Ch . 5

J30 1min (p ' 1 r ) pr -+ ( Bl

o 00 I :802 ) i f 2 e r l p

1- e + e 1 2

Also ( L 00 L , . There fore we have proven the p r following

( 4 )

Theorem 1 . We have

F : BO min (p ' , r ) i f pr -+ Lp ' r

Taking r = p ' we find in particular

1 < p < 2

I f r = p ( 4 ) leads to a weaker conclusion

than ( 2 ) , this in view of th . 7 of Chap . 4 . One can also • sq formulate results in terms of the Beurl ing-He rz spaces Kp

( see Chap . 3 for de finition ) and more generally the Lorentz• sq Beurling-Herz space K ( the obvious de finition ) . pr

( 6 )

Theorem 2 . We have

f : :B0q p i f l < p � 2 ,

We al s o have ( genera l i z ing ( 3 ) ) .

i f 0 < q < 00

1 1 8

O < q �

We invi te the re ade r to provide the obvi ous proof o f

th . 2 and a l s o o f the fo l lowing

Coro l l ary . We have

i f l < p � 2 .

Note that a we aker form of ( 5 } re sults from { 6 ) by taking

q p ' .

\ve now go on studying a s l ightly di ffe rent prob lem . For

certain rea son s it is convenient to interchange the role of

x and � . In other words we cons ider the inve rse Fourier

trans form F- l rathe r than F • Since we have perfect

symmetry we have o f course

( 1 I )

( 2 I )

- 1 F :

- 1 F :

L p

L p

-r L p ' i f

-r L p ' p i f

l � p � 2 (.!. + 1 1 ) p' p

l < p � 2 ( .!. + 1 1 ) p' p

119

Notice that sin ce p� 2 we have p ' � 2 . We now ask what

condition s have to be imposed on f in order that f should

belong to a Lebesgue or Lorentz space with an exponent

< 2 , e . g . f E Lq . Here is a first re sult in this sense .

Theorem 3 ( Be rnstein ) . We have

( 7 )

Proof : Let By Schwarz ' s inequality and

Plancherel we have

Taking the sum it fol lows that

... and

( 8)

00 l lf l l Ll� C \! �-""

f E L1 •

Theorem

-1 F :

I If I I

4 ( Szasz ) . We

n ( 1 1 p 2) , p

. B2

00 dx � C l:

\) = -00

have

-+ L i f 0 < P < 2 . p

1 2 0

Proo f : Adaption o f the above proof fo r th . 1 .

Remark . Us ing in te rpo lation one can al so prove ( 8 )

with l � p � 2 starting with ( 7 ) and F - l . L2 -+ L2 ( i . e . ,

Planche re l 1 s theorem) .

Re turn ing to th. l we see that , in view o f th . 5 of

chap . 3 , ( 7 ) en tai l s also

( 7 I ) - 1 F : i f l � p � 2 .

I f in ( 7 1 ) we repl ace � l n

• p ' by . p ' 00

( 7 I ) i s B B p p

F : Ll -+ L oo imp l ie s

n o longe r true .

Fo r ( 7 1 ) together with

� l • p ' -+ L and we know that such an e s timate i s be s t p o s s ib le • B oo p

Howeve r , the re is the fol lowing s ubsti tute :

Theorem 5 : \'Ve have

� co p , .o , l- -2

- 1 F : • p ,.. -+ Bp I I B 00

In particular hol ds (p

l o , -F -l . � () :B 2 -+ . l 00

l )

i f

Proo f : Use Co r . 2 o f th . 6 o f Chap . 5 .

proof i s equal ly simple ! )

Coro l l ary . Take n = l . Then holds :

(A dire ct

1 2 1

Proof : Use th . 7 o f Chap . 5 .

The next rather natural deve lopment is to substitute

for L1 in th . 3 the space • Q oo Bl

We argue as follows . To show that f E :8° oo we have to l ,

e stimate I I ¢ * f I l L • But in view of th . 3 we have v l

A A A Now cp f of course depends only on the restriction o f f to

v

the net R v

Let us the re fore for any open set Q

I I g I I = in f · sq Bp (Sl ) l lh l l • sq Bp ( rl)

set

where the in f is taken over al l h whose re striction to Sl is

g . Then i t is easy to see that

We have proven :

( 9 )

Then

Theorem 6 : Assume that

sup v

where

Corollary. In particular this is so i f

(Mikhlin condition )

h i s the smallest integer n > 2 .

We are now in a position to fil l in a gap le ft open in

1 2 2

connection with th . 3 o f Chap . 4 (namely that ( l ' ) entail s

( 5 I ) ) o

Theorem 7 . ( "Mikhlin " ) . As sume that for some s > n/2

holds

( 10 ) sup v

< ()()

Then conditions ( 5 ' ) o f Chap . 4 i s ful fil le d , i . e . , we have

( ll ) f I f ( x-y) - f ( x ) I dx � C , y E: K ( r ) for eve ry r > 0 JRn "K ( 2r )

Proof :

( i . e . ' f v each

we write

¢ f ) . v

()() f L: f where f = ¢ * f

v =- ()() v v v It clearly suffice s to e stimate for

f I f ( x-y ) - f ( x ) l dx , y E: K ( r ) JRn" K ( 2 r ) v v

Choose E: so that n 0 < E: < s-2. Then we get

1 2 3

A < f I fv ( x-y ) I dx + n f I f v ( x ) I dx \) JRn "- K ( 2 r ) lR "'- K ( 2 r )

< 2- 2 r- 2 f I x-y I 2 If v ( x-y ) I dx + 2- 2r-2 f I x I 2 I f v ( x ) I dx

where we at the last step have used th . 3 . We al so get by

the same theorem :

< I Yl f I grad � ( x-y ) I dx � C r l l t::f') l n . 2' l B2

< c r 2v 1 1 f v 1 1 • s � c r 2 v •

p2

Altogether we have thus proven

\) \) A < C min ( r 2 , ( r 2 ) -2 ) • \) -

I f A i s the expre ssion to the left o f ( 1 1 ) we get

00 A � E A < C < oo

v =- oo \)

and the proo f i s complete .

Remark . Condition ( 1 0 ) says e s sentially that

1 2 4

f E: :B 0 00 F P; . A s imilar remark appl ie s to condition ( 9 ) •

1 2 5

Note s

The first re sult o f thi s chapter , notably th . l and

th . 2 together with its corollary , be long to the folklore .

C f . e . g . Riviere-Sagher [ 9 4 ] . Th . 3 and th . 4 which indeed

go back to Bern stein and S zasz respective ly were first con

sidered in the context of T1 • See the re ference s listed in

Peetre [ 9 5 ] , [ 6 7 ] . Th . 3 goe s back to Zygmund and its

corol lary to I zumi-Izumi [ 9 6 ] ( c f . Peetre [ 5 2 ] ) . The proof

o f th . 7 i s e s sentially the one o f Peetre [ 9 5 ] . Regarding

the interpolation of the Mikhlin or ( better ) Hormander con-

dition ( 10 ) see also Johnson [ 6 9 ] .

Quotation : Your student ' s life i s not entirely without value ( though I suppose he will never understand why ) .

G . H . Hardy ( in a letter to M . Rie s z )

Chapter 7 . Multipliers .

The general situation considere d in thi s chapter may be

described as fol lows . Let X be any quasi-Banach space o f

functions o f distributions i n En • Often one assume s that X

i s invariant for translations n ( i . e . fEX = > f ( x + y ) X , y E R ) X and - to make it symmetric - invariant for multiplication by

characters ( i . e . fEX = > eixn fEX , n E Rn ) , but this is of course

not at all necessary . We than also assume that the norm is

invariant . We ask what conditions have to be imposed on a and

b in order to a ssure that

( 1 ) f E X = > a * f X ,

( 2 ) f E X = > b f E X .

I f thi s i s so we say that a i s a Fourier multiplier and b

an (ordinary) multiplier . The reason for such a terminology

is of course that the convolution a* become s an ordinary A

multiplier a (= F a ) after taking Fourier transforms , by the

formula ( see Chap . 3 )

F ( a * f ) = F a F£ .

1 2 6

1 2 7

I f X satisfie s the above invariance propertie s trivial example s

o f Fourier multiplie rs are Fourier tran sforms of bounded

measure s ( i . e . a E FA or a EA where A i s the space of bounded

measure s ) and of ordinary multipliers inverse Fourier transforms

o f bounded measure s ( i . e . or

More generally one can conside r multipliers from one space

X into another one Y .

We discuss the problem o f Fourier multipl iers first . Of

course with the after al l rather simple minded tools we are

equipped with ( interpolation spaces , etc . ) we cannot hope

to settle al l problems , but nevertheless a certain insight in

these matters can be gained .

Generally speaking given X let us denote the space o f all

inverse Fourier tran sforms of Fourier multipliers by CX or

C (X ) ( i . e . , a E C X iff ( 1 ) holds ) . Clearly C X i s a quasi-

Banach algebra . The quasi-norm is given by - 1 I I a! I c x = /. I I a * f I I x I I fl I x

and Y we w�i te C X , Y .

I f we have two space s X

Let ljts first consider the case X = Lp ( 1 � p � oo ) and let

us write , \orrupting the notation j ust introduced , Cp = C LP .

Let us now l i st some classical e lementary facts about the

space s

( 3 )

( 4 )

c : p

F -l L ( inverse Fourier transforms of L ) 00

A (bounded measure s )

1 2 8

( 5 ) c P c i f 1 + 1 1 p ' - p' p

c c -1 if 1 � p � pl � 2 A + + + F L p pl 00 ( 6 )

In view o f ( 5 ) one can o ften re strict onese l f to either the

case 1 �p � 2 or 2 � p � oo We have also the following

use ful

Lemma 1 . We have

( 7 )

( 8 )

provided 1 - = p

Proo f :

proof of ( 8 )

1-e -- + Po To see

e pl

+ c p

+ c p

( 0 �

the idea

is s imil ar but

e �1 ) .

let us prove ( 7 ) only. The

require s the three line theorem .

Let a s ( c , cp ) 8 1 • P o 1 Then by de finition we may write

I f

00 a = l: a

V = -oo V 00

with l: v =-oo

f s L it fo·llows t.hat p

a * f = 00 l: a * f .

v =- 00 v

Using the convexity inequality

1 2 9

which is an immediate consequence o f Thorin ' s theorem , we

get

I I a* f I I L < =

p

<

00 V'J_oo I I aV * f I I L <

00 L: I I V=-oo

p

) I c�-8 1 1 a) I � I I f I I L 0 p l p

00 2 -v 8 J ( 2 v, c 0, C l l l l f / / L < L: av = v=-oo

There fore a * f ELP and , by de finition ,

We can now easily prove

Theorem 1 ( Hirschman ) . We have

( 9 ) n -• p F ' ( l

1 -+

In particu�ar holds

\ � , 1 ( 1 0) F : B P -+

p

1 I �

-1 c P i f 2 p

1

> I � - i I c P i f

p

a E C • p

Proof : By ( 4 ) and th . 3 o f Chap . 6 we have

Also ( 3 ) give s

n . 2 , 1

F : B1 -+

c / . 1 f / / L < =

p p

1 3 0

F : L00 -+ c 2

Thus by appl ication o f ( 7 ) o f Lemma 1 we get

-+ c i f p 1 p

1- e e -1- + 2 .

On the otherhand by th . 6 o f Chap . 5

1 1- e e

l5 == =r-+ 00 •

Thus ( 9 ) fol lows . To get ( 1 2 ) we have to invoke the (obvious )

Lorentz analogue o f th. 5 o f Chap . 3 . The proof i s complete .

Remark . Th . 1 should be compared with the JIIJ.ikhlin

multiplie r theorem. If p == 1 (Bernste in ' s case ) the conditions

on a is almost the same but not really . In fact th . 1 doe s

not apply to the Hilbert transform o r the Calder6n-Zygmund

operators and of course the conclusion is not true e ither in

this case . (Mikhlin says only a s CP i f 1 < p < oo ) • On

the other hand it is possible to prove analogous interpolation

analogue s of Mikhlin .

Vve now give a simple necessary condition for a to be

in C p Theorem 2 . We have

c p

n -- 00 • p I -+ B (i. p

n - - 00 • p B ' p

Proof : Let

But

1 3 1

Then in particular we must have

< I I a l ie p

I I ¢v I I L � C 2 p

n - v p l

< C I I ¢v I I L p

There fore we al so get

and n - 00

o p I a E: Bp

n - v p l I I ¢ * a I l L � c 2 • v p

In view of ( 5 ) now also follows

The proof is complete .

Corollary . Let a E: c P . Assume that supp

and -1 0 . Th n a E: Lp L p I o

Proo f : Obvious .

A a

n .- - , 00

a E: B p p i

i s compact

Since ( 6 ) 1 say , • Q l Bl -+ c we have thus enclosed c p p

so to speak b tween two Be sov spaces . The bounds are the

closer the closer we get to p � l or p � oo • Now we want to

show C p i s almost a Be sov space but "modelled" on C p itsel f .

( ll )

Theorem 3 . Assume that

00

v �- 00 I I ¢ * a I I P ) l/p < oo where v c P

1 3 2

Then l v * a I I \) p

In other words holds

( 1 3 ) :B0P c -+ C -+ BQoo C p p p where

< ()()

l I � -

l I · p = 2

In view o f that we at once get ( using al so th l ) the fol lowing

Then

Coroll ary . Assume that

()() v=-ao

a t: C • p

where l l l p = l p - 2 1 .

Be fore giving the proof o f th . 3 i t i s convenient to settle the analogous question for that we now have a much sharper re sult .

Theorem 4 • We have

at: C Bsq <=> sup I I ¢ * a I I C < oo p \) \) p

It is remarkable

In other words : In particular C s;q depends only on p . ( That i t doe s not depend on s is obvious ) .

1 3 3

• sq Proo f : Assume that a E: C B • p Then i f f E: Bsq we p must in particular have a * •S oo f E B p Therefore for each v

holds

a * f l I L � C I I f I I sq p Bp

Apply this to lj;v * f where { ''' } 00 i s another sequence '�'v v = - oo A o f te st functions chosen in such a way that lj;v = l on R v •

Then fol lows

( 14 ) 2v s I I cj>: a * f l l L � C 2v s I I f I I L p p

which clearly implies

( 15 )

Conve rsely i f the e stimate ( 15 ) holds for a and i f f E: Bsq , p ( 1 4 ) fol lows �nd ; taking the sum we get a * f E: B=q and

• sq a E: C B • Th s complete s the proof o f th . 4 . p l h C •Bs

lq = •BO

l oo

• Coro lary . e ave

Proof :

c 8�q = s0 00

Now to the

Indeed i f p = l we get

C = BQ oo ' = J!O oo L = • 0 00 l 11 l Bl

Proof o f th. 3 : Assume that the estimate ( l l ) i s ful-

filled with some p • Then by an easy adaptation of the proof

of th. 3 we show that

( 1 6 ) a E Sq sql C B B if p ' p l + l q p

1 3 4

(Use Holde r ' s inequality ) . Now let f E L • Then assuming p l � p � 2 which i s no loss of generality • 0 2 f E Bp , so by ( 16 )

• Op a * f E Bp and a fortiori a * f E Lp . There fore a E C p ·

That again a E C entail s ( 1 2 ) we leave to the reade r to p verify .

We now proceed to give some e xample s .

First we remark that the cor . of th . 4 contains in part-

icular e x . 2 of Chap . 2 ( the Hilbert transform in Lips ) .

Indeed we have by now seve ral criteria for

e specially th . 4 of Chap . 4 and th . 5 of Chap . 6 .

the l atte r evidently apply in this case ( i . e . a ( x )

; ( s ) = i sgn s ) •

Next we consider the following

Example l . Suppose

Both of l = p . v . x'

a < s) = I s I n Y in a neighborhood o f oo , with n > 0 , y > 0

and C oo e lsewhere . Then it is easy to see that

I I ¢v av I I n < 1 -. p Bp

n v m ( -p - Y ) c 2

An application o f the cor. to th . 3 now yields a E C p

1 3 5

provided n On the other hand i f n and m 1 1 P > - p < -p p

then a ¢ c . It i s also possible to show that a s c in p p the limiting n but this much harde r . This shows case y = -

p

however that our re sults obtained in this rather straight

forward way in a sense nevertheless are almost sharp . We

return to thi s point in a moment .

PROBLEM. Are the exponents p and oo best possible .

At least it is known that c p • 0 00 c f B p i f p ,. 2 .

i s not even true that 0 00 c Bl

-+ p i f ,. 2 .

Now we give a quick survey of some re sults for e x , Y

extending the above once obtained for e x . By a repeated

corruption of notation we write C C L , L We p , pl p pl

I t

assume p � p1 for it is easy to see that indeed C = 0 p , pl i f p > p1 • We omit most o f the proofs or j ust give brief

hints .

Theorem 5 . We have

• s lq C BS B p pl

Proo f : T e same as

Theorem 6 . we have

s - s 00 • 1 c B P , P1

for th . 5 .

provided 1 max (p1 , 2 )

In particular holds

< 1 1 + p , 1 < p , pl <

min ( 0 , 2 )

• 0 00 B C p, pl

Proo f : The s ame as for th . 6 .

Theorem 7 . We have

Bsq -+ C i f 1 p p*p* p-* l l

l p*

1 3 6

Remark . This should be contrasted with the case

p

p 2 .

As we have remarked not even 00 -+ cp is true un les s

Proo f : By the Lorentz analogue o f the Be sov embedding

theorem ( th . 5 o f Chap . 3 ) we get

polation does the re s t .

Corol lary . We have

B-s , oo -+ l i f l p

Inter-

This may be considered as a very general form of the

Hardy-Littlewood theorem for the potential s

of Chap . 2 ) .

Theorem 8 . We have

-+ B n oo - P ' ,

Pl

Proo f : The same a s for th . 2 .

- s I ( see ex . l

Now we return to case of C X , more speci fically

1 3 7

c • p

We want to give a brie f discussion of the ce lebrated (Fourier )

multiplier problem for the bal l .

Let us adopt for a moment the standpoint o f classical

Fourier analysis . There the problem was to give a meaning

(other than the distributional ) to Fourier ' s inversion formula

f ( x ) ( 2 Tr ) -n

for , say , f s L1 or more generally f s Lp ( l �p �2) . The A

trouble i s that f i s not very small at oo Le t H be a

given homogeneous positive function and u some given

function with u ( O ) = l . Then one is lead to consider as

approximations to the integral , "means " of the type

A

( 2 Tr) -n J n m. I

A

f (� ) d �

I f a (t_; ) = u (H ( t_; i s a Fourier multiplie r in Lp ' fol lows

then by routine a guments

I I f - f I I L -+ 0 as r -+ oo • r p

A particularly important ca se is when u i s the characte ristic

function o f the interval [ O , l l so that a (U is the characte ri s t i £C

fun ction of the set E = { H (t_; ) _s_ l } ....:: nP . have

In this case we

f (x ) = f ( O ) ( x) r r ( 2 'IT) -n H ( r,/ � r e ix E, f( E,) d E, (partial

1 3 8

Fourier integral )

A somewhat more general case i s u ( A ) ( 1- A) � (with

t+ = max (t , O ) ) in which case

f; a) ( x ) = ( 2 'IT ) -n f eixt, ( 1- H �E, ) ) � f ( E, ) d E,, a2: 0

( Ries z means ) .

We agree to write � ( E, ) = ( 1- H ( E, ) ) a (with the obvious a r + interpretation in the limiting case a = 0 ) .

Two important special cases are :

a ) E i s a cube o r more generally a convex polyhedon .

b ) E i s the unit ball (with e . g . H ( E, ) = J t, J 2 ) or more

generally any strictly convex set .

It turns out that these two cases behave quite differently

and that it is the differential geometry ( curvature ) of the

boundary a E o f E that causes that di ffe rence .

The case a ) i s easily handled. We j ust notice that in

this case a = ao i s the product o f the characteristic functions

of a finite member of hal f spaces C::. JRn E,

consider the special case :

a ' ) E is a hal fspace

Thus i t suffice s to

With no los s of generality we may then assume that

E ={ E, 1 = 0 } so that 1 p . v . - • xl

and = -i

1 3 9

Thus i t simply suffice s to apply M . Rie s z ' theorem . We

conclude that a s cp i ff 1 < p < co

The case b ) i s much more compl icated . One can express

a0 ( x ) (as wel l as a a (x ) ) explicitly in terms of Bes se l

functions . Using the classical asymptotic formula for Be ssel

function s it then fol lows

C cos ( l x I + 8 ) n + 1

2 + 0 ( 1

n+3

lx I -2-

( One can also give a direct computation which perhaps better

reveals how the geometry of 3E comes in . ) I f we now invoke

the cor . o f th . 2 we get as a nece ssary condition for

s c : 0 > � - 1 where 1 =I � - l I The applie s ao p p 2 p

. same to

a a We now find that i f a s C then a > n a P P

1 2 " We next

ask if this is a sufficient condition . (This is (was ) the

mul tiplier n-1 problem) . Th . 1 readily give s a > -P- ( Stein) . . n-1 > S lnce -P- n

p - 2 or p < .:!:. 2 we are left with a gap .

has begun to be f ' lled up only recently . First Fe ffe rman

It

displayed a counte -example showing that a0 cannot be a

multiplier unle ss p = 2 . He al so showed a positive result ,

that if a > n p

1 2 and in addition 1 1 1 - > - + --

p 4 4n then indeed

Later Carle son-S j 6lin and Sj� lin in the special

case n = 2 and n = 3 respective ly re laxed the l atter con-

dition . In particular i f n = 2 the problem has been com-

ple tely settled. The fol lowing figure s illustrate the case :

1 4 0

We now give a simple proof o f the original re sult by

Fe ffe rman pre sumably due to Stein .

141

The key to the proof i s provided by the fo llowing :

Lemma ( Stein ) . Let S be the unit sphere in R1 . Then

we have

F Lp -+ L2 (S ) provided 1 < 2

where L2 ( S ) o f course denotes L2 with respect to the area

e lement dS� on s .

Proo f : De fine a_1 by

Then holds for a_1 a similar asymptotic formula as for a0 n-1 ( and a

a' a .:_ 0 ) , only the exponent is now -2-. It follows that

l" ff n-1 > n- . h a 1 E L -- To prove our assert1on we ave to - p 2 p e stimate

But i f f �::L then p

A 2 f I I a_ 1 = C < f * f , a_1 >

f * �::L i f q 1 q � - 1 , by Young ' s inp

equality. Hence by Holde r the above integral is finite pro-

vided

2 - 1 + p

1 1- -n -2- > 1 or .!. > � + 1

p 4 4n

1 4 2

The proof i s complete . Now i t is easy to e stablish

Theorem (Fe ffe rman ) . Assume a > � -1 ! > ! + 1 (where p 2 ' p 4 4n

l p I �

- � I ) . Then aa E CP .

Proof ( Stein ) : In view of the asymptotic formula

(explicitly given only when a = 0 ) we may as well suppose

s = a- n + ! > 0 and that a = a is defined by p 2 a

a (x )

Write

a ( x ) = a cos l x l ( in a neighborhood of oo ) •

l x l n;l + a

A = q, a , av = <jJ v a

with q, , { <jlv }�=O as in chap . 3 but for a change de fined in

lRn = lR� (not lR� ) . I t now suffices to show that

(1 7 )

with s > 0 . S ince supp ave Rv we may in considering a * f \) assume supp f C Kv • (A * f cause s no trouble ! ) We further

assume p < 2 . From Holder ' s inequality and Plancherel we get

c 2 n \) -p

c 2 n \) -p I I av * f I I = L 2

1 4 3

We next observe with Fe fferman that the main contribution to

�v ( � ) come s from the set R0 = { 2 -1< 1 � 1 � 2 } . There fore

we have

I I av * f I I L � C p

v� 1/2 2 P ( ! I �v ( � ) 1 2 d� ) sup ( ! I f ( 0 1 2

2 -1 <r<2 ( modulo Fe fferman )

1/2 ds � )

At last invoking the lemma and again Plancherel we get

n v -I I av * f I I L � C 2 p

p

v � -v (n+l + a ) v � < c 2 p 2 2 2 2

( modulo Fe ffe rman )

=

There fore ( 1 6 ) follows and he proo f i s complete .

Remark . It fo s from Fe f fe rman ' s theorem that for

radial functions th . 1 can be improved somewhat . I f

when n 1 d ' f a > - - - an 1 p 2 u = u ( x) i s a function with compact

support <: ( O , oo ) then setting � ( � ) u ( l � l > it follows that

n u E: p p

1

1 + E: p

while as th . 1 give s only

n-1 u E: p p

1

Notice again that n-1 p

+ E:

> n p

( E: > 0 ) = >

( E: > 0 ) = > a E: cp .

1 2 ( I f ao E: cp were true we

would have

u E A = >

but thi s is so only i f n

matters in Chap . 1 0 .

a E C p

1 4 4

l or p = 2 . ) See more on the se

In order not to confuse the reader we s tate explicitly

that the pre ceding discussion of the multiplier theorem for

the bal l , through important by i tse l f , was only a digre ssion

which had not much to do with our main theme , the study of

Besov space s . That i s why the previous theorem has not even

got a number .

We leave now our discus sion o f Fourier multipliers and

turn to our second ob j ect , that of (ordinary ) multipliers .

Generally speaking the set o f all mul tipliers in X will

be denoted by M X. (We only consider the case of j ust one

space . ) I f X i s a Banach space o f functions or distributions

in lRn it is easy to see that M X ' = MX whe re X' denote s

the dual space . There fore in discussing Be sov or po tential

space s we may re strict attention to the case s > 0 , the

case s < 0 thus being captured by duality . We omit the

discussion of the intermediate case s = 0 , which however

ought to be rather interesting by itse l f . Let u s also notice

that MLP = L00 , l � p � oo , as i s easy to verify .

Our first re sult is :

Theorem 9 . For any s > 0 ( and l � p � oo 0 < q � oo ) holds :

1 4 5

Proo f : Let { <Pv \>:=O and 4> be our usual te st functions ,

normal ized by

Let f t: Bsq p

f

00 I ¢ + 4> = 6 .

V=O V

and b t: Bsq •

00 F + I v=O

f v

00

b

Then we have

00 B + I bv v=O

,�, * f F '�'v ' 4> * f and s imil arly for b . Putting

g = bf we thus get

( 1 8 ) g 00 00 00 00

BF + B vEo fv + w�o bwF + vE o wEo bw fv I

We want to estimate ¢ A * g . (T e estimation of ¢ * g causes

no trouble

attention

e stimated

certain of

since i t is clear tha

to the terms ¢ A * b f w v in the same way ) . Since

the se terms wi ll be 0 .

W v A 2 + 2 ,;, 4 · 2 '

g t: Lp . ) Let us restrict

( The other ones can be A A

supp bw R and supp f Rv w v There are those for which

as can be seen by a s imple geometric consideration . Thus

we have roughly two cases to consider 1° v > A and

14 6

and 2° 11 � A o In case l 0 , tak ing first the sum over 11 ,

since I l L: b11 l l L � C , we get the contribution 00

I I fv I I L o

p

Simil arly in case 2° , this time taking first the sum ove r v ,

since I l L: fv I I L < C , we get p

Thus the over all contribution to

( 19 ) I I fv I I L + C 2 A s l: I I g 11 I I L p 11�A oo

Together with similar e stimate s for the other te rms we

easily see that

and thus gEBsq p 0 In an entirely similar way we can obtain

even sharper

Theorem

and de fine p

resul ts .

1 0 . Let o < s < � = p '

by n Then s = p

Bsq () L00-> M Bsq p p 0

assuming also l < p � oo

holds :

1 4 7

Proo f : The proof given for th . 9 has to be changed at

one point only . Name ly in case 2° we get , using Lemma 1 of

Chap . 3 , for e ach term , the e stimate

v (� - � . ) C I I blJ I I L I I fv I I L ,:S c I I blJ I I L 2 P P I I fv I I L

p p ' p p

v ( � - s+n ) p ' 2 -v s n \) p ' 2

Thus ( 1 9 ) has to be replaced by

( 1 9 I ) I I fv I I L + C 2 A s p

and the rest of the argument is the same .

It i s also po ssible to prove the following result.

Theorem 1 1 . For any s > 0 Bsq L i s a (quasi- ) Banach p 00 algebra . In particular Bsq is a (quasi- ) �nach algebra i f

q "" 1� In particula\ holds thus i n this s > � p case :

or if

Proof :

show that g

n s = p I

Let b and f both be in

\

Bsq L p (i 00 We want to

bf i s in the same space . We again use ( 1 8 )

but now we can treat b and f in a symmetric manner . Indeed

we get exactly the same e stimate in case 2° as in case 1° .

To be pre ci se in place o f ( 19 ) there appears now the contribution

1 4 8

( 19 " )

Remark 1 . A sl ightly weake r re sult than the one in

th . 10 can be obtained by interpolation by � 1

two endpoint re sul ts L = L and MB P p 00 p

leave the details for the reade r .

starting with n 1

-+ B p We p

Remark 2 . Not al l o f the above re sults extend to

Bsq Why? However p •

n particular thus BP

p

th . 11 indeed doe s generalize . 1

i s a Banach algebra .

In

Remark 3 . We have conside red only Be sov space s .

Analogous results hold for potential space s Ps . One has p

the

then to make use o f the Paley-Littlewood type repre sentation

of the norm.

\ve now ask how good are the above resul ts . We observe

that the function 1 certain ly is in MB;q whatsoever the

parameters are . At first sight this might be thought to be

a di sappointment , for 1 cannot belong to a Besov space (unless

p = oo ) • However we shall see that our results at least

locally are of precisely the right orde r .

We shal l introduce a new type of partition of unity in

( C f . e . g . the treatment in Chap . 4 and 5 . )

We choose a net of dis joint paral lel congruent cubes { I }

fil ling up JRn • We construct a family { ¢ I } of test functions

such that

ct>I E c""

supp cp i 2 I I

i f X E: I I

for each a •

We also construct a second such family { �I } such that

( 2 0 ) 1

We next de fine l inear mappings

In view of ( 2 0 ) we have

S o T id

It is easy to verify that

1 4 9

Here generally speaking i f { AI } i s a "bundle " o f space s

AI (ove r our net of cube s ) we denote by �p ( { AI } ) the space of

(s > 0 )

For S we have to reve rse the -> . In particular holds

150

By dualitie s we can include the case s < 0 and by a new

interpolation the case s = 0 . We may summarize all this in

the following commutative diagram :

.Q,p ( { Bsp ( I ) } ) p

or by saying that B;p i s a retract of .Q,p ( { B;p ( I ) } ) . From

thi s we can readily re ad off ( this corre sponds to th . 4 in

the case of Fourier multipliers ! )

Theorem 12 . We have for any s :

b E: MBsp < => p sup I I cp i b I I < 00 I MBsp

p

We leave it to the reader to supply the particulars .

In particular we get :

Corollary . I f s > !:.. then p

b E MBsp <= > p sup l l b l l sp I B ( I ) p < 00

1 5 1

PROBLEM. To extend the above to the case of Bsq with p general q .

Remark . The above , notably th . 1 2 and its corollary , s generalize to the case of P , 1 < p < oo • p

We conclude this chapter by giving an application to

partial di fferential equation s .

Example . Let A be a partial differential operator of

degree m in lRn which is uni formly elliptic in the sense

that the following a priori e stimate holds

Here I I · I I s I I · I I l < p < 00 , or I I · I I The

constant C depends on u but not on I . Then follows

I I u I I s < c I I Au I I s -m + c I I u I I s _ 1 •

One can use this to show the fol lowing regularity theorem :

I f u E Xs-l and Au E Xs-m then u E Xs , where we have written

Xs Ps or Bsp . As an application of the appl ication we mention p p

1 5 2

the following re sult : I f A is formal ly self adj oint then A is se l f-adjoint in the Hilbert space L2 ( i . e . the corre sponding

minimal ope rator is se lf-adj oint) .

1 5 3

Notes

Theorem 1 can be traced back to Hirschman ' s paper

[ 9 7 ] ( c f . the discussion in Peetre [ 9 5 ] ) . For inter-

polated version ' s of Mikhlin ' s theorem see likewise

Peetre [ 9 5 ] , and Littman [ 9 8 ] . Th . 2 probably belongs to

folklore , cf . e . g . Johnson [ 6 9 ] . ( I myself learned it

from Hormander (personal communication ) . ) Th . 3 i s

mentioned in Peetre [ 1 8 ] . The counterexample to C � B000 C p p can be found in Ste in-Zygmund [ 9 9 ] and also in Littman-

McCarthy and Riviere [ 8 0 ] . Be fore that I stated it as

an open problem in both [11] and [55] but nobody seems to

have noticed i t ! Th . 4 i s s tated ( in a diffe rent form) in

[ 9 9 ] but it really goe s back to Hardy-Littlewood ( c f .

Taibleson [ 15 ] , Pee tre [ 1 8 ] , [ 5 5 ] . Concerning the ( limiting

case Y = % of ex. 1 see Fe fferman-Stein [ 3 6 ] . \ The multiplier problem for the bal l has its or�'in

the work of Bochner ( see Herz [ 1 0 0 ] ) and L . Schwartz [ 10 1 ] .

Stein ' s resul t ( a > (n-1 ) / p ) is in [ 1 0 9 ] . A brie f summary

of the present situation including new proofs of the Carleson-

S j olin and S jolin re sults can be found in [ 10 3 ] . Our treat-

ment is al so taken over from [ 10 3 ] . Ste in ' s lemma occurs

already in Fe fferman ' s original treatment [ 1 0 4 ] . Let us also

remark that the analogue of the multiplier problem for certain

classical unidimensional expansions ( Legendre se rie s etc . ; cf .

Chap . 10 ) was completely solved much earlier ; see e . g .

154

Pol lard [ 10 5 ] , Wing [ 1 0 6 ] , Newman-Rudin [ 1 0 7 ] , Askey-

Hirschman [ 10 8 ] , Herz [ 1 0 0 ] .

The fact that potential and Besov space s are algebras

for mul tipli cation

folk theorem leve l .

i f s i s suf ficiently large is in the

The insight that � is an algebra i f p k > � i s said to originate from Schauder but I do not know p the e xact re ference . The homogeneous analogue of th . 11

( see remark 2 ) i s in Herz [ 6 6 ] . Multipliers for potential

space s have been considered by Strichartz [ 1 0 9 ] . He has in

particular the same partition of unity and he note s the

localization property ( th . 1 2 ) . The application to partial

differential equations is from Peetre [ 1 1 ] ( c f . Browder

[ 11 0 ] ) .

Quotation : Why don ' t you quote me ?

Anonymous

Chapter 8. Approximate pseudo-ide ntities .

1 5 5

I n the previous discussion the de finition of Besov space s ,

used the very particular test functions { � } 00 ¢ '�' v v=O ' (or { <Pv } oov=- oo in the homogeneous case ) , forced upon the

reader via the heuristic argument given in Chapter 1 . Now

we want to re lax the conditions imposed on the te st functions .

Thi s will give us a much greater degree o f flexibility in

many situations . Remember al so that in Chapter 1 we mentioned

equivalent de finitions using finite di ffe rence s or the Poisson

integral , specifically /':.. hf ( case a . ) and d U t () t ( case f . ) . We

now want to tre at this from a rather general point o f view

pertaining to approximation theory . I By an approximate identity one usually means a se�uence

{ ar } such that ar -+6 as r -+ oo , in some sense or oth�.

Then a * f -+ f holds in a corre sponding sense , under su · table r assumptions on f . To fix the ideas let us always assume that

a i s obtained from a given function by dilation . r

rn cr( rx ) or �r ( E, )

Then we must have

J cr ( x ) dx JRn

1 or � ( O ) = 1

15 6

Let us now drop the latter requirement . Then we get an

approximate pseudo-identity -- we have to have some name for

it so this is as good as any other . Let us give some example

of how approximate pseudo-identitie s arise . Let or be any

approximate identity { restricted by the above homogeneity

assumption , by convenience ) . Then 0 T - o i s an approxi-r r pseudo-identity . or { I; )

A 5_ ) mate We have = T { - 1 . Clearly r now � * f -+ 0 . More generally we may consider T given by r

� { ];. ) -r

for some k .:_ O ; the C a are suitable factorials . It i s

conceivable that this o tends to 0 the faster the bigger r k i s . This leads u s to the general problem which i s the one

which will be considered in what follows :

mate pse udo-identity and give n f E: Lp , what

the rate of convergence of l l o r * f I I as Lp OUr first re sult in this sense reads

Theorem 1 . As sume that ·-s 0 E: B1 1 and

some s . Then

Given an approxi-

can be said about

r -+oo

f E: Bs oo for I p

{ 1 ) 1 1 ° r * f I l L O { r-s ) as r -+ 0 or ()()

p

holds . In particular -s oo • 0 0 sB1 n

- s co • 1 Bl with

the condition on 0 i s fulfilled if

s E { s0 , s 1 ) .

1 5 7

Proo f . We could use one o f the multipl ie r theorems of

the previous chapter but for simplicity we write down a direct

proof . I f { � } 00 i s our sequence o f te st functions obeying '�' v v=- oo the normali zation assumption

we clearly have

Thus we obtain

00 I <P * <P = o \) =- oo \) \) I

00 \) =- 00 ( <P * 0 ) * ( <P * f ) v r v

But i f r :::: 2 lJ then <P * 0 i s essentially <P + * 0 • v r V J.l ( The

reader should himsel f make this point more rigorous . ) There-

fore we also get

00 \) =- 00


- V S 2 I I <P * 0 I IL < c \) 1 --s r

Next we prove a re sult in the opposite sense . Here we

need a Tauber ian condition on 0 •

( 2 )

Theorem 2 . Assume that 0 E B- s oo and that l

{ cr "I 0 } _::_) R ( l )

I f f E S ' and if ( l ) i s ful filled then f E Bs oo p •

1 5 8

Proo f . Choose r = 2 v • By the famous theorem o f

Wiener ( 2 ) implies that we can divide <Pv by ar 1 i . e . we have

Then

<P = l/! * 0 v v r with I l l/!) I L .::_ C . l

<P * f = l/J * ( 0 * f ) v v r

al so holds and we at once obtain

I I <Pv * t i l L .::_ l l l/J) I L l i ar * t i l L .::_ C 2 -vs p l p

There fore • s 00 f E B and the proof i s complete . p More generally we have

Corollary. Assume that l 1 2 1 • • • 1 N ) and that

( 2 I ) N A V { 0 . "I 0} R ( l ) •

j = l J

I f f s s ' and i f ( l ) 1 with 0 replaced by 0 j 1 i s fulfil led

for e ach j ( j = l 1 • • • 1 N ) then • s 00 f s B • p

159

Remark 1 . We have in the foregoing characterized the space matter • Sq B • p

( 1 ' )

Bs 00 , thus taking g p of routine to extend

Then ( 1 ) o f course has

= 00

the to

However , it re sult to the be replaced

1 dr ) g < oo r

by

is only a case o f general

In what follows for convenience we go on considering the special case g = oo only . This is also the most natural one from the point of view of approximation theory .

Remark 2 . Let us return for a moment to theorem 1

taking for simplicity s = 0 , p = oo • In view of theorem 2 we see that 0 in ( 1 ) can not be replaced by o It is not even possible to assert that crr * f (x ) tends to a l imit at any point x E: JRn

• Here is a counter-example . Take n = 1

and assume also for simplicity that supp ;C:R ( l ) but ; ( l ) = 1 f 0 . Choose f of the form

f 00 L: v =l

ix2v e f \)

with l l f ) IL oo .::._ c , supp fvC K ( � ) . The convergence of a * f ( x ) at one point x E: m. would then imply the existence r o f lim f ( x ) but it is easy to arrange that this limit never

\)->-00 \) exists .

Now some concrete illustrations for th. 1 and th. 2 , along with the latter ' s corollary .

160

0j ( E J is; .

Example 1 . == e J - 1 ( j == l , • . • , n ) i . e .

- 8 * f ll te . f ( t -1 The jointly oj r == 0te . o . == == r ) • o . ' J r J J J

satis fy the Tauberian condition ( 2 ' ) . Also O . E: A J Indeed that o . E: A ( bounded measure ) J

· -1 co i s sel fevident and to

i S . show that o . E: B we J· ust remark J co that (e J -1 ) / s . is the J inverse Fourier transform o f the characteristic fun ction o f

an interval . There fore , the opera tor ll t can be used to e . characterize

In particular

• s co J Bp as long as 0 < s < 1 , i . e .

s 0 ( t ) as

• s co B == Lip i f 0 < s < 1 . p s

t -+ 0 or co ( j ==l , • • • , n )

Example 2 . ,., is . 2 o . ( S ) == ( e J - 1 ) ( j =l , • • • , n ) . J This case

is entirely similar . We find , as long as 0 < s < 2 ,

fE: 8 s co<=> I l ll2 f I I p te . L J p

Example 3 . 0 ( s ) = I t;, le- I t;, 1 . This time we find , as long

as s < 1 ,

co

Here (as in Chapter 1 ) u denotes the Poisson integral o f f .

This settles then cases a . , d . and f . o f chapter 1 . The

reader can now certainly display by himsel f the proofs in the

remaining case s o f the l i st a . - j .

16 1

We can now also think out as general i zations various

new cases .

Example 4 . A

0 . ( €; ) J

Now we must take k - 1 < s < k . We then obtain a de scription

of Bsq using the kth order Taylor remainde r . p Next an example pertaining to approximation theory .

Example 5 . We take a = T - 8 where T i s an approximate A

identity , in the sense that T ( O ) = 1 . More speci fical ly i f

H ( €; ) i s a given homogeneous positive function , say

H ( €: ) = l €: lm, we fix attention to the case ( c f . chapter 7 )

A

T ( €; ) u ( H (€; ) ) ,

with u ( O ) = 1 ( to assure �(0 ) = 1 ) . I f a u ( A ) = ( 1- A ) + we

obtain the Rie sz mean s , i f u ( A ) - A e the Abel-Pois son means .

Assuming that in addition u ' (0 ) f 0 ( thus a kind o f second

order Tauberian condition ) and imposing i f nece ssary conven• -sl ient regul arity assumptions on u , we may veri fy that a s B1

i f s < m . We obtain in this the fol lowing resul t : I f s < m -s • Soo then l i ar * f I lL O( r ) , r __,_ oo i f f f s Boo • In other

p words , somewhat loosely speaking , we get an increased orde r

o f convergence by increasing the regularity of f . But. i f we

are approaching s = m no matter what we do we cannot get

beyond that l imit . We say (with Favard) that the approximation

i s saturated with the satura tion index s = m. We il lustrate

it by the figure be low :

1 6 2

.5

Leaving the specific example aside we prove now some of

the s implest re sults connected with saturation :

Theorem 3 . Let H ( � ) be a given homogeneous function , 00 A

positive and C for � 1- 0 and de fine A by A f = H ( � ) f (O

(i) Assume that

0 ( � ) = n c� ) H ( � ) , 11 s 11.

Then i f f s S ' with

( 3 ) I I o * f I lL r p

Af s L it fol lows that p

0 ( r -m) , r -r 0 or oo ( = ( 1 ) with s = m)

( ii ) Conversely i f ( 3 ) holds true and i f 11 ful fils the

Tauberian condition n (O ) f. 0 then Af s Lp provided

1 m and if again n (0 ) f. 0

then f must be a polynomia l .

Proof : ( i ) We have

0 * f r

There fore we get at once

r -m 11 * A f r

I I 0 r * f I I L ..':. r -m I I \ I I A I lA f I I L ..':. C p p

-m r

( i i ) By Wiene r ' s theorem there exists w c: A such that

"' "'

n ( t; ) w ( t; ) l in a neighborhood of 0 .

16 3

Let <Pc: S be a given test function with � ( 0 ) = l and with A

supp contained in a sufficiently small neighborhood of 0 .

Then we have

"'

 * w * ( 0 * f ) r r r

In other words the family { \- * A f } r> O i s uni formly bounded

in Lp . By weak compactness and <Pr + o in the di stributional

sense , it follows easily that Af t: L or A • p

( i i i ) Now the s ame argument gives

I I <Pr * A f I lL :5._ C rm-s p

- E: C r 1 s > o

16 4

I t fol lows that A f 0 . There fore supp f { 0 } and f i s

a polynomial .

Remark . Perhaps we now al so understand better why Lip1 i s not a Besov space . In fact ( the argument of ) theorem 3

shows that Lip1 = w: 1 which is of course wel l-known . In

the same way we find Lip1 = w1 l p p 1 < p < oo 1 as we 1 1 as

w\ which is also wel l-known .

Let us return to the study of Be sov space s proper . We

wil l now make use of the circumstance s that we can allow test

functions { <Pv }:=- oo having compact supports . For re ference

let us state the fact that we are going to use as a theorem.

( 4 )

( 5 )

( 6 )

Theorem 4 . Let { <Pv }voo=- oo be a sequence of fun ctions in

such that

Jx a sup ()() \) 2v s i i <Pv * f i i L < oo •

p

Proof : Use th . 1 and 2 .

Example 6 . The s implest way o f obtaining such a sequence

1 6 5

i s by dilation starting with a given one ¢ E: S , satisfying ( 5 ) ;

¢, } x ) = 2\Jn ¢ ( 2 \! x ) .

We give now several appl ication s o f the new definition

of Besov space provided by th . 4 .

Example 7 . Let lJ be a positive Radon measure in mn •

Then lJ E: .8:, (n-d) ' oo where 0 < d < n , i ff for eve ry ball

K = K (x , r ) in lRn with center at x and radius r holds

( 7 ) d r

Indeed by th . 4 and ex . 6 we have lJ �:: .B:( n-d) ' oo i ff

( 8 ) ! ¢ ( 2 \! (x-y) ) d lJ ( y ) .::_ c 2 -\J d

for a given ¢ E: S satis fying ( 5 ) • But since s < 0 condition

( 5 ) i s in fact void . There fore we may assume that ¢ is

positive with ¢ ( 0 ) = l , say , and supp ¢ i s contained in the

unit ball K = K ( O , l ) . I t i s now readily seen that ( 7 ) and

( 8 ) are equivalent . Me asure s lJ satisfying condition ( 7 ) ,

sometimes known as d-dimensional , play a great role in potential

theory , in connection with d-dimensional Hausdorff measure .

Example 8 . (proof o f th . 4 o f chap . 4 ) . We wi sh to prove

that conditions ( 5 ' ) , ( l l ) and ( 1 2 ) of chap . 4 imp ly

(We know by now that this indeed imp lie s A a �:: L ! ) 00 We have

to estimate

I I ¢ * a i i L = ! I ! ¢ (y ) a (x-y ) dy i dx v l v

1 6 6

where ¢v s atis fies ( 4 ) - ( 6 ) , with k = l . We bre ak up the

integrals into parts , one with integration over mn'K-v+l •

the other with integration over K 1 • Using ( 5 ' ) o f chapter 4 -v+ and ( 5 ) we obtain

f f ¢ (y ) a ( x-y) dy i dx v

I !K · ¢v (y ) ( a ( x-y) -a ( x ) ) dy i dx < - v

< c 2v n JK ( J I a ( x-y) -a ( x ) I dx ) dy - v nP ·, K -V+l

< c 2 vn !K dy � c < 00 -\)

Simi larly using ( l l ) and ( 12 ) of chap . 4 and ( 6 ) we obtain

f i f ¢v (y ) a ( x-y) dy I dx = K-v +l

f I f ( ¢\! ( y ) - ¢v( x ) ) a (x-y) dy I dx � K_v +l K_v +x

< C 2- v (n+l ) f K I y I I a ( y ) I dy dx < -V

< c 2- v (n+l ) 2 - v f K dx < C < oo -v+l

These two estimates together yield

and ·O oo a t: B1 •

I I <Pv * a I I L < C < oo 1

1 6 7

Example 9 . (proof of th . 7 of chap . 5 1 completed) . We . 1 00

wish to prove that V -+ BP 1 i . e . 1 the third -+ in ( 1 1 ) of p p chap . 5 . It clearly suffice s to show that i f

1 f E V p then

• -- 1 00 g = f ' E B p I

p We choose the family of dis joint intervals

Ik in a special way , namely taking

(k=O , :t_l , :t_2 , 2\)

. . . ) .

Since

<Pv * g ( x ) = J <jlv( x-y) f ' ( y ) dy

= f <Pv' ( x-y) f ( y ) dy =f¢v ' ( x-y ) ( f ( y ) - f (x ) ) dx

it is e asy to see that

sup X E l k

I <Pv* g (x ) l

I t follows that

I I <P * \)

< c 2

g i l L ::._ ( p

\) ( 1 p

< c 2 v sup a ,bt: I k

l f ( b ) - f ( a ) 1

1 00

g (x ) l ) p 2 -v ) p 2:: ( sup I <P *

x t:Ik \) v=- oo

1 ) 00 2:: sup

v=- oo a 1b Eik

<

< c 2

which is what there was to prove .

Example 1 0 . The technique of ex . 9 can be used in

1 6 8

connection with several other clas ses of space s re lated to

Vp . We intend to return to this in the last chap . 1 2 . Here

we consider only the space Lp A. of Stampacchia . �'Je pre fer

here the notation Bs ; P • (We have Lp A. = Bs ; P with

A.= n + s p , p = P • ) We say that f E: Bs ; P i f f for each ball

K = K ( x0 , r) holds

1 ( 9 ) £:. I f ( x ) - n ( x ) I P dx ) P .::._ C r s

for some polynomial Tr , depending on K , of degree < k , where

k i s a fixed intege r > s , and C < co independent of K . We

claim that

( 1 0 ) s · p • S co B ' -+ Bco •

To prove ( 1 0 ) it suffice s to notice that ( 9 ) i s equivalent

with

( 11 ) s s�p I I a r * f / I L .::._ C r

co

where a runs through the set o f all functions

a t: L p I

1 + 1 p p' = 1 > with I I a I I = 1 L P '

satisfying conditions

1 6 9

corre sponding to ( 4 ) and ( 5 ) ; of course , the connection

between o and o is o { x ) = r r j ust essentially the converse

1 X -n o (

-)

. r r of Holde r ' s

plain that ( 1 1 ) in particular implie s

Indeed this is

inequality . It i s

for our sequence {1\ ) of test functions . o S CD This proves f E: B CD .

I f s > 0 the -+ in ( 1 0 ) can be reve rsed , in other words we

have B s ; P = B�oo irrespective ly of P . (Campanato-Meyers ) .

I f s = 0 we can only prove that s . p B ' is independent of P

( John-Nirenberg) . Thi s is the famous space B . M . O .

remark that obviously f E: L00 implies f E: B . M . 0 .

we have the embeddings

L CD -+ B . M . O . -+

Let us

so that

Now we want to introduce some maximal functions related

to our spaces .

Let us begin with a general orientation . Maximal

operators usually arise in conne ction with a . e . convergence

of families of l inear ope rators . To be specific let { Tr } r > O be a fami ly of continuous linear operators Tr : E -+ F where

E is any abstract quasi-Banach space and F a quasi-Banach

space on a measure space s-6 carrying the measure J.l ; it i s

assumed that l h l ..:::_ l g l , gE:F=>h E: F ( i . e . , F is a "quasi-Banach

function space " ) . Then we de fine the corre sponding maximal

operator by

M f ( x ) = sup I T r f ( x ) I f E E r > 0

1 7 0

Then H is no longer linear . But if we assume that H : E+F

at least is continuous and i f we know that lim Tr f ( x ) exists

a . e . when f belongs to some dense subspace E0 of E then we

may conclude that lim Tr f ( x ) exists a . e . for e ve ry f EE . r >O

Indeed to this end let us set

N f ( x )

Then N : E + F i s continuous a t 0 ( since N f < 2 M f ) and

N f ( x ) = 0 a . e . for f E E0 • Our claim is that N f (x ) 0

a . e . for fEE . Indeed for any E > o choose f0 E E0 so that

I I f - f0 I I < E • Then i t follows that

I I N f I I 2_ C ( I I N ( f-f 0 ) I I + I I N f 0 I I ) 2_ C I If-f 0 I I + 0 < CE and

N f = 0 . There is also a converse o f the above result ( the

Banach-Saks theorem) .

Example 1 1 . Taking X n lR , fl Lebe sgue measure , let

f ( x ) 1 me as . K ( x , r ) f K ( X , r ) f ( y ) dy •

Then M is ( e s sentially) the Hardy-Littlewood maximal operator .

The Hardy-Littlewood maximal theorem simply says that

M : L1 + L1 oo and M: Lp + LP if 1 < p < oo • As a consequence

171

we obtain T f (x ) -+ f (x ) a . e . r for f E: L , l < p < oo , which is p - -

Lebesgue ' s ce lebrated theorem on differentiation of the in-

de finite integral .

Example 12 . Another famous maximal operator i s the

Carle son maximal operator related to the a . e . conve rgence of

h F . . l t e our1er ser1e s on T .

After this general remark let us return to our Besov

and potential space s . Using Paley-Littlewood type theorems

it is not hard to prove that

f E: Ps => sup l 0 * f l /rs E: L p r> O r p

under suitable assumptions on 0 • We now wish to prove

stronger results where the sup range s over a whole family

of te st functions 0 • Here is a resul t in thi s sense :

Then

Theorem 4 . Let n l sup P r > O sup l 0 * f I Irs t: L 0 r p oo

l p

holds where 0 runs through the set of all functions

Lp ' ( !_ + l l ) with l l 0 l l L , l , supp 0 C. K ( l ) 0 E: p' = p p

s n

and a. dx i f l a. l� s . I f n there is similar fx 0 ( x ) = 0 p > - a - s

result with p = 00 •

Remark . By the conve rse of the argument of example 1 0

we can also e xpre ss the re sult as fol lows :

•s f EP => sup in f p r> O TI f

K ( x , r )

1 7 2

l l f (y ) - n (y ) I P dy ) P /rsE L • p oo

I t fol lows readily that we have the Taylor expansion :

l (meas . K ( x , r ) f K (x , r )

( f ( y ) - C ( y-x )a D f ( x ) ) a a

Proo f : We want to e stimate

It suffice s to take x0 = 0 , r = l . S ince we have -s f = I g with g E: Lp . Thus we are faced with the e xpre ssion

f 0 ( -x ) f ( x ) dx f a( - x) f 1 g ( Y ) dy dx l x - Y l n-s

l f a( -x ) f ( n _ s I X - Y l

" C ( -x ) a D 1 ) g (y ) dy J /.., n-s � I a I� s a a I Y l

Here we have used the Taylor expansion for l l x - Y l

n s at the

point y . We divide the integral into two parts . First we

integrate over the set { l x I < } I Yl }

be e stimated by

The inner integral can

C l x l k+l f I g (y ) I dy < C I x is M g ( O )

I Y I > 2 l x l l z l n-s+k+l

where M is the Hardy-Littlewood maximal operator ( see

example 1 0 ) and k the intege r part o f s . Thus we get the

1 7 3

bound

C J I x I s I a ( x) I dx M g ( 0 )

In order to e stimate the integral over the complementary

set { l x l � i I y I } we conside r each term by i tsel f . First

come s the term

J a (-x ) f 1 g (y )

n-s dy ) dx = J a ( -x) h ( x ) dx l x i > 2 1 Y I I x - y l

Here by the theorem of Hardy-Littlewood ( see chap . 2 ) h E L p

and we get the e stimate

1 l l o i i L • l l h l l L 2_ C I Io i ! L , ( f l g (y ) ! P dy ) P

P P P K ( 2 ) 1

(M I g I p ( 0 ) ) p •

There remain the terms corning from the Taylor expansion . As

long as l a l < s there arise no complications and we re adily

get e stimates of the type C ! l o l l L 11 g ( 0 ) • However 1 i f 1

! a l = s and thus k = s we must use an auxiliary fact from

the Calde ron- Zygrnund theory not mentioned in chap . 4 1 name ly

that under suitable as sumptions on a holds

J a ( y ) f ( x-y) dy -+ J a ( y ) f ( x-y ) dy a . e . I Y I� E

for eve ry f E L 1 1 2_ p < oo p

1 74

and that we have good e stimates for the corre sponding

maximal functions . Espe cial ly if p > l ( and we are in this

situation ) this is not very difficult . We leave the details

to the reade r .

1 75

Notes

In writing this chapter we have obtained much inspiration

from the beauti ful work o f Shapiro [ 2 0 ] , [ 2 1 ] ( see al so

Boman-Shapiro [ 6 0 ] ) . (We are awfully sorry that we have not

been able to write in an equally lucid manner ! ) The use o f

Wiener ' s theorem i n particular i n connection with theorem 3

stems from Shapiro . We re frain howeve r from making a more

detailed comparison . We mention also Lofstrom [ 1 1 1 ] , [ 5 9 ] ,

[ 1 12 ] . Regarding the problem o f saturation we re fer to the

work of Butzer and his associates ( " die Butzer Knaben " ) , see

e . g . Butzer-Ne ssel [ 1 1 3 ] . The counter-example in remark 2 i s

due to Spanne [114 ] . I t has the following application to

partial differential equations : I f u is the bounded solution

of an e l liptic equation of order > 2 then its pointwise

boundary value s need not exist . Thi s should be contrasted to

the case o f second orde r equation ( Fatou ' s theorem etc . ) .

C f . Spanne [ 1 15 ] . In this context see al so Strichartz [ 1 16 ] .

Concern ing the use o f d-dimensional measures see Adams [ 1 1 7 ] ,

[ 1 1 8 ] ( c f . Peetre [ 5 6 ] ) . The spaces L P A we re introduced by

Stampacchia in [ 1 19 ] . The pre sent treatment fol lows the

survey article by Peetre [ 1 2 0 ] . The space B . M . O . was first

treated by John-Nirenberg [ 1 2 1 ] , whose paper al so contains the

independence o f p ( this i s the John-Nirenberg lemma ) . Here

" B . M . O . " i s usually interpreted as ( functions of ) "bounded

mean oscil lation " but it really stands for my children

Ben j amin , Mikaela and Oppi . The fame o f B . M . O . rose enormously

17 6

when Fe fferman-Stein [ 36 ] a few years ago identified B . M. O . n+l as the dual of the Hardy space H1 { m+ ) { see Chap . 2 and l l )

Theorem 4 i s mode l led on a resul t by Calderon and Zygmund [ 1 2 2 ]

{ see also Ste in [ 14 ] ) . The missing e stimate for maximal

functions can also be found in Peetre [ 1 2 3 ] . A good introduc-

tion to the entire subject o f a . e . convergence { Banach-Saks etc . )

are al so Cotlar ' s note s [ 1 2 4 ] . We further mention Garsia ' s

little book [ 12 5 ] which also contains a discussion o f Carle son ' s

work { re fe rred to in ex . 1 2 ) .

Chapter 9 . Structure o f Besov Space s .

In this chapter we will consider our space s from the

point of view of topological vector space s . More preci se ly

we wish to determine their i somorphism classe s .

We shall deal not only with spaces de fined in the whole

of mn but al so with space s defined in an arbitrary open

subset It of mn • General ly speaking i f X i s a quasi-Banach

space o f functions or distributions in 1Rn we de fine X ( lt )

as the space of the re strictions to It o f the elements i n X ,

i . e . f sX ( It) i f f there exists g s X such that f i s the

restriction o f g to It • The corresponding quasi-norm we

de fine by setting

I I f I I X ( It) = in f I I g I I X •

In other words X ( rt) gets identi fied to a quotient o f X :

X (rt ) :::

where XF denote s the subspace o f X consisting o f those

funct ions or distributions in X the support of which is con

tained in F . In the case X = � ( Sobolev space ) thi s was p k done already in chapter l . h'e obtain the space W ( It ) . p

Similarly taking X = Ps or Bsq we obtain the space s Ps ( rt ) p p p

In what fol lows we shall mostly take l < p < oo • Our

re sults in the extremal cases p = l and p = oo will be rather

1 7 7

incomplete . It i s maybe intere sting to mention the

fol lowing . I f n = 1 Borsuk proved that al l the spaces

Ck ( I ) , k > 1 , are isomorphic to each other and thus to

C ( I ) (= C0 ( I ) ) the space of continuous functions on the

1 7 8

closed unit interval I . On the other hand , i f n > 1 Henkin k n proved that the spaces C ( I ) , k � 1 , are not even uniform

retracts of C ( In ) which space i s known to be isomorphic to

C ( I ) . The proof i s not very difficul t . First one replaces

In by Sn ( the n-dimensional unit sphere ) . k I f C ( I ) were a

uni form retract o f C ( I ) i t must be injective . I f one con-. h . d k ( n ) k-1 ( n ) h TSn � s s�ders t e mapp�ng gra : C s -+ C TS , w ere .,_

n the tangent bundle of S , then there must exist a uniform

retraction M : Ck- l ( TSn ) -+ Ck ( Sn ) , i . e . we have M qgrad id .

On the other hand by a theorem of Lindenstrauss one can

arrange that M is linear and using invariant integration al so

invariant ( for the group S 0 ( n ) ) . But such an invariant

linear M cannot be continuous . We have a contradiction . A

similar result holds also for the space s � ( In ) ( Stypinsk i ) .

It i s there fore intere s ting to note that one nevertheles s has

constructed a Schauder basis in � ( In ) as well as in Ck ( In )

(Cie sie lski-Domsta and Schonefeld ) .

After the se remarks concerning Ck ( Q) and � ( Q) let us

fix attention to the case 1 < p < oo • For a while we also

take Q= lRn •

1 7 9

We begin our di scussion with the space P; = P; ( JRn) •

By cor . 1 of th . 8 of chap . 3 we know that the isomorphism

class of P; doe s not depend on s . In other words all the

Ps . h . ( n ) space s p are 1 somorp 1c to Lp = Lp JR • But by a theorem

of von Neumann ' s all spaces Lp (s-2 ) where st is a general

measure space carrying a non-atomic measure are isomorphic

to each other and in particular to the space Lp ( I ) where I

i s the unit interval IR . Thus our contention is

Theorem 1 . Let l < p < oo . Then s P ::: L ( I ) . p p Next we consider B;q = B;q ( JRn ) , 1 < p < oo , 0 < q < 1 .

Again we know that the isomorphi sm class at least does not

depend on s . But othenvise the situation is much more com-

plicated . To formulate the result is however simple . Let

A q i s the space o f infinite p matrice s a = @v k } such that

00 l:

V =O

( Later on we also use the space � � which is the subspace

of A q consisting of those matrices a which are re stricted p

0 i f k > m v

Here m = { m)� =O i s a given sequence in JN with sup mv = oo . - q One can show that the i somorphism class o f A p

depend on m so we can supress m in the notation . )

doe s not

Then we

1 8 0

have

Theorem 2 . Then

In part icular

We know already a weaker resul t heading in the same

direction , namely that B;q is at least a retract of tq ( Lp ) .

We must now re fine thi s result . We col lect here some pre-

liminary material on retracts , which all basically goe s back

to Banach .

Let us consider general quasi-Banach spaces A and B . We

say that A is a retract o f B ( in symbol : A « B ) i f there exist

continuous linear mappings T : A + B and S : B + A such that

S T = id . Thus we have the commutative diagram

We say that S is a retraction of T ( and T a section of S ) .

Cle arly « i s transitive . We say that A is stable iff

A « B , B << A => A ::: B ( i somorphism) .

The following criterion for stability of A i s now o f para-

mount importance for us .

Proposition . Assume that for some q ( 0 < q .2_ oo ) holds

tq (A ) < < A . Then A i s stable .

Proof . Let u s first assume that A �t q (A ) . Let B be

any space with A <<B , B << A. Then we may al so wri te

We note that for any E

£ q ( E ) ::l £ q ( E ) ED E

181

holds ( s ince JN and JN {oo} are equipotent ! ) . Using this we

readily get

Thus it fol lows in thi s special case , that A is stable . For

the general case note that for any E

£q (£ q ( E ) ) ;;: £ q ( E )

holds . There fore by the special case £ q (A) a t any rate i s

stable . But then it follows from £q (A) << A ( assumption ) ;

A « £ (A) ( trivial ) that A � £ (A ) . There fore A is stable . q q The proof is complete .

We note some equally important corollaries , with or with-

out the complete proo f .

Corollary 1 . Assume that A :::: £. q ( E ) or A :::: Lq ( E ) .

Then A is stable .

Proo f . We have already noted that £ q ( tq (E ) ) :::: £. q ( E ) .

But it is equally simple to prove that t q ( Lq ( E ) ) :::: Lq ( E ) .

1 8 2

Thus in either case A :::: tq (A) and we apply our proposition .

Corollary 2 . Assume that A « £ q for some 1 < q < oo

and that dim A = oo • Then A :::: £. q .

Proo f . We may assume that A is a complement subspace

of A . In view of the propos ition it suffice s to show that

A con tains a complemented subspace B isomorphic to t q . But

this fol lows readily by a classical construction of Banach ' s ,

which we choose to omit however . The proo f is incomplete .

Coro llary 3 . A q i s stable . p Proo f . Apply cor . 1 with E = tp Corollary 4 . � q

p is stable . I f 1< q < oo ' p .,. q , none -of the A q - q t p £, isomorphic . Also the spaces A ' or q are p ' p isomorphism class of � q

p doe s not depend on the sequence

m = { m)� = o ·

Proof . For s tabil ity i t suffice s to note that

� q which is pretty obvious . p Of the remaining

assertions only the non-isomorphism of A q p and � q causes p

some trouble . We show that every subspace of � � contains a

subspace isomorphic to tq . Thi s we do by the construction

omitted in the proof of cor . 2 . However the analogous

statement for A � is fal se . Then the two space s cannot be

isomorphic . The proo f is highly incomplete .

We return back to more concre te matters . Be fore

approaching the proo f of th . 2 , we have to prove al so the

following lemma . For any c lose d subset we denote

by Lp [ G ) the subspace o f L p = Lp ( lRn ) made up by those

A

f E Lp such that supp f c G . (Note that we use brackets [ ]

and not parenthese s ) I in orde r to avoid confusion with

1 8 3

Let l < p < oo . Then for any cube holds

L [ F ) � £ • Indeed it is possible to construct an i somorphism p p H : Lp [ F) -+ 9-p such that the operator norms I I H I I and I I H

- l I I

are < C where C is independent of F .

Proof . By reasons of homogeneity we can take F to be

the n-dimensional unit cube , i. e . , F = In where I [ - 1 , 1 ]

i s the unit interva l . By an inductive argument it i s e asy

to reduce one sel f to the case n = 1 , in which case we will

prove the following more precise statement :

00 ( 1 )

1 00 I f ( 2 nm ) l P ) P .::_ I I f I l L .::_ c2 ( l: p m=-oo

1 l f ( 2 nm) I P ) P , fELP [ I ]

Here c1 and c2 are certain constants , o f whi ch c1 can be

chosen independent of p (even p = 1 , oo i s pe rmissible ) . In

other words H is the mapping

H : f -> { f ( 2 n m ) } oo _ • m-- oo

To prove the first half o f ( 1 ) it suffice s to prove the

following inequal ity

00 ( 2 ) L:

m=- co

1

I cjJ * f ( 2 nm) I p ) p < C I I f I I L , p

1 8 4

cjJ E: s ,

with C depending on cjJ only . Again to prove ( 2 ) we need only

to consider the extremal cases p = 1 and p = oo and apply the

Schur interpolation theorem ( = the corre sponding special case

of M. Rie sz-Thorin ) . The case p = oo being trivi al , we

turn at once to the case p = 1 . In this case ( 2 ) follows

from the triviality :

00 00 L: m=- oo I c)J.f f ( 2 nm) I < f

JR l cjJ ( 2 nm-y) l l f (y ) l dy L: m=- oo

j ust b y noting that

sup y E: :IT<

00 L: I cjJ ( 2 nm-y ) I .2_ C sup

m=- oo yE::JR

00 1 m=-oo 1+ l 2rrm-y l 2

It remains to prove the second half o f ( 1 ) . To thi s end we

write

f D * f ,

where D ( " Dirichlet kerne l " ) i s de fined by the requirement A

that D should be the characteristic function o f I . ( Thus

D = a0 in the notation used in the connection with the

multiplier theorem for the ball in chap . 7 . ) For any g then

the identity

00 � f ( X ) g ( X ) dx f ( 2 nm ) D * g ( -2Tim )

m=-oo

holds . Using Holde r ' s inequality , ( 2 ) applied to D * g

1 8 5

and the fact that D E C p (by M . Rie s z theorem on con j ugate

functions ) we then obtain

1

I f JRf ( x ) g ( x ) dx I :5_ 00 I

I - , I ( D*g ( ( -2nm) I p ) p :5_ m=-oo

1 00 -

< ( I I f ( 2nm) I p ) p C I I D * g I I L < m=-oo p '

1 00 < ( I I f ( 2 Tim) I P ) p C I I g I I L m=-oo p '

The converse of Holde r ' s inequality prove s our point . The

proof i s complete .

Now finally we are ready to prove theorem 2 .

Proof o f th . 2 : VJe consijer two families o f cube s :

Note that

H� = { � I I t: · I :5_ 2 2 v+l ( j =l , • • • , n ) } , J ' = 2 In J

v = 0 , 1 , . . . ) .

2 H " C R , 2 R c_ H ' v v v v

( I f M i s any set c.JRn and A > 0 then AM denote s the image

of M under the dilation x -+ A x . ) I f we can show that

( 3 )

( 4 )

Bsq « Q,q ( { L [ H ' ] } ) ED Lp [ J ' ] p p \)

we conclude from the lemma that

and then from cor . 3 o f the proposition that

e stablish ( 3 ) we use our usual tes t functions

{ 1/! } oo\! =O , '¥ to obtain mappings

()() S ' · (F {f } 00 ) -+ '¥ * F + l: 2-\!s 1/J\!* f :

•

, \) \!=0 \) =0

I f we impose the usual normali zation assumption

()() = 0

1 8 6

we clearly get the crucial re lation S ' v T ' i d . Similarly

to get ( 4 ) we de fine mappings

T" :

S " : f -+ {

2 -vs f

O n * f } oo \) V=O

9-q ( {L [H " ] } ) -+ B8q p \) p

1 8 7

where D " i s so de fined that D " i s the characteri stic function

It i s trivial that S " o T" = id . The proo f i s com-

plete .

We also note the fol lowing

Theorem 3 . goo oo We have B00 z 9- •

Proof : By a theorem of Lindenstrauss i f A < < 9-oo and

dim A = oo then A z .Q, Also i t is known that 00 L = L ( JRn ) z 9-00 , a result due to Pelczynski . Now 00 00 B8q << 9-q ( L ) al so i f p = oo . In particular then p p

goo oo Boo < < 9- ( L00) Z L00 and we are through .

Next we consider the analogue s of the above results

when the n-dimensional torus Tn is substi tuted for llin • Thus

we have the space s p8 = P 8 ( Tn ) and B8q= B8q ( Tn ) which are p p p p de fined in a completely analogous way ( see chapter 3) . I f

we a s usual identify a function o n � with a periodic

function on JRn we see that

B8q L ( T ) . p n

All the above proo fs go through . The only signi ficant change

occurs in connection with the lemma. More speci fically

1 8 8

corre sponding to ( 1 ) we wil l have

1 r

I f < 2nm ) l p ) p < ! I f i l L ( T ' ) ( 1 ' ) cl ( I -- < m=-r r p

1 r

i f ( 2;m) l P ) P, < c2 1 . This has the effe ct

that in theorem 2 in the final end Aq has to be replaced p by �q

p • Our contention may be summarized as follows :

Scholium . Theorems 1-3 hold al so in the case of Tn ,

�q q with the exception that A should be substituted for A p p • In particular i f q � p the spaces Bsq ( llin ) and Bsq ( Tn ) are p p not isomorphi c .

Final ly we conside r arbitrary bounded open sets rl CJRn .

(Compact manifo lds with or without boundary would al so do . )

Recal l that for any space X we had de fined X ( Q ) . The only

assumption on Q relative to X will be the following :

( * ) There exists a continuous linear mapping S : X ( Q ) + X

such that i f f EX ( rl ) then Sf E X i s an extension o f f .

I n other words denoting by T : X + X ( rl ) the re striction map ,

s i s a section o f T . Thus X ( Q ) gets identi fied to a comple-

mented subspace of X . I n particular X ( rl ) < < x . Regarding

( * ) it is satis fied (with X = Bsq or ps p p i f rl satisfies a

00 kind o f " cone condition " • The case o f a C boundary wil l

be brie fly treated i n the Appendix , B .

1 8 9

I f ( * ) holds and i f in addition X i s stable for mul ti-

plication by function s in V , i . e . V -+ M X then i t is easy

to see that X (Si) < < X ( Tn ) , X ( Tn ) < < X ( rt ) . So i f we know that

X (Tn ) i s stable it fol lows that X ( rt ) < < X (T ) . n we may apply this in the case X

thi s in a new

Scholium . Assume that rl satisfie s ( * ) with

1 < p <

l < p < co , o < q _::. co .

()() I

In particular

We summarize

or

i f

I n conclusion we note that from our structure theorems

it follows that the space s in all case s we have inve stigated

do posse s s a Schauder basi s . This i s said to be o f some

intere st in que stions of numerical analysis . We remark also

that in view of En flo ' s famous re sults not every separable

Banach space has a basis .

190

Note s

Henkin ' s result is in [ 1 2 6 ] . Where Stypincki ' s i s

publi shed I do not know . A Schauder basis in space s W� ( In )

or Ck ( In ) has been constructed by Ciesielski-Domsta [ 1 2 7 ]

and Schone feld [ 12 8 ] . Th . 1 certainly belongs to folklore .

Th . 2 was found by me in April 1 9 70 . A proo f was later pub-

lished by Triebel in [ 1 2 9 ] ( c f . [ 2 2 ] ) . As was already noted

all the functional analysis background comes from Banach , see

in particular the last chapter of his famous book [ 1 3 0 ] .

Regarding the proposition ( and al so its corollary 2 ) see

Pelczynski [ 1 31 ] . The lemma i f n = 1 is a well-known result

of Marcinkiewicz ( see [ 8 3 ] or [ 1 7 ] ) . The analogue of th. 3

for the unit interval i s due to Ciesielski [ 1 3 2 ] . He also

proved the corre sponding analogue of theorem 2 with q = oo

[ 1 3 3 ] . For Lindenstrauss ' result used here see [ 1 34 ] and

Pe czynski ' s [ 1 35 ] . In the final sentence we have made

allusion to En flo ' s counter-example to the approximation

problem in Banach space [ 1 36 ] .

Chapter 1 0 . An abstract generali zation o f Be sov spaces .

In the previous chapters we have deve loped the theory of

Besov spaces and we have given some applications to it . In

thi s chapter we shall consider seve ral generali zation s o f

Besov space s . We noted alre ady in Chap . 3 that in the de fini-

tion of Besov spaces we could have used instead of Lp '

1 �p � oo , a general translation invariant Banach space X of

functions or distributions on llin • These more general spaces

were denoted by Bsqx . (Recall that thus Bsq= Bsq L ) p p · S imilarly we de fined general potential space s PsX . (Now

In this chapter we intend to go a little bit

farther . The space X will be an arbitrary abstract space with

no connections with llin whatsoeve r . However some additional

structure is then needed . We remark that in princip le this

chapter is independent o f the remainder of the lecture note s .

I t i s only because o f l azyne ss that we have not written out

al l the proofs , re ferring instead to the entirely similar

proo fs in the concrete case of llin . Let us now de scribe

the precise situation .

Let there be given a Banach space X . Let there also

be given a Hilbert space N and a positive sel f adjoint ope rator

A in N . ( In what fol lows we i n general re frain from mentioning

N specifical ly . ) It is also assumed that both X and N are

continuously embedded in some l arge Hausdorff topological vector

space ( unspecified) , XnN being dense in both X and N . I f u

\ 1 9 1

)

1 9 2

i s any bounded ( Bore l ) function o n [ O , oo ) then u (A ) has a

precise meaning as bounde d operator in N . Namely if

{E A } A � O is the spectral re solution associated with A then

u (A) = !� u ( A ) dE A •

In particul ar the res triction o f u (A) to X N . It may now

happen that u (A) can be e xtended by continuity from x n N to

the whole of X . I n this case u (A) thus has a meaning as

bounded operator in X too . We shall in particular be con-

cerned with the case when all the ope rators

uni formly bounded in X

sup I I u <� ) I I < oo t >o

A u ( t ) , t > 0 are

( I I · I l is of course the operator norm ! ) I f thi s is the case

and i f u is normali zed by requiring u ( O ) = l then a

routine density argument shows that

for eve ry f s X. ( Now I I · I I is the norm in X ! ) Thus the A u (t) serves as a kind of approximate identity . In the special

case when X is a space of measurable functions in some locally

compact space � equipped with a positive measure � we may

also s tudy the ques tion of pointwise convergence , i . e . , we

ask under what conditions on u do we have

1 9 3

where by pointwise convergen ce we mean e i ther a ) conver-

gence a . e . (with re spect to w ) or b ) locali zation ( con-

vergence at continuity points , Lebesgue points , etc . )

To get any farther we shall as sume that u (�) e xists t for some p articular u . We shall take

u ( ;\ )

i . e .

( 1 - ;\ ) � {( l- ;\ ) a i f A � l

0 elsewhere a � 0

u (�) = ( l - �) a t t + ( the Ries z means ) .

More precisely let us make the following

De finition l . \'Ve s ay that A is of Bernste in type in

X , of exponent a , i f

( l ) sup I I ( 1 - �) � I I < oo . t >0

We al so say that A is of exponent >a if A is of e xponent S for every S > a etc .

We give a l i st of operators of Bernstein type .

Example l . I f X = N then A i s of Bernste in type of

exponent 0 . Obvious .

Example 2 . I f X LP ( nP ) 1 � p � oo (with N

and A i s given by

( 2 ) Af (x) = ( 2 n ) -n f n eix!; H ( i; ) f ( !; ) d!; , m:

19 4

where H ( !; ) i s a given homogeneous positive sufficiently diffe rentiable (outside {0 } ) function , then A is of Bernstein type

I t -

o f exponent a > ( A-1 ) / p where and in what follows � I = � . A typical example is H ( !; ) = l s l 2 = s i + · · · +

in which case A = - � • I f the set {H ( !; ) � 1 } in addition i s strictly convex it is conceivable that the bound can be improved to max (n/ p- 1/2 , 0 ) but this has not yet been proved in all general ity ( see Chap . 7 ) .

Example 3 . I f again n X = Lp ( JR ) and A is given by ( 2 ) but with H having a di fferent degree of homogeneity in diffe rent coordinate dire ctions , i . e .

1 m n = c

then A is of Bernste in type of exponent a .:_ (n-1 ) I p • A typical example now would be

ml which case A = D1 + • • • +

are even positive integers ) .

m1 m H ( !; ) = !; + • • • + !; n in 1 n m D n ( i t is assumed that the m . n J

Example 4 . I f X = Lp (� ) , 1 < p < oo , and A i s given by

Af ( x ) ( 2 n ) -n

1 9 5

with H as in Ex . 2 then A is o f Bernstein type o f exponent

a > n/P - 1/2 .

Example 5 . I f X = L ( rl) , 1 < p < oo , where rl i s an p = =

n-dimensional sufficiently diffe rentiab le manifold with bound-

ary , carrying the measure � determined in terms of local

coordinates x ( x1 • • • xn ) by a density w ( x ) , i . e . d� = s ( x ) dx ,

and A a formally sel f-adjoint (with re spe ct to w ( x ) ) e lliptic

partial di ffe rential operator , then A is - under suitable

assumptions on the boundary conditions - o f Bernstein type ,

o f exponent a > ( n-k ) I p where k i s a constant . In compact

mani folds (no boundary ) thi s was e s tabl ished with first

k = } and later k = 1 by Hormande r , in fact as a byproduct

of his work on the asymptotic behavior of the spectral function .

The above probably a lso extends to the quasi-e lliptic

case ( c f . Ex . 3 ) . What can be s aid for other partial differ-

ential operator ( s ay , formally hypoel liptic one s ) i s not clear .

I f we in e x . 5 spe ciali ze to n = 1 (ordinary differential

ope rators ) but allow certain singularitie s at the boundary

we obtain a number o f classical expan sion s .

Example 6 . Thus

rl = ( -1 , 1 ) , w (x ) = 1 , A d 2 d dx ( 1-x ) dx ( Legendre operator)

corresponds to expan sion in Legendre polynomials . Here A i s

o f Bern ste in type o f exponent a > max ( 2/ P - 1/2 , 0 ) .

1 9 6

Example 7 . More general ly

( -1 , 1 ) , w (x )

d dX ( Gegenbauer operator )

corre sponds to expansion in Gegenbauer ( ultra-spherical ) polynomials . If v = 1/2 we get back Legendre polynomial s . Now A i s of Bernstein type o f exponent a > max ( ( 2 V + 1 ) / P - l/2 , 0 ) . Note that if v = n;l (n integer ) then A comes by separation o f variable s from the LaplaceBeltrami operator in sn ( the unit sphere in lRn+l ) .

To some extent the above resul t for Gegenbauer poly-nomials extends to the case of Jacobi polynomials

A )1 2 V-l/2 (with ( 1-x) ( l+x ) in place of ( 1-x ) ) •

Example 8 . I f

rl = ( 0 , 1 ) , w ( x )

A -2 'J -x d dX

2 V X

2 V = X

d dX ( Be s se l operator )

now re stricted with a boundary condition f ( l ) or more generally f ' ( 1 ) + H f ( l ) = 0 , we get expansion in Fourier-Be ssel re spectively more generally Dini serie s . He re A i s again o f Bernste in type o f exponent > max ( (2 V +l ) / P -1/2 , 0 ) . I f n-1 v = --2- (n integer ) then A

comes by separation o f variable s from Laplace operator in the unit ball Kn o f lRn ( restricted by suitable boundary conditions

1 9 7

on the boundary sn-l ) .

Example 9 . I f S"l = ( 0 1 00 ) but w and A are the s ame as in

Ex . 8 (no boundary condition s ) analogous results hold. Now

we have to deal with the Hankel tran s form .

Remark . In mos t of the above examples it i s possib le to

modi fy the original (natural) weight w a little bit without

the property o f A being of Bernste in type getting lost , only

the exponent has to be changed . E . g . already in the case of

ex . 1 it i s possib le to replace w (x ) = 1 by w ( x ) = l x i A I t would be tempting to try to prove a general resul t in this

sense .

Returning to the general c ase we now show that for

operators A o f Bernste in type the operators u (�) exist and

are uni formly bounded in X for quite a few functions u . Thus

A admits a rather extended spectral calculus in X ( generalizing

the v . Neumann spe ctral calculus in the original Hilbert space

N ) •

Theorem 1 . Suppose that A is o f Bernstein type o f

exponent a • Then holds

( 3 ) sup j j u (�) j j � c j j A �u l l al* t>O Bl

Conversely i f ( 3 ) holds then A is of Bernste in type o f

exponent > a •

Here generally speak ing sq * B p are the Besov space on

the multiplicative group JR� = ( 0 , oo ) . In the s ame way we

1 9 8

denote by P; * , �* , L; being the potential , Sobolev, Lebesgue space s respectively in :IR� • Since dA/A is the Haar measure on :IR� thus holds

Similarly A £-A being the invariant derivative in :IRx we have

+ I I A df I I + • • . d L * p s We re frain from stating the de finition s in the case o f Pp

The groups :IR = ( - oo , oo) being isomorphic ( the canonical i somorphism is provided by the exponential mapping A -+ e A ) , all the previous re sul ts obtained for :IRn

can be carried over to the case o f :IR+ •

Proo f (outline ) : Because of the expre ssion to the right in ( 3 ) is multiplicative ly invariant , we may take t = 1 ,

i . e . , it suffice s to prove

( 3 I ) l l u <A ) I I � c i i A � I I al* Bl

Consider first the case a integer . We write Taylor ' s formula in the form

u ( A ) 1 - aT

( -1 ) a+1

( t ) dt

/'" ( 1- �)a ta+1 u ( ( a+1 ) ( t ) dt 0 t + t

I f we ( formally) repl ace A by A we there fore have

u (A ) = ( -l ) a+l

Then we get at once using ( 1 )

< c �� Since w a *

1

ta+l l u ( a+l ) ( t ) J dt c J J t du l l T � dt a* wl

+- Bal * 1 ( 3 I ) follows in this special

dt T

case .

1 9 9

I f a

is not an integer we can proceed in an analogous way i f first

we de fine u ( a+l ) ( A ) by

u ( a+l) ( A )

the formula

tk k ! u ( k ) ( A ) dt

T

k being the integer part o f a , with a suitable normali z ation

con stant C a Thi s was the direct part . The conve rse fol lows

readily i f we j ust note that the right hand side of ( 3 ) i s

finite if u ( A ) = ( 1- A ) ! , f3 > a .

We now give t>vo simple consequence s of th . 1 .

Theorem 2 . I f A i s of Bernste in type o f exponent a then

it i s al so of exponent f3 when f3 > a

2 00

(We have already implicitly as sumed thi s in the fore-going discussion . )

Proo f ·. Thi s follows f Bal * + BS l * rom 1 1 when S > a . Theorem 3 . Let x0 and x1 be two Banach space s

satisfying our initial assumptions with the same operator A . Suppose A is of Bernstein type of exponent a . in x . ( i=O , l ) . l l Then A is of Bernstein type of exponent > a = a0 ( l- 8 ) + a1 8 in X F (X) , F being any interpolation functor of exponent

-+ -+ 8 • In particular we may take X = [X ] 8 or X = ( X l e q . Here we have put X = {x0 , x1 } •

Proo f : By th . l we have

l l u (A ) l l x . , x . � l l ci >- 1 1� 1 1 a l* B l

l By interpolation ( u fixed ! ) it fol lows

( i=O , l )

1 1 >- � 1 1 a l * B l l

which 1n turn implie s

But

l l u (A) l l x , x <

2 0 1

and we are through.

Next we discuss somewhat the role played by the parti

cular function ( 1 - A ) � . It turns out that in place o f

( 1- A ) � we can use say a function u such that v sati sfies

( 4 )

Here

(� ) I < C ( l+ I E.: I ) a v ( i; )

du A

v = A d1 and v its mul tipl icative Fourier transform ,

i n other words the Me llin trans form ,

A v ( i; ) A-i i; V ( A) d A / A .

I f u ( A) ( 1- A) � then

A v ( i; ) r ( a) r ( 1- i 0

r ( a+l - i E.: )

so that ( 4 ) certainly holds in this case . We shal l not give

the detai l s and mention j us t the corre sponding re sul t in the

scalar-valued case on m. , the proo f o f which will be left as

an exercise for the reade r .

Theorem 4 . Let f E S ' ( JR) such that v * f E L 00 for

some v whose usual ( additive ) Fourier trans form sati sfie s

( 4 ) • Then f E B- aoo 00 We mention also a variant o f th . 4 , of a somewhat di fferent

nature .

Theorem 5 . Suppose that v * f ( x) for some v , satis fying

for some a > 0 the condition

( 4 ) c , < =

admits an analytic continuation g (x + iy) when I Y I < a such that

Then f s B- a, oo 00

lg (x + iy) � � __ c:_ __

( a - I Y I a

2 0 2

The re ader will have no di fficulties in supplying the proof .

Remark . Note that ( 4 ) and ( 4 ' ) are e s sential ly condi-tions of a Tauberian character .

Again th . 5 is connected with the fol lowing results for operators of Bernstein type - a counter-part o f th . 1 .

Theorem 6 . Let A be o f Bernste in type o f exponent Then holds

( 5 ) I I w R ( w> l l < c 1 ( cos !>

l+ a ' 8 arg w , I G I < Tr

-1 where R (w ) = (A + w ) i s the resolvent o f A . Conversely i f ( 5 ) holds true A i s of Bernstein type o f exponent S > a •

Proo f : ( incomplete ) We prove on ly the trivial part . I t suffice s to apply th . 1 to the function

u ( A ) ( 1 + e-i e It ) - l

For simplicity let us also take a intege r . S ince A wR ( w ) = u (x) and

u ( a+l ) ( A )

we get

and ( 5 ) fol lows .

( a+l ) 1 e-i ( a+l )8 ( l+ e -i 8 A ) - ( a+2 )

c ( cos �) l+a

2

A variant of th . 6 i s

2 0 3

Theorem 7 . Let A be o f Bernstein type o f exponent a .

Then holds

( 6 ) I I e-TA I I ;s C 1 8 = arg T , I 8 I < � •

( co s 8 ) l+ a '

Conversely i f ( 6 ) holds true A is of Bernste in type of ex-

ponent (3 > a.

We note that th . 7 make s a bridge to the theory of

distribution semi-group s emanating from Lions .

Remark . I f the spectrum of A consists of the positive

integers alone A = 0 , 1 , 2 , • • • it i s often more natural to con-

sider Ce saro means in place of Rie s z means ; for our purpose

this is irrelevant . We recal l that , according to a clas sical

theorem by M. Rie s z , Ce saro surnrnability and Rie sz surnrnability

are e ssentially equivalent .

2 0 4

After these preparations we are finally in a po sition to give our promised generali zation of potential and Besov space s .

Let A be an operator o f Bern stein type in the Banach space X. For s implicity let us assume that the spectrum o f A keeps away from 0 .

De finition 2 . For any real s we set


that

I I f I I s p X

Let {X } 00 be a sequence o f C00 test functions such v v=- oo

X ( A) � 0 iff A E: int R where R = [ 2 v- l , 2v +1 ] \) \) \)

I ( A) I c 0 ' f A R h R = [ 2 - E:) - l 2 v , ( 2- E: ) 2 v ] XV � E: > 1 E: V E: w ere VE:

We o ften also normalize our sequence by

00 v=-oo X v ( A) = 1

2 0 5

()() Example 1 0 . I f X i s any C te s t function such that

X ( A ) i 0 i f f A E int R_1

then it suffices to take

De finition 3 . For any real s and 0 < q� oo we set

()() { f l v=- oo


()() I

v=-oo

Example 1 1 . I t i s not hard to see that in the case o f

Ex . 2 we get back our old space s Ps and Bsq The only formal p p • change to be noticed i s that we have now res tricted ourse lve s

to very particular te st function s { ¢ } , name ly essential ly \) such which are radial fun ctions . This i s o f course in

e s sential in most case s .

Example 1 2 . In the case o f Ex . 3 we obtain the " an isotropic"

(or "mixed homogeneous " ) analogue s of Ps and Bsq , much studied p p by the Soviet mathematici an s and also othe rs .

Example 1 3 . In the case o f Ex . 5 with a compact mani fold

� (without boundary ) we again obtain ordinary Sobolev and

Besov space s , at least i f 1 < p < oo ; the s ame space s regardless

of whatever A i s .

We state now some o f the basic propertie s o f PsX and Bsqx .

2 0 6

Since the proofs are entirely parallel to those o f our previous treatment ( see notably Chap . 3 ) we omit the proofs , leaving them as exercise s to the reader . First we state an interpolation theorem.

Theorem 8 . We have :

s q s q = ( B 0 OX , B 1 lX ) eq

So much for real interpolation . The correspoding que stion for complex interpolation wil l be postponed for a moment .

Next we state an analogous approximation theorem. For any g let us de fine spec g ( spectrum) as the smal lest closed set such that for any u with s upp u contained in the complement of the set in que stion holds u (A ) g = 0 . We then can de fine the "best approximation " in A. o f f with elements g with compact spectrum as fo llows .

E ( t , f ) = inf l l f-g l l · spec g C ( 0 , t ]

I t is now easy to verify the fol lowing Theorem 1 . ( Jackson-Bernste in ) . Let s > 0 . Then

q dt l/q ( f� ( ts E ( t , f ) ) t ) < co .

2 07

Remark . From the proof of th . 1 which we j ust omitted

one gets the following two inequalitie s :

( 7 )

( 8 )

E ( t , f ) < c t -s I I f I I PsX

1 1 £ 1 1 s � C ts l l f l l x ' f s X with spec f C. ( O , t ] p X

In the classical case ( 7 ) and ( 8 ) corre spond to the inequali-

tie s of Jackson and Bernste in respe ctive ly . (A dual form o f

( 7 ) appears sometime s i n the literature under the name o f

Bohr ' s inequali ty . )

Now we consider an approximation theorem o f a somewhat

di fferent nature . We place ourselve s in the situation o f

th . 2 where A acts s o to speak i n two space s x0 and x1 .

Theorem 1 0 ( Stein ) . Assume that A i s o f Bernste in type

in both x0 and x1 • Let T be a l inear operator such that

( 9 ) I I Tf l l x . � l

Then holds :

11 · c t 1 l ! f l l x . i f 1.

f EX . with spec f C( O , t ] ( i==O , l ) . 1.

( 10 ) I I Tf l l x � c t11 I I f l l x i f f EX with spec f C(O,t ]

-+ -+ Here X == F (X ) , with X == { x0 , x1 } , F being any interpolation

functor o f e xponent e ' and Tl == ( 1- 8 ) Tlo + e 11 1 .

Proof : Consider Tt == T u (�) whe re u ( A ) == 1 i f AE ( O , t ] .

Then holds

n . I I Tt l l x . , x . < c t 1 ( i

l l

There fore by interpolation

or

0 ' l ) •

I I Ttf I I X < c t n I I f I I X for any fE X .

As sume that spec f C(O, t] . Tf de sired inequality ( 10 ) fol lows .

2 0 8

and the

Let us give an application o f th . 10 , to Markov ' s in-equality.

Example 14 ( Stein) . The inequality in question reads :

( l l ) (Markov , with C l )

i f f i s an ( algebraic ) polynomial o f degree � n and denotes the norm in X = L00 (-1 ,1) For comparison we write down a variant o f Bernste in ' s inequality

( 1 2 ) l l ( l-x2 ) 112 f ' l l < C n l l f l l ( Bernstein , with C l )

which follows from the usual Bernstein ' s inequal ity ( for

2 0 9

trigonome tric polynomial s ) i f we make the substitution

x = cos 8 • I f we consider the Legendre operator ( see Ex . 6 )

A d (1 - i) dxd - dx

we see that

d ) 2 + d dX X dX

Thus we see that , assuming ( 12 ) , ( 11 ) is equivalent to

( 1 3 )

But A is o f Bern stein type . Thus ( 1 3 ) - and thereby Markov' s

inequality ( 11 ) - fo llows from ( 8 ) . Invoking now th . 3 we

see , the case p = 2 being trivial , that A i s of Bernstein

type in Lp ( -1 , 1 ) , 2 < p < 001 too . Thus ( 11 ) holds also in this

case which i s Stein ' s generali zation of Markov ' s inequality .

Analogous re sults hold for Gegenbauer and even Jacobi poly-

nomials ( see Ex. 7 ) .

Be fore leaving approximation theory let us point out

that al so various other problems can be tre ated in the pre sent

abstract framework , e . g . the que stion s pertaining to the

phenomenon o f saturation ( see Chap . 8 ) .

Our next concern wil l be with "mul tipliers " . Imitating

the procedure in the case of Fourie r multipliers ( Chap . 7 ) we

set for any space Y (embedded in the s ame large space that X

( and N ) were embedded in )

sup i iu (A) f iiY I i f l l y

210

and consider the correspoding "multiplier space " M Y . We will only be concerned with the cases Y = PsX and Bsqx so we right away abuse the notation to

It i s plain from the de finition that M and M do not depend on s . That does not depend on q either wil l be seen in a second . I f A i s o f Bernstein type in X o f exponent a we get a sufficient condition for u to be a multiplier on PsX with the aid o f th. 1 :

al* E: Bl => u E: M

Note also that we have the embedding M + M , i . e . we have

-u c: M => u c: M .

The following theorem give s a complete characteri zation of the space M in terms of the space M •

Theorem 1 1 ( Hardy-Littlewood) . As sume that A i s of Bernstein type in X . Then

( 14 )

u E M <=> s'0p / I X\! u I I M

< oo

It has the fol lowing immediate

Corollary . Let A be o f exponent a • Then

sup t >0 I I v u I I a+ 1 1 * B I

1

2 11

To get any farthe r we must put stil l more re strictive

conditions on A . A compari son of ( 3 ) and ( 14 ) sugge sts the

following

De fin ition 2 . We say that A is of Marcinkiewicz type

in X , o f exponent a , i f

( 15 ) sup I I u (I) I I < C sup t >O \! I I X \! u I I a+ 1 1 * B I

1

Remark . Why we choose the name of Marcinkiewicz should

be pretty obvious , and al so why we previously chose that o f

Bern ste in .

Clearly i f A i s o f Marcinkiewicz then A i s also of

Bernstein type , o f some exponent a • But the converse fail s .

Expre ssed in symbol s we have :

Marcinkiewicz => Bernstein <of

From Cor . o f th . 1 1 follows that that i f A is o f

Bernste in type in X then i t is of Marcinkiewicz type in BsqX

2 1 2

(but of course only of Bernstein type in PsX ) . An important case o f operators of Marcinkiewicz type

is obtained when A i s an el liptic (or more general ly quasi-elliptic ) partial diffe rential operator on a manifold � and X = L ( m with 1 < p < oo 1 see Ex . 5 . p

The nice thing about operators of Marcinkiewicz type i s that we have a theorem o f Paley-Littlewood

Theorem 1 2 (Paley-Littlewood) . Let A be o f Marcinkiewicz type in X Lp ( m I 1 I I L: v = - oo

1/2 ( 2 \is I X (A ) f I ) 2 ) I I < oo provided \! X

Proof : Use the Khinchine-Littlewood inequality (see remark 1 1 Chap . 4 ) .

PROBLEM. Extend to the case 2 � p < oo .

Here i s one standard consequence of th. 12 . Theorem 1 3 . In the hypothes es o f th. 1 2 we have

provided 1 < p� 2 ,

provided 2� p < oo •

Proo f : The case 2 �p < oo follows from 1 < p � 2 by duality .

�Vi thout proof we also state ( c f . Th . 7 )

1 < p < 2

Theorem 1 4 . Let A be o f Marcinkiewicz type . Then we

have

2 1 3

Remark . An intermediate class of operators which we

might call operators of Mellin type , i s also o f some intere st.

They are de fined by the requirement that

( 1 6 )

o r equivalently

l l u (A ) I I < C JJR ( 1 + l t l > 13 1 � ( t ) l dt

where u denotes the Mellin ( = multiplicative Fourier ) trans-

form of u . We have

Marcinkiewicz => Mellin => Bernstein <-=f <f=

Note that for the proof of th . 1 4 actually on ly ( 1 6 ) is

needed.

Remark . In some of the above resul ts , in particular those

concerned with the appli cation to approximation theory , the

assumption that A should be of Bernstein type or of

Marcinkiewicz type i s unne ce ssarily re strictive . In fact it

would have been sufficient to assume that in place of ( 3 )

the re holds the inequality

sup l l u (�l l l < c l l u l l u t> O 00 *

2 14

where U is a space of C functions on :ffi+ containing sufficiently many e lements . E . g . one could take U to be a Gevrey class .

We leave the general theory and consider in some greater detail the two case s corre sponding to ex . 2 and ex. 4 . In particular we are going to prove the re sults which were mentioned there .

We start with the case o f JRn (Fourier integrals ) . In particular we thus take

Af ( x ) ( 2 n ) -n A H ( i:; ) f ( t; ) d t;

where H ( t; ) may be any homogeneous positive function of degree n but we shall pay our most intere st to the case H ( t; ) = l t: l 2 ( spherical case ) . C f . the discussion of the multiplier problem for the ball in Chap . 7 .

With r = A m let us write

We have

( 1- �) a f (x ) A +

a ar * f (x ) .

( 2 n) -n Je ix t; ( 1- H ( t; ) ) a d t; A +

2 1 5

Note also that

( 1 6 )

With an interpolation technique ( c f . th . 3 of Chap . 6 ) it

i s easy to see that

( 1 7 ) < c l 1 + l x I a+l

The exponent a + 1 cannot be improved in general . In the

spherical case H ( � ) = 1 � 1 2 and more general ly i f the surface

{H ( �) = 1 } i s strictly convex it is possible to improve the

exponent to a + 1 + n;l .

( 1 7 I )

In thi s case thus holds

1

l+ l x I a+l + (n-1 ) /2

It fol lows from ( 1 7 ' ) that a aEL1 i f n-1 . a + 1 + -2- > n , 1. . e . n-1 a > --2- (Bochne r ' s critical index) . Because o f the homogene ity

holds then a ar E L1 and moreover

sup I I a; I I� c < co r

We conclude that

h t A . f B · f > n-l l.. n L ( th so t a l. S o ernste1.n type o exponent --2 - 1 e

L l 6

spherical or more general the strictly convex case ) . In

view of Plancherel ' s theorem i t i s al so clear that A is of

Bernstein type of exponent 0 . There fore we can apply th . 3

and conclude that A is of Bernstein type > ( n-1 ) / p where

1/ p = l l/p-1/2 1 • In Chap . 7 this was done using Hirschman ' s

theorem. By the discussion o f the multiplie r theorem for

the ball in that chapter we also know that the above exponent

is not the best possible .

In thi s case we can also conside r pointwise convergence

( usually a . e . ) . ( C f . what we said about this is the beginning

of this chapte r ) . With the aid of ( 1 7 ' ) one shows readily ,

stil l under the assumption a > n- 1 -2-, that

sup I f� ( x ) I < C Mf ( x ) r

where M is the maximal operator of Hardy-Littlewood ( cf .

Chap . 8 ) . From the maximal theorem o f the se authors we now

infer

By the den sity argument pre sented in Chap . 8 , it follows now

that f a ( x ) � f ( x ) a . e . , r � oo , for any r n-1 and a > -2- .

With other methods (we re turn to this point in a few minute s )

one can show that f� ( x ) � f ( x ) a . e . , r� oo for eve ry fsL2

2 1 7

and a > 0 . With a suitable modi fication of the argument

of th . 1 1 one can next conclude that fa ( x ) 7 f ( x ) a . e . r 7 oo , r 1 1 for every f s Lp 1 1 (n- l ) (p - 2 ) . The se

results ( actually its analogue for Tn which case is somewhat

harde r) emanate from Stein .

S h . n o muc concern1ng JR • Be fore we ente r into the dis-

cussion of the case of Tn (Fourier serie s ) we recal l the

Poisson summation formula which claims

( 1 8 ) L: f ( x + 2ny ) Y s zn

The most general condi tion for the validi ty of ( 1 8 ) i s that

f s U Bs1oo ( i . e . f s ' in the notation of L . Schwartz ) . The Ll

convergence has to be taken in the distributional sense .

For the proof one has to take the Fourier trans forms of both

members .

Conside r now the operator

(Af ( x ) =) A f ( x ) ( 2 n ) -n

(We use the symbol ' to emphasize that we stay on Tn . ) Let

us write

�a (x ) * f ( x ) r

( 2 n ) -n

2 18

with

( 2 1T ) -n

We do not have any more the s imple relation ( 16 ) so that it is not possible to reduce to the case r = 1 . However from Poisson ' s summation formula ( 1 8 ) we conclude that

�a (x ) r I n rn K ( r (x + 2 1T y ) ) Y E Z

n-1 i f a > --2- ( spherical or more generally strictly convex case ) . Moreover it i s not hard to see that

We get thus

so that A is of Bernstein type of exponent n-1 . > -2- 1n L1 • As be fore using interpolation ( th . 3 ) we also get that A i s of Bernstein type o f exponent > (n-1 ) / P where 1/P = j l/p-l/2 j . It i s al so easy to carry over the considerations concerning pointwise convergence . if and

The result i s that a > (n-1 ) { 1/p-1/2 ) .

• a f (x ) -+ f ( x ) a . e . r

We say a few words about the limiting case a = n-1 -2-(Bochner ' s critical index ) . As Stein has shown the rel ation

219

n-1 fr --z- ( x ) -+ f ( x ) a . e . doe s not hold in general if f E: L1 •

This is a generali zation of the same re sult in the real ly

more difficult special case n = 1 . The case n > 1 depends

on an old re sul t of Bochner ' s which says that • n-1 fr -2- (x ) -+ f ( x )

at points of regul ari ty doe s not hold in general i f f E: L1 .

( This in contrast to what is true in :rnn 1 namely that n-1

fr --2- (x ) -+ ( x ) at Dini points i f x E: L1 . ) The se are also

positive re sults . He mention that Ste in showed that n-1

f -r ( x ) -+ f (x) at Dini points if f E: L 1 p > 1 or even r P f E: L log L . Some simpler results in thi s direction can be

treated by the interpolation technique deve loped here . E . g .

we can prove the re sult j ust started with 2 f E: L ( log L) •

Up to now we have mostly been conce rned with spherical

and more generally the strict ly convex case . Now we say

something about the general c ase ( no assumption on the

differential geometry of {H ( �) = 1 } ! ) . We have already noted

that we have only the weaker e stimate ( 1 7 ) in place of ( 1 7 ' ) .

There fore we must proceed differently . In the case of :rnn we

apply simply directly Hirschman ' s theorem ( th . 1 o f Chap . 7 ) .

This shows that A i s of Bernste in type o f exponent a > (n-1 ) I P in the general case too . This can be extende d to the case o f

Tn . For pointwise conve rgence a . e . similarly one has to invoke

a Paley and Littlewood type re sult ( see th . 2 of Chap . 4 ) .

But the bound obtained in this way i s a bad one : 1 1 a > n (p - 2 ) .

Finally we spend some words ( to be exact a few hundred )

on the que stion o f pointwi se convergence a . e . in the L2 case .

2 2 0

The following may be considered as a modern treatment of some c lassical topics connected with orthogonal serie s .

We take X = N L2 ( � ) whe re � i s any measure space and we just assume that A i s sel f-adjoint positive . Thus we are back in the trivial situation of ex . 1 . We have the following

Theorem 1 5 . Assume that Then holds

f� l u (�) f (x ) 1 2 d: ) 1/2 < oo a . e . for eve ry fEL2 ( � ) .

Proof . Using the spe ctral theorem we get

I I u(A ) £ I I

which relation of course plays the role of the Plancherel formula . I t suffice s now to invoke Fubini ' s theorem.

Theorem 16 . Assume that Then holds

l 1 X U E: B2 with 2 u ( O ) 1 .

u (�) f ( x ) -+ f ( x ) a . e . as t -+ oo for every f E L2 ( � ) ,

provided we know that this relation i s true for every f EX0 where x0 is a dense subspace of

Proof : By interpolation . Remark ( Hormander ) . It is also possible to give an

even simpler proo f o f th . 16 by starting with the observation that trivially

2 2 1

A . � sup I ( :E) � f (x ) I < oo a . e . for every f€L2 ( rl ) , � E: JR t

1 Indeed this also shows that.

-1 replaced by f E: F

x L1 .

2 1 X f E: B2 in th . 16 could be

Cf . Bernstein ' s theorem ( th . 3 of

Chap . 6 ) and remark .

He consider some example s :

Example 15 . I f U ( A ) = ( 1- A ) � , a > 0 then certain ly 1 1 2 u € B1 I t fol lows that

f� ( x ) = ( 1- �) a f ( x ) � f ( x ) t + a . e . for

This is a re statement of a classical re sul t o f Zygmund ' s

in the case o f orthogonal serie s . We used it already above

in the case o f Fourier inte grals and Fourie r serie s .

Example 1 6 . I f in Ex . 16 we take a = 0 , i . e .

we have

u ( A ) = (i i f A < 1 lo i f A > 1

1 2 , 1 X u E: B1 and the conclusion 0 ft ( x ) � f (x ) a . e . i s

not true either in general . However in the case when the

spectrum is discrete , say A = 1 , 2 , • • • , there is a way out.

Namely first observe that u (E ) obviously is constant if t is

in an interval between two consecutive intege r s . There fore

we may as well take t integer too , thus t = 1 , 2 , . . . Now

we see that

2 2 2

A u ( -) = v (A ) with t {1 i f A < t-1

v ( A ) = v t ( A ) = A - t if t-1 < A < t

1 2 Now obviously v E: B2

sup ! ! v i i � c log t . t >o

0 if A � t 1 *

and it is possible to show that We are thus lead to the conclusion

sup j ft (x ) j < C log t a . e . i f ft:L2 ( st ) t> O

This i s essentially the content o f ( the easier side o f ) a clas sical re sult by Menchov-Rademacher .

Example 1 7 . Let us return to � ( the case o f Fourier serie s ) • Then by the same construction as indicated in Ex • 16 one can show that if n > 1

n-1 ! ! :ft ! I L � C t -2- if f t: L1 = L1 (Tn ) 1

( I f n = 1 the corresponding re sult holds with log t ) . We remark that ft is of course nothing but the partial sum of the Fourier serie s of f , the spherical one i f

2 2 3

Note s

Almost al l the material o f thi s chapte r i s taken ove r ,

in somewhat updated form , though , from my mimeographed notes

[ 1 3 7 ] ( 19 6 5 ) . My papers [ 1 3 8 ] , [ 1 39 ] should perhaps also be

mentioned . Rel ated ideas , i . e . an abstract (operator ) setting

for this type of classical analysi s , can be traced e lsewhere

in the l iterature . Let us mention Littman-McCarthy-Riviere

[ 8 0 ] , Stein [ 82 ] , Fisher [ 1 4 0 ] as wel l as work by the people

of the Butzer School ( re ference s may probably be found in

[ 11 3 ] . Regarding the classical expansions (ex . 6 - 9 ) we have

l i sted already a number o f re ference s in connection with our

discussion of the multiplier problem for the ball in Chap . 7

( see [ 1 0 0 ] , [ 1 0 5 ] - [ 1 0 8 ] ) . For Hormande r ' s work on the

asymptotic behavior of the spe ctral function see [ 1 4 1 ] , [ 1 42 ]

( c f . al so [ 1 3 9 ] for a less succe ssful attempt , and Spanne [ 1 4 3 ] .

Regarding less preci se forms of th . 4 and th . 5 see [ 1 38 ] ,

[ 1 39 ] . Distribution semi-groups were introduced by Lions [ 14 4 ]

and have been studied b y many authors ( some auxil iary

re ference s can be found in [ 1 3 8 ] ) . Here we mention especially

the paper by Lars son [ 1 4 5 ] , because he use s Gevrey functions .

Th . 10 and the appl ication to Markov ' s inequality ( ex . 14 )

are from Stein ' s thesis [ 1 4 6 ] .

The treatment of Fourier inte grals and Fourier serie s

i s inspired b y the work o f Stein [ 1 0 2 ] , [ 1 4 7 ] ( See also Stein-

Weiss [ 3 7 ] , Chap . 6 ) . Bochner ' s classical paper is [ 1 4 8 ] .

See al so the survey article by v . Shapiro [ 1 4 9 ] . That ( 1 7 ' )

does not hold in the general not strictly convex case was

2 2 4

noted in [ 15 0 ] . Concerning the general case see also [ 9 5 ] n .. . . and what concerns T Lofst.rom [ l l l ] .

For the classical theory of orthogonal serie s see Alexits [ 15 1 ] . The sketch given here fol lows [ 1 5 2 ] . The same type of methods can also be used in the case of the pointwise convergence a . e . of the di ffusion semi-groups of Stein [ 82 ] . Regarding ex . 1 7 see H . s . Shapiro [1 5 3 ] .

Chapter 1 1 , The case 0 < p < 1 .

Now we shall extend our theory in yet another direction .

In the previous treatment our Besov and potential space s were

always assumed to be mode lled on Lp with 1 �P < ()() We wish

to extend the discus sion to include the range 0 < p < 1 .

First we shall answer the que stion : Why make such a

generalization ? Strangely enough I mysel f ( circa 1 9 70 ) was

lead to consider the case 0 < p < 1 through the mediation of

some non-linear problems in approximation theory . To fix

the ideas let us take n 1 and let us consider approximation

with spline functions with variable node s . By a spl ine

function of degree < k we mean a function f with compact

support such that

Z:: A o ( x-a )

where the right hand side thus stands for a di stribution which

is a finite linear combination of point masses A placed at

the points a ( the nodes of f ) . The restriction of f to any

interval that doe s not contain any node s i s a polynomial of

degree < k . The space o f all spl ine functions i s denoted by

Spl and the subspace o f those with at most N node s will be

denoted by Spl (N ) . Note that Spl ( N ) i s not a vector space

( in general we can only say that

Spl (N1 ) + Spl (N2 ) C: Spl (N1 + N2 ) ) whence the non-linear

character of the problem. It quickly became clear to me that

the question of characteri z ing these function s f on lR admitting

2 2 5

2 2 6

a best approximation with spline functions of given order of accuracy necessitated the introduction of Besov type space with 0 < p < 1 . Although the extension of Bsq to p thi s case has not yet been made , let us , to illustrate the point , state without proof one result in this sense .

1 , min ( l , p ) Theorem 1 . Let Then we have

f s J3P where p 1 k �p < 00

( 1 ) inf l l f-g i i L oo = gsSpl (N )

0 ( N 1 p } , N -r oo .

1 -, 00 Conversely if ( 1 ) holds true then f s sP 00

We go on asking. How can such an extension be made ? In chapter 1 we started with Sobolev space � - However in p the de finition we required taking derivatives in the usual (L . Schwartz ) distributional sense . But Lp is not a space of distributions if 0 < p < 1 . In fact since L� = 0 it cannot be embedded in any locally convex space . There fore we must proceed differently . A way out would be to try to de fine � ( in view of the density theorem) as the abstract p

00 completion of some space of C functions ( say , V ) in the metric I I f I I k " But even this fail s . In fact by a counter

Wp example of Adrien Douady al though there exists a natural mapping a true embedding .

this will not be a monomorphism, i . e . not One can also show that Wk ::: L so that by p p

the same token as above � can in no way be realized as a space of distributions . The situation seems to be hopeless !

2 2 7

A t least where Sobolev space s ( and potential space s ) are

concerned . Turning to Besov space s the situation at once

becomes much bet te r . I n 19 7 0 I \\TOrked out a rudimentary

theory of Besov space s at leas t when n = 1 using one of the

"discre te " de finitions with fini te diffe rences ( ac tual ly the

Taylor remainder doe s about the same service ) . There

turned up however the rather unpleasant re s triction 1 s > - -1 p ( or s > n ( .!. - 1 ) for general n ) . p In the summer of 1 9 7 3 I

happened to come a cross the work of Flet t . I noticed that

he had done immensely bet ter using the Hardy-Li t tlewood

approach with harmonic functions in the case o f T1 • I then

saw that also the method used here in our treatment of the

case 1 < p < oo wi th the tes t functions { cp } �=-oo could be carried

ove r with only minor change s to the case 0 < p < l . The key

turns out to be a classical theorem on entire functions of

exponential type due to Planchere l-Polya ( 19 3 7 ) the importance

o f which I had overlooked . At the s ame t ime I began to

study the work of Fe fferman and Stein on the Hardy space Hp .

I t then became reve aled that the true reason for al l the

obstacle s mentioned above i s that i f 0 < p < 1 then the Hp are the good space s to be conside red and not Lp . Of course !

After the Fe fferman-Riviere-Sagher interpo lation theorem made

its appearance eve n the resul ts on interpolation of Besov

space s could be carried ove r • • •

Let us now put an end to this almo s t Wienerian confe ssion

2 2 8

and also switch back to the first person plural , in accordance with normal decent habi ts in mathematical prose .

It appears mose convenient to start with a quick review of the basic properties of Hp spaces in several variables ( Fe f ferman-Stein-Weiss theory ) .

First we have to de fine them . The space s Hp { D ) where D i s the unit disc � C were already mentioned in chapter 2 .

The de finition of Hp ( E! ) is analogous . We now want to de fine Hp ( E�+1 where E�+l = {t > 0 } :i.s the "upper " hal fspace in JRn+l whose generic point wil l be denoted by x = ( t , x ) = ( t , x1 , • . . , xn ) or whenever it is convenient by x = ( x0 , x1 , . • • , xn ) The heuristics leading to the de finition in the n-dimensional case is as follows . An element of f E: Hp ( lR! ) is an analytic function f = u + iv. Thus it is in particular complex valued . But what is a complex number? It is nothing but a pair of real numbers -- a vector . There fore f too can be identified with a vector fie ld ( u , v) satis fying the Cauchy-Riemann equations :

_Cl� d y _a__y_

() X 0 .

This leads us to consider as an n-dimensional (or perhaps (n + 1 ) -dimensional ) analogue vector fields u = ( u0 , u1 , • . • , un ) satisfying the generali zed Cauchy-Riemann equations (which really go back to M. Ries z ) :

au . ( 2 ) J ;::: ax.-1

au . l ax:

J ( i , j ;:::O , • • • , n ) ;

au . _J ax .

J

2 2 9

0

or in more conci se vector analysi s notation

rot u 0 , div u o .

Note that ( 2 ) in particular implie s that each component u . J

n+l We now say that u E Hp ( lR+ ) has to be a harmonic function .

where 0 O

where I u I = /1 u0 12

+ • • • + 2 l un I • We also drop the assump-

tion that the components u . were re al and consider thus from J

now on complex valued vector fields as wel l . Usual ly we can

identify u with the ( di stributional ) boundary value s f of

its t component , i . e . f (x ) = u0 ( 0 , x) . We wil l there fore

write HP ;::: Hp ( lRn ) in place of Hp ( R�+l ) and consider thi s as

space of distributions on lRn . Actually thi s de finition is

a good one only i f 1 p > 1 - -n The reason is the following .

I f n ;::: 1 a basic property of analytic functions which is used

is the fact that if f is analytic then log l f l is sub-

harmonic . I f n > 1 and 1 p > 1 - n proved the fol lowing substitute :

Stein and Weiss have

( * ) I f u satisfie s ( 2 ) then

l u l p is subharmonic . I t turns out that ( * ) i s the key to the

entire theory . To be able to include the case of any p > 0

2 30

we must consider higher generali zed Cauchy-Riemann equations . \'Ve consider instead (kth order ) symmetric tensor fields U = (ua ) ' where a thus i s a multi-index with I a ! = k , in place o f ( 2 ) , satisfying

dU QJ" ( 2 I ) fJ = ---ax.-1

n I i=O

k-1 , i , j =O , l , . • . , n ) ;

0 ( I s I = k- 1 ) .

Also ( 3 ) has to be modi fied to :

l ( 3 1 ) sup ( f I U ( t , x ) I p dx) p < C < oo

t >o

where I f n-1 p > n+k- 1 we than have the crucial

result : ( * 1 ) I f U satisfie s ( 2 1 ) then ! u ! P i s subharmonic . The above de finition of Hp thus was a la Hardy-Littlewood ,

via harmonic functions . Fe fferman-Stein however managed to obtain a pure ly "real variable " characterization of Hp , using approximative identities only . Let a E: S with cr ( O ) f. 0 .

Then

( 4 )

holds .

f E:H <=> sup ! or * f ! E: Lp P r >O

(Thi s should perhaps be compared to the Hardy and Littlewood maximal theorem. See our discussion in chapter 8 . )

Even more , for a sui table neighborhood A of 0 in S

( 4 I ) f E H <=> sup p 0 E 0 * f I E L r P

2 3 1

holds . Using ( 4 ' ) i t i s possible to extend the Calder6n-

Zygmund as well as the Paley and Littlewood theory ( see

Chapter 1 ) to the case of Hp . Another resul t which follows

from ( 4 ' ) i s the Fe ffe rman-Riviere-Sagher interpolation

theorem for Hp (mentioned in chapter 2 ) .

( 5 ) (H , H ) Po P1 8 p i f 1

p 1 - 8 + ( 0 < 8 < 1 ) .

Another maj or achievement of Fe ffe rman-Stein not directly

related to ( 4 ) or ( 4 ' ) is the identi fication of the dual of

H1 (mentioned in chapters 2 and 8 ) :

( 6 )

( To the dual of H when 0 < p < 1 we return below . ) This p ends our review of H space s . p

In orde r to avoid any ri sk of confusion let us al so state

explicitly that

H L i f 1 < p < oo p p

This follows immediately from Calderon-Zygmund theory (or i f

n = 1 from M . Riesz theorem on conj ugate functions ! ) .

2 3 2

We also insert the fol lowing

Remark . In de fining H we did exclude the case p = 00 • p With ( 6 ) in view and othe r facts too we are however lead to

the contention that the only reasonable de finition of H 00 is :

H00 B . M. 0 .

We now finally give the formal de finition of Besov

space s . Let {<1\) 00\) = - 00 and be as in chapter

formulas ( 4 ) - ( 1 1 ) ) .

De finition l . Let s be real , 0 * f i l L + ( l:

p v=O

3

De finition 1 . Analogous de finition of Bsq • p

( see

We de fine

We al so have the obvious Lorentz analogue s Bsq and Bsq . pr pr Thus these de finitions on the surface look identical with the

corre sponding ones i f 1 � p < oo ( see chapter 3 ) . However one

important word of clarification must be said. Namely that

although f is only a ( tempered ) distribution * f and ¢ * f \) are both distributions and functions , in fact entire functions

of exponential type . Thus I I * f I I L and p

I I ¢v * f I I L are for p

2 3 1

any f well-de fined , i . e . positive real numbers or + oo •

By force of this the de finition certainly i s a meaningful

one .

It also turns out that in the case 0 < p < l the

distinction between Bsq and p Bsq is of a far more serious p nature than in our previous encounters .

By reasons mentioned above we re frain from de fining

potential space s . I f they neve rtheles s have to be de fined

they should rather be mode l led on Hp ' not Lp .

Now we want to deve lop the basic theory o f Be sov space s

with 0 < p < l . In our treatment in chapter 3 we started

with a certain lemma l . I t turns out that thi s lemma goe s

through i n the pre sent case too but i t s proof has to be

modi fied entirely . The reason i s that the Minkowski in-

equality doe s not apply anymore . We need al so some other

propertie s of en tire functions of exponential type so let

us give a rather penetrating study . For any set G c...JR2 and

0 < p ..:::_ 00 let us set ( c f . chapter 1 0 )

A

{ f I f E: S' , supp f G , ¢ * f E: L } p

whi ch space we equip with the induced quasi-norm I I f I I L • By p

the same token as above at least the de finition i s a comple tely

meaningful one . What fo llows i s now a kind of miniature

theory of Bsq when there i s , roughly spe aking , on ly one v p pre sent .

2 34

Lemma 1 . Let 0 < p .2_ p1 < oo

embedding

L [ K ( r ) ] -+ L [ K (r ) ] . p I\

Moreover the inequality :

n ( l p

Then we have the

f E: Lp [ K ( r) ]

holds . Thus this i s part of lemma l o f chapter 3 only . Proo f : In view o f the homogenity it suff ices to take

r = l , and also in view o f Holde r ' s inequality pl = 00 . I f in addition n - l this i s j ust the classical re sult of Plancherel-Polya that we mentioned. We re fer to the litera-ture for the proof . Now it is easily seen that the clas sical re sult at once extends to the vector valued ( to be more accurate , Banach space valued) case . It is now easy to reduce oneself to the case n = l , by induction over n .

Lemma 2 . Let 0 < p .2_ co • Then for any compact G C lRn

we have the ( continuous ) embedding

Lp [ G ] -+ S ' •

Proo f : The embedding in que stion i s nothing but the composition of the embeddings L -+ s ' . 00 and The continuity of the former follows from lemma l with p1 = co •

2 3 5

Lemma 3 . For any compact G , L [ G ] i s complete , thus p a quasi-Banach space .

Proo f : Thi s i s a routine exercise in functional

analysis ( see the

the detail s . Note

Lemma 4 . Let

i s compact , then

version is : I f

and G R ( r ) then

proof of theorem 1 in chapter 3 ) •

however that lemma 2 i s needed .

f E Ck 1 A

f E S' with supp f C G ,

f E L [ G ) provided p > !.:. A more p k

I a l < k

n ( � - 1 ) -t C r

.

We omit

where G

quantitative

Proo f : By partial inte gration in Fourie r ' s inversion

formula we find

I f <x > I

or

1 f <x > 1 < c

n- l a l -t < C r

n-t r k I 1 + ( r l x l >

whence the de s ired inequality.

Lemma 5 . I f G i s compact then we have for any 0 < p < co the embedding

S [ G ] -+ Lp [ G ] •

Proo f : Obvious consequence of lemma 4 .

2 3 6

So much for embeddings . The next two lemmas de scribe the dual of Lp [G ] .

Lemma 6 . Let G be compact . Let M s ( Lp [ G ] ) ' . Then for any G1 with G1 C int G there exists g s L00 [G ] such that M ( f ) = <g , f > i f f s Lp [ Gl ] .

Proof : Again this is a functional analysis exercise . In view of lemma 5 and the Hahn-Banach theorem there exists h s S' such that M ( f ) = <h , f > • De fine g by the formula

A - A g ¢ * h where ¢ = l on G1 but supp ¢ CG . Set M1 ( f ) = <g , f > By lemma 8 , which we have not yet proven , we have M1 s ( Lp [G ] ) ' too . It suffices now to show that

A A Choose f so that f

g ( O ) l

<g , f >

1 on supp g . Then we get

l A A

--n / g ( U f ( � ) d� ( 2 TI )

Thus ( 7 ) fol lows with C = I I f i l L p

Then

Lemma 7 . Let g E L00 [ G ) . De fine M by M ( f )

M E ( Lp [ G ] ) I ' 0 .

We have also to con sider multipl iers . The following

lemma is a substitute for Minkowski ' s inequality i f 1� p� oo

Lemma 8 . For any G , Lp [G ] , 0 < p < 1 , i s a quasi-Banach

algebra for convolution . More preci sely , considering the

special case G = Q { r ) = the cube o f s ide 2r and cen ter 0 ,

i f a E LP [ Q ( r ) ] , f E: Lp [ Q ( r ) ] then

have the inequality :

< C r n ( 1 -1 ) p

a * f E Lp [ Q ( r ) ] and we

Proo f : Let us again take r = 1 and wri te g = a * f .

Then for any E > 0

g ( O ) = 1

Continue a ( � ) and f ( � ) , restricted to Q ( l+ E ) , to a periodic

function with period 2 ( l + E ) and expand the re sulting periodic

function in a Fourier serie s . We get

2 3 8

" f ( � )

and similarly for a ( � ) . By Par seval ' s formula ( for Fourier series ) we then get

g ( 0 ) TI Y f ( - l + E ) .

i . e . we have "discretized" the convolution . Now it follows readily that (by the p-tr iangle inequality ) :

l l l g ( x) I l L :: !n o : l a ( l?'s > l p l l f ( x- l7Tls > I I � ) p

p 2 p

But by the same result by Plancherel-Polya referred to in the proof of Lemma l it can be inferred that

l ( L: \ a ( 1?'s > I P ) P 2 c l l a ! I L •

p

The proof is complete . It is now easy to prove the counterpart of the remaining

parts of lemma l of chapter 3 .

2 3 9

Lemma 9 . Let f E L [K ( r ) ] . Then for any multi-index p a , D f E L [ K ( r ) ] holds and we have the inequality a p

Proof : Write

Lemma 10 . Let

Da f = a * f with a suitable a E LP [ K ( 2r ) ] . 0 Then for any a, I f s Lp [ R ( r ) ]

and we have the inequality

Proof : Similar .

Finally we note the fol lowing

Lemma 1 1 . We have the embedding

In fact the topology induced in Lp [ R ( r ) ] by Hp agree s with

the one induced by Lp ( i . e . the topology for Lp [ R ( r ) ) which

we have been concerned with ) and we have

I I f I l L < l ! f i ! H 2 c I I f I l L p p p

Proo f : As always we may take r = l . For s implicity

take n-1 Let f E: Lp [ R ( 1 ) ) and let we p > --n u = ( u0 , u1 , . • . , un )

be the vector fie ld sati sfying the generalized Cauchy-Riemann

24 0

equations ( 2 ) determined by the boundary condition

u 0 ( 0 , x ) = f ( x ) , i . e . we have

{;:0 ( t , /';; )

We have to veri fy that

sup l l uj ( t , x ) I I L 2 C < oo t >o p

To this end we write again for a fixe d t , u . ( t , x ) = a . * f (x ) J J

with suitable a . ( depending on t) . I t suffice s to verify that J

sup I I a j I I L 2 C < oo t >o p

which can be readily done invoking lemma 4 . We leave the

detai l s for the reade r .

Coro l lary . I f we in de finition l substitute H for p we obtain the same space s . In other words : Bsq H • sq

p Bp •

L p

Remark . The same in the case o f de finition l i s not true .

Thi s il lustrate s a point referred to already , namely that

the space s Bsq and Bsq behave quite di fferently . p p We have ended our survey of Lp [G ] . After this thorough

sq • sq background it is e asy to develop the theory o f Bp and Bp .

Since most of the proo fs are entire ly paral lel to the previous

ones in the case 1 2, P2 oo ( see notably chapter 3 and also to

a lesser extend chap . 4 - 8 ) , we s tate al l results for B;q only J.. Sq and leave the modifications necessary for � to the reader . p

2 4 1

First we insert however an example .

Example 1 . Let f = o ( Dirac function ) . Then

n ( ! -1 ) , oo f E B P and thi s is the bes t resul t in the sense that p

f ¢ Bs ,q i f p s > n ( ! - 1) or p s = 1 n ( p- - 1 ) , q < 00 This i s seen

exactly in the same way as in the case of ex . 2 of chap . 3 .

Notice that the critical exponent 1 n ( - - 1 ) changes its sign p at p 1 . The significance of this wil l appear later .

Theorem 2 . B;q is a quasi-Banach space . I f 1 .2_ p ..:_ oo

1 .::_ q .::_ oo it i s even a Banach space .

Proo f : I f 1 ..:_ p .::_ oo this i s j ust th . 1 o f chap . 3 . The

s ame proof goes through only in one point we have to invoke

lemma 2 ( the corresponding fact for 1 ..:_ p .::_ oo was so obvious

that we had no need to state it on that occasion ) . Strictly

speakin g , we need also the analogue o f lemma 2 of chap . 3

but the extension of it to the case 0 < p < 1 causes absolute ly

no diffi culty , so we leave it out .

Theorem 3 . We have the embedding S -+ Bsq p •

dense in Bsq i f p , q < oo p Proof : This fol lows at once from lemma 4 .

Theorem 4 . We have the embedding Bsq-+ S ' . p Proo f : Same as for th . 3 of chap . 3 .

Theorem 5 . Let s1 < s or s 1 = s , q1� q .

the embedding

Also S is

Then we have

2 4 2

Proo f : This is entirely trivial ( c f . th . 4 o f chap . 3 ) .

Theorem 6 . Let s1 .2_ s , p1 .:::_ p , n n s- P = s 1 - p-;_ Then we

Bsq slq have the embedding + B p p •

1 Proo f : Use lemma 10 ( c f . th . 5 o f chap . 3 ) •

Thi s was the analogue o f the Be sov embedding theorem

( th . 5 o f chap . 3 ) . Now we should have come to the analogue

o f the potential embedding theorem ( th . 6 o f chap . 3 ) • But

we have no potential space s so we consider instead embedding

into Lp • Now something happens ! 1

Theorem 7 . Let 1 1 p .2_ p1 , s = n ( p - p ) . Then there exi sts

a natural mapping

natural mapping

Bsq + L p plq Bsq + L • p pl

Bsq+L p " But these mappings p monomorphisms ) i f s < n ( 1

p

1 and , a fortiori , i f

Al so i f s > 0 then we

are not true embeddings

1 ) n ( 1 1 ) 1 or s = p They it the other hand i f s > n ( 1 1 ) are on or p 0 < q .2_ 1 . ( The n ( 1 - 1 ) 1 1 < q <oo i s thus case s = -

p doubt . )

q.2_ pl a

have

( i . e . not

= oo q .

n ( 1 -1 ) s = -

p le ft in

We illustrate the latter point in the fol lowing diagram : 1 p

1// I I I s=n c! -1 ) p

s

Note that the critical line i s the same as in the approach

based on finite di fferences , re ferred to in the beginning o f

thi s chapter .

2 4 3

Proof : We normalize the test functions in the usual 00

way : � + I ¢v= 8 v =l

By lemma 1 we have

whence

C 2 Vs I I ¢v* f I I L p

< c I I f I I i f - Bsq p * 1 p

1 p* 1 )

It follows that � * f + N I v =1

¢ * f v has a l imit in L as pl N -+ oo . Thus we have obtained a "natura l " mapping

Bsq-+ L under the above restriction on q . Using inter-P pl polation ( see theorem 1 0 below) we get B;q-+ Lp1q .

remains the que stion whether this is an embedding

or not . I f s > n ( � - 1 ) or s = n ( � - 1 ) , 0 < q ..::_ l

There

monomorphism

we have

also a mapping Bsq-+L and thi s must be an embedding , because p lq Llq is a space of distributions . From this the monic charac-

ter of Bsq -+ L p plq i f ( l l ) s < n p - or

readily can be read off . On the o ther hand

s = n ( � -1 ) , q = oo we have to produce a

coun ter-example . We simply take f = 8 ( cf . example 1 ) . Then

it is readily seen that

N <P * f ( x ) + L:

v =l <P * f ( x ) -+ 0 \)

244

if x � 0 . Thus we have an f � 0 - a distribution - which

by our mapping is sent into the function 0 .

Thi s i s thus a new phenomenon . Again we could have

avoided all complications i f we had con sequently worked with

Hp in place of Lp .

PROBLEM. To de scribe more direct ly those fun ctions in

L which come from distributions in Bsq under the mapping plq p

Bsq -+ L o f th . 7 ( in particular thus in the cases of non-p plq uniquene ss ) .

Theorem 8 . For every n we have Jn

where J = 11- l.l •

Proo f : We leave thi s as an exe rcise for the reader ( c f .

th . 8 of chap . 3 ) .

Corollary . Al l the spaces Bsq with given p , q are p isomorphic .

PROBLEM . Determine the i somorphism class of B;q ( c f .

chapter 9 in the case 1 .2_P .2_ oo ) • I n particular doe s Bsq p

posse ss a basis?

Da for

can

Theorem 9 . For eve ry multi-index a we have

Bsq -+ Bs- J a J , q Conversely i f for some k , p p D f E: Bs-k , q a P

all J a J �k then f Bs ,q E: p • Also fE Bsq i f f for some k we

write f - I I L: D f a < k a whe re a

p f E: Bs+k , q . a p

2 4 5

Proo f : Another exe rcise ( c f . th . 9 of chap . 3 ) .

Now we come to interpo lation . Since our space s are

quasi-Banach we have only the real method at our di sposal .

Here i s the re sult (which was alre ady used in the proo f o f

t h 0 7 ) 0

Theorem 10 . We have

Bsq i f s p

Proo f : Although theorem 7 o f chapter 3 was formulated

with potential space s , its proof really goe s through otherwise

unaltered . We leave the detail s to the re ade r .

Regarding the corol laries of th . 7 o f chap . 3 we notice

that cor . 3 now is devoid of sense , because the Sobolev space s

are not de fined in our case . Consequently also the proof o f

cor . 4 breaks down . We are thus faced with the fol lowing

unsolved

PROBLEM. Are the space s Bsq invariant for a local C00 p

change of coordinate s?

For Esq thi s obvious ly i s not true ( even i f 1� p� oo ) • p In view of the deve lopments of chapter 1 0 the following

problem i s also of some inte re st .

PROBLEM. To extend the Planchere l and Polya business

( lemma 1 , etc . ) to the case of eigenfunctions o f an el liptic

partial diffe rential operator A on a , say , compact manifold Q •

2 4 6

I n other words , doe s the pre sent theory for 0 < p < l have

any counter-parts for other deve lopments than j ust the

Fourier trans form?

The following theorem on the other hand we only can • sq prove for Bp •

where

Theorem 1 1 . We also have

· ( ) . s oqo · s lql As ,mln q , r � ( B , BP ) e r � pr Po l

s =

B s , max ( q , r ) pr

1- e + e ( O < S < l ) qo ql

Proo f : In view o f lemma ll we have the following

commutative diagram , analogous to the one s in chapters 4 and

5 :

Thus interpolation of A;q is reduced to inte rpolation of £sq (Hp)

I f we now take into account the Fe fferman-Riviere-Sagher the

( see ( 5 ) ) , we readily get the re sult reque s ted j ust by

invoking theorem 4 of chapter 4 .

Next we would like to treat the analogue of the Jackson-

Bernstein theorem in approximation theory ( c f . theorem ll of

2 4 7

chapter 3 ) . For any ft: L , O < p .:s_ oo let us se t p

E ( r , f ) I l f-g l I L ( "be st approximation " ) p

It i s a legitimate problem to ask for whi ch functions ft:L p

( 8 ) E ( r , f ) - s O ( r ) 1 r -+ oo

holds where s i s a preassigned number > 0 . Since Lp is not

a space of distributions if 0 < p < 1 we encounter the same

type of di fficulty as in th . 7 . (And again a way out would

have been to use Hp on the onse t , and not Lp .

problem would have been another one too . )

But then the

I f

Theorem 1 2 . Let f t: Lp and assume that ( 8 ) holds true .

s > n ( .!. -1 ) there exists an p � soo f t: B such that f i s in the p

image o f f under the mapping Bsoo-+ L of th . 7 . p p Conversely

i f � s 00 f t: Bp , s > 0 1 and f i s the image o f f under the same

mapping then ( 8 ) holds true .

Proo f : Take 0 < p < 1 1 since l .:s_ p .:s_ oo we know already .

Pick up a sequence {g }00 v v =0

De fine

I I f I I < C 2 -v s 1 -gv L p

f l im 'J-+00

with

g t: L [K ] • v p v

(with limit in the sen se o f I J ) Then we get

00 I I <t> * \) f I l L 2 c ( \J�O I I ¢\)* ( g v+ A. I i P -g\J+ A.-1 ) 1 L )

p

Using lemma 7 we see that

A.n ( ! - 1 ) I I <Pv* (gv+ A. -gv+ A.- 1 ) I l L < C 2 P

p-A. (n (.!_ -1 ) -s )

< C 2 -vs 2 p

1 Since s > n ( -1 ) we then get p

and � s oo f s Bp •

Converse ly if � s 00 f s B p we set i f

�

g q, * f +

We readily get

00 I I f-g I l L 2 L:

p v =N+l

N l: v=l

"' * f '1' \) •

I I <t>v * f I I i > p 1 00 2 -v sp ) p -Ns < c ( L: < c 2

v =N+l

The proo f is complete .

r :::

1 -p

< c

p

-s r

2 4 8

1 p

PROBLEM . The case 1 s < n ( - - 1 ) . - p

Next we inve stigate the dual . Although L ' = 0 p '

2 4 9

· Oq O < p < l , by the theorem o f Day , Bp , o < p < l , being a distri-

bution space , has a nice big dual .

Theorem 1 3 .

O < p < l , O < q .2_ 1 .

s = n ( 1 -1 ) , p

Proof : By th . 6 we have B�q -+ B�sl . There fore

( B�sl ' -+ ( B�q ) ' . But by th . 1 2 of chap . 3 we know that

( B�sl ) ' � B:00• Thi s prove s hal f o f the statement . For the

remaining hal f we invoke lemma 6 . By Hahn-

Banach we have at any rate M ( f ) = <g , f > with g s S • . Lemma

6 now shows

There fore gsBsoo 00 The p roof i s complete .

Turning our atten tion to Fourier mul tipl ie rs instead ,

we have the following

Theorem 14 . We have

In parti cular the

Proof : I f

latter n ( !

a EB p p

space - l ) oo

are te st functions such that A

supp � then lemma gives

i s a

and A

1J! = \) 1

0 < p < 1 .

quasi-Banach algebra .

f sBsq and i f { 1)!\) } co\) = 0 ' p A A

in supp ¢\) ' '¥ =1 in

'¥

vn ( 1 - 1 ) ! I <P * a * f i ! L � c 2 P \) p

2 5 0

which apparently implie s a * f s B�q and a s C B�q· Conve rsely

i f a s C B;q it is easy to see that we must have ( c f . proof

of th . 4 of chap . 7 )

A A Choosing f 1 in supp <Pv we get

and n ( 1 - 1 ) , oo

a E B p p

Vn ( 1- .!.) < c 2 p


The following coro llary of the proof i s o f some intere st .

n ( .!. -l ) oo Corollary l . We have C H + B p i f 0 < p < l . p p

Proo f : Clearly C H + p C Bsq ( cf . proof of th . 4 o f p chap . 7 ) .

n ( .!. - l ) oo But • sq C B p

• p Bp by the argument of the proof o f th. 15 .

We also mention

Corollary 2 . Assume a satis fie s for some 0 < p < l

vn ( 1 - 1 ) sup 2 p

\) < 00

then a E C Bsq . p Proo f : Use S zacz theorem ( th . 7 of chap . 7 ) .

2 5 1

We also mention another corollary , which should also be

compared to some of the re sults of

Corol lary 3 . We have

chap . 7 1 - - l ) co p -+

i s

i f O 0 . co p Proo f : This generalizes th . 9 of chap . 7 and indeed

the same proo f extends to the pre sent case .

Remark . In the case l .:':._ p .:':._ co we could obtain information

about mul tipliers in the case s < 0 j us t by dual ity from the

corre sponding resul ts i f s > 0 . I f 0 < p < 1 thi s does not

apply anymore . However some results on multipliers can be

obtained using the last part o f theorem 9 . Name ly theorem 9

b E M Bs , q provide d p b E M Bs+l ,q p , implies that

D . b E M Bs+l ,q ( j =l , • . • , n ) . But the final result i s not so J p

neat so we re frain from s tating it e xplicitly .

He now make a direct confrontation with Hardy classe s Hp .

This i s completely analogous to the treatment in chapter 4 .

But we have not formulated explicitly either the Paley-Littlewood

or the Calderon-Zygmund theory so we j us t state the result

without proof .

Theorem 1 6 . We have :

i f O < p 2._ 2 .

H p

2 5 2

Recall also that :8°2 -+ H -+ :B0P i f 2 .2_p < oo (and that p p p Lp i f l < p < oo ) • I f we agree to put H00 = B . M . O . the

latter re sult remains valid for p = oo too .

However we di scuss in some more detail some o f the con-

sequence s of th . 16 .

Coro l lary 1 . We also have

provided

Proo f : We fix attention to the case 0 < p ..::_ 2 1 because

i f 2 < p < oo 1 since then Hp Lp ' we could simply apply the

re sults of chap . 3 . From theorem 6 (or rather its analogue

for Bsq ) we infer p

Next by interpolation we obtain

The de sired re sul t follows upon invoking ( 5 ) and th . 1 0 (We

apologi ze for having used p1 in two di ffe rent sense s ! )

Corollary 2 .

O < p < l .

S oo We have (H ) ' :::: B 1 where p 00 s = n c! -1 ) , p

Proof : From corollary l we obtain at once

2 5 3

But in view of theorem 1 3

B S ao ()()

The proof i s comple te .

Finally we take up for di scussion the que stion of using

other approximative pseudo-identitie s ( c f . chap . 8 ) .

First we e stablish the analogue o f th 1 of ch 8 . 1 n ( p -1 ) -s , p

Theorem 1 7 . As sume that o E: BP and

0 < P < 1 . Then we have

as r -+ 0 or oo

Proof : The proo f o f the said theorem goes through almost

unaltered . We only have to invoke lemma 8 in place of

Minkowski ' s inequality .

As a con sequence ( c f . chap . 8 , ex . 1 and ex . 3 ) we can

prove that

( 9 ) • s ()() f E:B p ( j =l , • • • , n ) i f

max ( n ( � - 1 ) , 0 ) < s < 1 .

2 5 4

Corresponding re sults hold of course also for general q .

Now we ask for the conve rse . The proof of th . 9 o f chap . 8

breaks hopelessly down . Fortunately i t is possible to rescue

the case by treating each of the two case s separately each

time by a different special method . We begin with the case

of ( 9 ) .

Theorem 1 8 • 1 Assume that max ( O , n CP - 1 ) ) < s < 1 , 0 I /:,t f I L p e j p

( j = l , • • • , n ) .

Proof : One direction is of course ( 9 ) so we concentrate

on the other. Let thus

it is easy to see that this is indeed equivalent to the

assumptions of the theorem. If { ¢ }"" is one of our \) v=-oo sequences of te st functions we have to estimate the L -quasip norm of

¢v * f ( x ) = J ¢v( y ) f ( x-y ) dy

= f ¢v( y ) ( f (x-y) - f ( x ) ) dy .

The idea is to approximate the latter inte gral with the

fol lowing discrete sums :

L: n y sZ y f 0

2 5 5

J wk { y ) <P\) ( z ) d z ( f ( x- 2yk ) - f ( x ) )

where -k Wk (y ) are the parallel cube s of s ide 2 with one vertex at :z k • 2

Indeed since as is readi ly seen

lim k -+ - 00 0

for any f E S ' we have

<P * f \) 00

and it suffice s to e stimate the di fferences Sk+l- sk . Writing

L: n y EZ yf"O L: e W f ( 2 + ) <Pv ( z ) dz ( f ( x- \++el ) - f (x- Xk) )

k+l Y e 2 2

w·here the inner sum i s extended over all the 2n vectors e of the form e = e . + J l

+ e . with l < j 1 < • • • < j < n , we find Jp - p-

! l sk+l -sk I l L .2. c ( z p ys zn l

( sup I v ( z ) I ) p2-kn ) p

n ysZ wk (y ) yf"O

-k ( s-n ( l -1 ) vn ( l- .!.) < c 2 p 2 p min ( l , 2 (k-v ) A )

2 5 6

where A is a number at our di sposal . From this i t. readily

follows that

where , in order to assure convergence , we need 1 s > n ( - 1 ) . p The proof is complete .

Now we turn our attention to the case of ( 10 ) .

Theorem 1 9 . Assume that s < 1 , 0 < p < l . Then

( Here u = ut is the solution of Laplace equation 6 u = 0 · JRn+l · h b d d · d d b f · · h d 1n + Wlt oun ary ata prov1 e y , 1 . e . 1n ot er wor s

the Poisson integral of f . )

Proof : Again i t is only one direction which matters .

Assume thus

Let us write with t � 2- v

where \j!v i s given by

with v au t at ·

Again we want to di scretize . Writing

we get

I <Jv ( x ) 1 .::_ C

where we have put

It follows that

1 l l g I I < ( L: tnp ( ljJv* (yt ) p ) p l l vt* I l L < v L - n p y EZ p

2 5 7

sup l vt (x+ e) I · I e 1 .::. t

The proof is thus complete i f we can prove the following Lemma 12 (Gwil liam) . We have

Again lemma 12 is a :3imple consequence of the following Lemma 1 3 . Let h be any harmonic function de fined in an

open set (J) C :ffin+l and let p > 0 . Then

1 l h ( x l i .::_ C ( n+l r

1

J I h ( Y l I P dy l P K ( x , r )

2 5 8

holds where K ( x , r ) i s a ball c <D , with radius r and center

at x E: CD. For the proof o f lemma 1 3 we re fer to the literature .

Remark . While as in the case of th 18 we treated • sq sq Bp ( and not Bp ) j ust for convenience , we do not know i f th . 1 9

i s true anymore in the case of Bsq . Thi s would require an p analogue of lemma 1 3 for metaharmonic functions .

That we have been obliged to treat th . 1 8 and th . 19

by two entirely di fferent methods is rather annoying. They

can ' t be j ust completely unrelated . We conclude there fore by

stating it as a

PROBLEM . F ind a general condition on a which implie s

-s 0 ( r ) .

2 5 9

Notes

The writer ' s work on non-linear approximation theory and Be sov spaces with 0 < p < 1 re ferred to is [ 4 7 ] . For the proof o f th . 1 see [ 1 5 4 ] . For the Douady counter-example see [ 1 5 5 ] . Flett studies Be sov (or Lipschit z ) spaces on T1 using the Hardy-Littlewood approach with harmonic functions in [ 5 3 ] .

The pre sent approach to Besov space s with 0 < p < l was announced in [ 15 6 ] . Most of the re sults have counter-parts in [ 5 3 ] . The theorem of Plancherel-Polya , underlying lemma l , is discussed in Boas ' book [ 1 5 7 ] , p . 9 8 . (Another proof can actually be based on the ideas of Fe fferman-Stein [ 36 ] ) . The idea of the proof o f lemma 8 is classical , too ; see notably books in approximation theory , e . g . [ 2 3 ] or [ 2 4 ] . Al l the re sults used on Hp spaces are from Fe ffe rman-Stein [ 36 ] , the only exception being the interpolation result ( 5 ) which is from Fe fferman-Riviere-Sagher [ 4 9 ] . For an overall introduction to Hp see also Stein-Weiss [ 3 7 ] , chap . 3 and Stein [ 14 ] ,

chap . 6 . Cor . 2 of theorem 16 was first proved by Walsh [ 35 ] .

It is the analogue o f the Duren-Romberg-Shields result [ 3 4 ]

for D . Note that ( 3 ) was incorrectly announced in [ 15 6 ]

(with 0 < s < l , instead o f max ( 0 , n ( l - 1 ) ) < s < l ) . The diffi--p culties in the proof o f th . 18 are the same as in the inte-gration of functions with value s in local ly quasi-convex topolo-gical vector space . Concerning thi s latter topic see [ 4 7 ]

where some re ference s can be found . The idea of the proof of th . 19 , including lemma 1 2 , is taken over from a classical paper by Gwilliam [ 1 5 8 ] . Lemma 1 3 is due to Hardy-Littlewood

2 6 0

i f n = 1 and to Fe ffe rman-Ste in [ 3 6 ] for general n . I f

p = 1 i t i s j ust the classical me an value property for

harmonic functions .

Chapter 1 2 . Some strange � space s .

In this brief final chapter we shall - to the bewilderment of the reade r , we are afraid - indicate several new generali zations of our spaces . Trace s of them can be seen here and there in the preceding chapters . But in no case has a systematic study been made and the following l ine s should j ust be considered as a rough dra ft for a general pro-gram.

1° We begin with the space s Fsq of Triebel ( see Chap . 4 ) . p Recall their de finition :

{ f I f E S ' and I I <P * f I I L + p

00 + I I ( L:

v =-oo

• sq In the same way the "homogeneous " space s F can be de fined . p As a matter fact Triebel considers only the case 1 < p < oo , 1 < q < oo and the extension to the full range 0 < p � oo , 0 < q � oo thus remains to be carried out . In particular Triebel proves interpolation and duality theorems for Fsq which are p analogous to the one s we have encountered for Bsq . In any case p it is easy to see that

2 6 1

2 6 2

This obviously extends th . 1 of Chap . 4 , because we have

Be side s , the cases g = 2 and g = p also the case g oo

i . e . g oo

the space s Fp , might have some intere st .

Remark . At second thought , maybe the notation P8g p

might have been pre ferable for F8g . p 2° \ve have encoun te red the space s L p A. o f Stampacchia

which we choose to denote by B s ; P ( see Chap . 7 m ex . 1 0 )

where usually l 0 I 1 0r* f l I L < C r-s 00

and that we proved Here , and in what follows ,

the quanti fication 11 0 11 means that 0 runs through some pre -

assigned set of te st functions , in the pre sent case determined

by the conditions :

( l ) I I 0 I I L , < l , s upp 0 p

K ( 0 )

J x a 0 (x ) dx 0 i f I a I � k (where k is an integer > s )

I f we here substitute L for a general space X and also 00 compare with the de finition of B8g with p general g we are

thus lead to consider space s B 8gX (or perhaps better B 8gX )

de fined by a condi tion o f the type

2 6 3

v 0

In particular i f X = Lp we have the spaces To specify that the te st functions are the particular ones appearing in ( l ) we write sq · P B , p • The interest of the l atter primarily come s from the fo llowing result which is easy to prove .

Theorem l . We have

CX , B . M . O .

In particular taking X L we thus have 00

CLOO ' B . M . o .

But by Fe ffe rman-Stein ( see ( 6 ) o f Chap . l l )

so we have the following : Coro llary . We have :

( The latter space i s thus independent of ) . Thi s should be compared with the corollary of th . 4 of

Chap . 7 stating that

2 6 4

3° Next we recal l the space s V o f functions o f bounded p pth variation in the sense of Wiener ( see th . 7 in Chap . 5 and

e x . 9 in Chap . 8 ) . We have

1/p f t. V <=> ( L: / f (b ) - f ( a ) / P ) ;;, C < oo p I

for all famil ie s {I } of di s joint inte rvals I = ( a , b ) C lli .

They are real ly j ust a special case o f certain more general

space s N P A of Stampacchi a , also generaliz ing the space s

LP A ( = Bs ; p ) . These spaces are de fined by a condi tion of

the type

( 2 ) ( L: Q

1 r n+s

Q

for al l famil ie s {Q } o f di s j oint cubes Q TIP of side r0 ,

n0 being a polynomial of degree < k (k > l) , depending on Q . p If n = 1 and p = 00 , s = 1/p

if p = ()() we have the space s

to expre ss thi s 1n terms of

we apparently get

Bs ; p ( = Bs o0 ; p ) p . our approximative

back v p " Al so

We would l ike

pseudo-identities

or . But somehow it does not quite match . Anyhow it is not

diff icult to show that ( 2 ) implies

( 3 ) sup 0 sup r

r s I a r * f I s Lp oo •

Thi s should be notably compared with th . 4 of Chap . 8 which s 1 1 s

says that ( 3 ) i s implied by f t. Pp , with p p - n · We

are thus lead to conside r space s F sqX de fined by conditions

of the type

sup 0 dr l/g - ) E: X . r

2 6 5

In particular we have the spaces F sg ; P as wel l as the ir p Lorentz counter-parts F sg ; P • Thus th . 4 of Chap . 8 can pr be rephrased as

soo · p F I poo l ' p

l p

l -n

Notice also that the Fe fferman-Stein characteri zation of Hp ( see ( 4 ) and ( 4 ' ) o f Chap . l l ) can be interpreted as

where we have not speci fied the set of test functions . The insight gained under the headings

be st summarized in the following table :

-B

I F

�

0 0 l - 3 can maybe

Final ly we would l ike to mention very brie fly two more generalizations :

4° I f one studie s the type of sets of te st functions entering in the Fe fferman-Stein characterization of Hp ( see ( 4 ' )

o f Chap . l l ) one i s lead to the idea o f introducing Be sov type

2 6 6

spaces where also powers o f l x l figure i n the de finition . For example , one might consider conditions o f the type :

00 I I <P * f i i La

+ ( v�O p

where L;v denote s Lp with re spect to the weight ( 1+ 2\J i x l ) a , i . e .

This clearly pose s a lot of new problems . 5° In Chap . ll we de scribed the dual of H when 0 < p < l p

as a Be sov space ( see Th . 1 3 of the said chapter ) . In the case o f the di sk D we have as a limiting case o f Hp= Hp (D ) a s p + Q the Nevan linna class N . The closure of nice functions in N i s the Smirnov class N+ . ( Smirnov i s also the name of an American vodka but there might be no deeper connection . ) The dual of N+ was recently identi fied by Yanagihara . There ari ses for us now the que stion whether there i s an analogous theory of Besov type space s , even in JRn

•

6° Maybe one should use Beurl ing distributions instead of ordinary (tempered ) distributions . Maybe there i s even a connection between 5° and 6° .

2 6 7

Notes

Space s related to the spaces Fsq of Triebel [ 7 3 ] have p also been studied by Lizorkin [ 1 5 9 ] . Some of Triebel ' s re sults ( for l < p < oo ) are extended to the ful l range 0 < p < oo in Peetre [ 1 6 0 ] where there i s also given an appli-

Ooo cation of the space s F to a type o f problem in approximation p theory first studied by Freud [ 2 6 ] . Th . l was stated in [ 1 2 0 ] .

The work of Fe fferman and Stein we have been re ferring to is of course [ 36 ] . The space s N P A appear e . g . in Stampacchia [ 1 6 1 ] . One o f the works of Yanagihara i s [ 1 6 2 ] . Regarding Beurling di stribution s see Bjorck [ 1 6 3 ] .

A . On the trace .

Appendix

n-1 n Identi fy :m with the hyperplane x1 =0 in :m and

2 6 8

conside r the operation o f taking the re striction to :mn-l

o f functions in :mn ( " trace " )

The purpose o f thi s appendix is to prove the following

theorem which goe s back to the work o f Aronszajn , Babic ,

S lobodecki j , Gagl iardo , Stein , etc . Thi s theorem can al so

be considered as an optimal case of the Sobolev Embedding

theorem as stated in chap . 1 ( the spe cial case n1 = n-1 ; the

general case n < n 1 fol lows from it easily by induction ;

in view of the Extension theorem and the Invariance theorem

it suffice s to consider the case when � , is a linear sub

manifold of � = :mn ) •

( 1 )

( 2 )

where

Theorem. We have

s-. T ( 13sq ) B p p

s-. T c Ps ) B p p

1 p ' q

1 p' p

1 S > - 1 1 < p < oo 1 1,2_ q .2_ oo . p Proof ( Outline ) : We note that ( 1 ) and ( 2 ) in particular

imply that

T :

T :

We shall construct a

. ( 3 ) S : B

. ( 4 ) S : B

1 . s- P-,p

-+ B p

mapping s such

1 s- p 'q -+ Bsq p

1 s- - ,p p -+ .ps p p

that

which is also a section o f T 1 i . e . ToS

2 6 9

id . We point out

that we do suppose that S i s independent of s . Clearly this

is more than is needed for the theorem. It is sufficient to

prove ( 2 ) 1 be cause ( 1 ) follows at once from ( 2 ) by interpolation .

We there fore fix our attention to ( 3 ) . We shall indicate by

different methods . 0 Method 1 (abstract ) . We consider a general Banach

-+ couple A = {A0 1A1 } . We re cal l the following wel l-known

resul t (Lions-Peetre ) : An element a s A0 + A1 admits a repre

sentation of the form

where w

a = w ( O )

w ( t ) ( 0 < t < oo ) satisfies

1

U� i ! w ( t l l l � dt ) p < oo 1

00 I

w ( s ) denoting the derivative of order s , s intege r , i ff

-+ a E (A) 8 P , e l l - -sp

2 70

It i s ho>-Jever not difficul t to extend this re sult to the case

s non-integer > �; one then has to consider w ( s ) as a

fractional derivative a la Riemann-Liouville . In one case we

-+ (A ) e P

l • s- P p' B p

Using the N.ikhlin multiplier theorem ( see chap . 4 ) we see

that i f f E Ps and a = Tf one can take w defined by p w (t ) = f ( t , x2 , • • • , xn_1 ) . Thus ( 2 ) follows but not dire ctly

the stronger statement embodied in ( 4 ) . However an analysis

of the general abstract re sult reveal s that at least in our

particular special case one gets a section S satisfying the

de sired continuity conditions and which moreover does not

depend on s . Whence e f fective ly ( 4 ) .

Method 2° (via a differen tial equation ) . This treatment

is based on an idea of Lizorkin ' s . We shall base ourselve s

on the following lemma which wil l not be prove d .

Lemma . Consider the boundary value problem

l" + I f 0 i f t > 0 dt ( I .;::::;;:-; t xl ) .

f = a i f t 0

Then holds :

f E: .ps <=> p

l . s- p p' a E: B p

where l < p < co but s i s arbitrary real .

2 71

Let us however veri fy that the theorem fol lows from the

lemma . In view of the Extension theorem ( see Appendix B )

may a s wel l replace JRn by the hal f space JRn = + {x1 > 0 } .

f E: .ps admits then , i f s > l the unique repre sentation - I p p

with both f0 and f1 in P; where in addition

Clearly

Therefore follows s-.

verse ly i f a E: B p

0 i f t 0

from the l p' P

it

0 i f t > 0

lemma that

is clear by

s-Tf E: Bp

the same

1 p' p Con-

token that

we

Each

a = Tf with f E: Ps for some f. This prove s ( 2 ) but we also p get readily the s tronge r statement ( 4 ) by de fining S with the

aid of the formula Sa = f where f i s precisely the solution

2 72

of the boundary problem .

2 7 3

B . On the extension theorem. - --We begin by proving the Extension theorem as formulated

00 in chap . l . We thus as sume that � i s bounded with a C (or j ust " sufficiently" differentiable ) boundary and we want to prove t

every f E: vi< ( �) i s the restriction to � of some g E:v}< = Wk ( JRn ) . p p p In view of the Invariance theorem (cf . again chap . l ) it suffice s to prove the same thing if � Let thus f E: � (]R� ) and de fine g by

g ( x ) � (.�f00(X ) i f x1 > 0

!0 </J { A) f ( - A.x1 , x2 , • • • , x1 ) d A i f x1 < 0 "-.

where ( A) d A l

� f� ¢ p) A d A = -1

I f <P ( A ) A 2 d A = l

\ � . . It is clear that where JR.� = {x I x1 < 0 } •

2 ()g d g g , dX I ---2 t • • • l ax1 = 0 } I

But in view of ( l ) i t now fol lows that JR.n-1 = {x I xl

n- 1 approach JR al l have the same trace on irre spective o f whether we By Green ' s theorem we have

g f � +

from or

+ g <P

2 74

for al l te st functions <P E V( JR� ) , and simi larly for higher

order derivatives . Addition now give s

f JRn g 3 ¢

ax1 ag �n axl

<P

It fol lows that 3g i s dxl the distributional derivative of g

2 which thus belongs to Lp . In the same way we find _IL5!2 t:: Lp , . • •

a x1 Hence g E: l.f ( JRn ) • p Since the re striction of g to

JRn in f , we are through . +

We now notice that the exten sion we have constructed is

independent of k ( at least i f k i s bounded ) , and i t i s al so

clearly linear continuous . Thus we have k independent

commutative diagrams of the form

This is o f importance i f one wants to interpo late . Indeed

one shows easily e . g . that

s q B l l ( rl ) ) p 8 q

i f s = ( 1- 8 ) so + 8 s l , 0 < 8 < l ,

at least i f one , as sugge sted in chap . l , de fines

2 75

We shal l al so discuss the following problem. When

( i . e . for which value s of s ) is it possible to extend f by

0 outside rl • Again we may take rl :ffi� • Given f let us set

ff ( x ) i f x1 > o h (x )

'-0 i f x1 < 0

We shal l show that i f 0 < s < ! , l 1/p fol lows from the existence of the trace

( see Appendix A) . using the Hikhlin multiplier theorem

( see Chap . 4 ) one see s that it suffice s to consider the case

n=l . As a norm for h in Ps i f 0 < s < 1 one can take p

h (x+t ) - h (x) ts

dtt I I L ( :ffi ) p

Since h 0 i f x < 0 there are two terms to be e stimate d :

( 2 )

and

( 3 )

f ( x+t) - f ( x ) s

f ( x+t) ts

dt t I I L ( :ffi ) p -

Here ( 2 ) causes no difficulty . To estimate ( 3 ) we use

interpolation ( cf . e . g . the treatment of potential s in chap . 2 ) . Let p0 < p < p1 • Set

Tf ( x ) !� f ( y) dy (y+x ) s+l

Tkf ( x) f f ( y ) 2k < < 2k+l (y+x ) s+l _y

In view of Holder ' s inequality we get

where 1 g

1 + s . p Hence

dy •

1 ) p ! I f i l L ( i=O , l ) p

or

k ( 1 2 p

Since Tf

.!. ) k ( l _ !_ ) Po J ( 2 Po P1

T :

follows now

Lg -+ L poo

or after another interpolation

T : L -+ L gp p

2 7 6

2 7 7

But the potential embedding theorem ( th . 6 o f chap . 3 ) says

that

Hence

.ps -+ L p qp

T : .ps -+ L p p

Thus the expre ssion in { 3 ) can be e stimated in terms of

I I t I I . s . p p

We have shown that

2 7 8

C . On the partial regularity o f vector valued functions .

We consider functions F def ined in

JRn with value s in a given Banach space v. If W E V ' we

have the scalar valued function <w , F > • The problem we

are going to di scuss is what one can s ay about the regularity

of the function F i s we know some thing of the regularity of

some o f the functions <w , F > in some direction n O ;# h E JR .

First we have to make precise what we mean by regularity

in direction h ;# 0 . Consider in :rn.� the sets

I I I I v-1 I I

v+ 1 E (h ) = { � h � � 1 } , Hv ( h ) = { � 2 � h � � 2 } ( v=O , l , • • • )

and test functions � and { �v }�=O satis fying analogous con-

ditions as those in Chap . 3 , with E (h ) and {Hv ( h ) }�=O taking

the role of the usual K and {Rv }� =O " E . g . a typical case

would be

where � is given .

De finition . We set

-1 L: v=-oo

1/q Bsq (h ) = { f l f E S ' , I I � * f i l L + ( E ( 2 v s i i <P * f i l L ) q ) < P p v=O v p

( Besov space in the direction h)

The reader will probably have no di fficultie s in proving the

fo llowing

i ff Lemma . Let s > 0 l � p� oo , 0 < q � oo f E Bsq ( h ) p for all ( It j usti fie s the terminology only . ) Now we can announce our main result .

2 7 9

Then f E Bsq p

Theorem . Let A be a subset of V ' x lR� such that every w E V ' and every 0 � t; E IRn we may write with sui table scalars c .

J

( l )

where Let F s , p ,

Then

for be q

n w = 2: c . w . j=l J J

some h . J E lRn holds (w . , h . ) E A J J

and h . t; J

a function in lRn with value s in V such where s > 0 , l � p � 00 ' 0 < q � 00 holds

for all (w , h ) E A

< w , F> E Bsq for al l W EV ' . p

� O ( j=l , • . . , n ) . that for some

Remark . I f V is finite dimensional then it is seen that ( l ) is equivalent to the following condition

( l ' ) I f for some pair ( v , U E V x lRd holds <W , v > h t; al l (w , h ) E A then v = 0 or t; = 0 .

Proof : Let 0 � t; E IRn and w E V ' . Let us write t;

0 for

2 8 0

f = <w , F > , f . = <w . , F > . It i s e asy to see that it is J J sufficient to prove that

( 2 ) ()()

I I � * t i l L + ( L p v=O

where � and { ¢v }�=O are test functions such that the set

{ � UJ f. 0 } contains K0 and the set { ¢v ( E; ) f. 0 } contains

the bal l with radius 2v s and center at 2v i; , s being

sufficiently small > 0 .

view o f ( 1 ) we have

( 3 ) <P * f \) n l:

j=l

( Use j ust a partition on unity . ) In

c . ¢ . * f . ; � * f J J J

n l:

j=l c . � * f .

J J

Util i zing the fact that hj f. 0 i t i s now e asy to produce � and { ¢v } �=O such that ( 2 ) holds with f repl aced by f j .

There fore ( 2 ) itse l f i s a consequence o f ( 3 ) . The proof is

complete .

D . Pseudo di fferential operators in Be sov space s .

In the foregoing we have been concerned both with

2 8 1

ordinary multipliers and with Fourier multipliers , i . e . we

have considered linear operators of the special types

Tf b f and Tf a * f

( see notably Chap . 7 ) . Now we want to merge the two type s .

To begin with let us consider finite linear combinations of

the type

Tf Z b . ( a . * f ) l l

With the aid of Fourie r ' s inversion formula we can write

Tf ( x ) 1 ( 2 TI) n

This leads us to consider quite generally operators of the

type

( 1 ) Tf ( x ) 1

We say that such a T i s a pseudo di fferential operator with

symbol o = o ( x , � ) . Such an appellation is chosen because

in the special case when o (x , � ) is a polynomial function in

� for fixed x, T actually is a (partial ) diffe rential

operator . One can show that the symbol adequately reflects

2 8 2

the properties of the operator . Indeed i f T1 and T2 are

pseudo differential operators with symbols 0 1 and 0 2 re spectively then clearly T1 + T2 has symbol 0 1 + 02 but

one can show that T1T2 too is a pseudo differential operator

and that its symbol is 0102 , up to a certain error term

( i . e . pseudo di fferential operator commute approximative ly) .

We refer to the literature for detail s . Here we will be

concerned with the action of pseudo differential operators in

Be sov spaces .

First we take 1 .::_ p .::_ oo Let us rewrite ( l ) in the form

of an integral operator

( 2 ) Tf ( x ) f k ( x , y ) f ( y ) dy

with the kernel given by

( 3 ) k ( x , y ) = l f ei ( x-y ) E;, 0 ( x , E;,) dE;,

By partial integration we obtain

( 3 I ) ( x-y) a k (x , y ) = l ( 2 n) n

Our basic assumption will be one of the Mikhlin type

( 4 )

( . a By our convent1ons , D

< for all a , B

acts in the t;, variable s , o 8 on the

2 8 3

x variable s . ) For simplicity let us also assume that cr ( x , � )

vanishes for � s K ( l ) ( unit ball ) . Let us write

co

T E Tv v=O

where T i s an pseudo di fferential ope rator with symbol \)

00 { ¢\! } v =O being one of our sequence s of test function s . From

( 3 ' ) and ( 4 ) (with S = 0 ! ) we readily obtain

kz..:being the kernel corre sponding to Tv . ( Note that this

implies in particular

Let now 1 �p � co

c 1

l x-y l n

Then follows

. )

{ �v }� =O being a second sequence o f tes t functions , with

� ( � ) = 1 i f \) It follows that

co

I I Tf I I L � E I I T f I I L � c p v=O v p

2 84

we have e stablished the following

Lemma l . I f ( 4 ) holds than for any

To proceed farther let us recall the following facts

( see Chap . 3 , th . 9 ) :

10

2 0

Here k i s

Lemma

f E Bkq<=>D f E BOq p s p

f E B-k , q<=> f l:

for al l I s I ,;S

with p I S I ,;S k ns fs

any intege r > o . Then we can prove

2 . I f ( 4 ) holds then T : Bkl -+ Bkoo p p

k

f E BOq s p

for any

l ,;S P ,;S oo , k integer > o .

( 5 )

Proof : By Le ibnitz ' formula we obtain

s l: C a , a n s ' + s " = s fJ 1-'

1 f e ix E: D a 1 G (x , l: ) ( i l: ) a n f ( l: ) d � ( 2 n ) n �-' �-'

s l: C S ' S " T S ' ( iD ) S " f

where thus the T S ' again are pseudo diffe rential operators

satisfying ( 4 ) . o Oq By 1 we then have D s f E BP •

By ( 5 ) and Lemma l it fol lows that D S Tf E LP . The re fore

again by 1°

Lemma 3 .

koo Tf E Bp •

I f ( 4 ) holds than

l ,;SP _,;S oo , k intege r > 0 .

Proof : We rewrite ( 5 ) as

( 5 I )

kl -koo T : B� -+ Bp for any

2 8 5

By induction we obtain

( 6 )

where the s 13 ,. are pseudo di fferential operators satisfying ( 4 ) .

Let f E B�kl . Then by 2 ° f = L: D i3 f i3 with f E B�kl . By

( 6 ) and Lemma 1 it follows that T n 13 f E Lp . There fore again

From Lemma 2 and Lemma 3 we now e asily get by interpola-

tion ( Chap . 3 , th . 7 ) .

Theorem 1 . I f ( 4 ) holds then T : Bsq + Bsq provided p p 1 ,;S P ,;S oo , 0 < q ..2_ oo , s real .

We now turn our attention to the case 0 < p < 1 . As in a

similar context in Chap . 11 thi s wil l be done via discretiza-

tion . We have the formula

1 2 ( v+l ) n ( 1/J * f ) v

- v ( x- TI2 y)

Using this it is not hard to see that Lemma 1 is valid for

0 < p < 1 too . But this does not help us much , for Lp is not

a distribution space . We would like to have an e stimate for

I I tP;. * T f I l L instead . p

To get thi s we imitate the procedure

used in the proo f o f th . 9 of Chap . 7 . Let T� be the pseudo v di fferential operator corresponding to the symbol

0� ( x , t_; ) ¢v ( t_; ) where 0� (x , t_; ) J cp ( x-y) 0 ( y , t_; ) dy �

2 8 6

Then we have

Luckily in � A* T f now enter only terms with either

v � A or Jl � A . The contribution of the former type of terms

amounts

The terms of the latter type cause some trouble . Essential ly

we get a contribution of the type

where CJl is a constant depending on the constants Ca� in

( 4 ) ( i . e . S = � ) , but for the symbol oJl • We there fore assume

now

( 4 I ) I Da D s oJl ( x , I; ) I � c� I I; 1 - a for all a , S , Jl s

Z (C Jl 1/p

with ( ) p ) < c < oo for al l a , S . as a s

( We remark that this is e ssential ly a condition at oo . ) I f

thi s i s so one argument shows that T : This is

the good analogue of Lemma 1 . I t i s now easy to prove the

analogous of Lemma 2 and Lemma 3 too . We are content to

2 8 7

write down the end re sult

Theorem 1 ' . I f ( 4 ' ) holds ( in place of ( 4 ) ) then the

conclusion of th . 1 extends to 0 < p < 1 too .

We conclude by indicating an application which shows

what pseudo di ffe rential operator are really good for.

First we remark that what we have conside red until now

really \vere only symbol s of de gree 0 . In the same way we

can treat symbols of any degree m , i . e . ( 4 ) is replaced by

( 4 " ) for all a , S ,

with a corre sponding modi fication of ( 4 ' ) . The conclusion

is then o f the type

Now to the application promised .

Example . Let A be an elliptic partial dif fe rential

operator of degree m with C 00 coe fficients , to simplify ,

say , constant outside a compact set . Then it i s possible to

find a pseudo di ffe rential operator T o f degree -m such that

T A id + S

where S i s a pseudo diffe rential operator of degree - 1 . I f

th . 1 ( or th . 1 ' ) i s applicable , we then may conclude that

from f t:Bs-l q A f t: Bs-m, q p I p

regulari ty theorem .

follows f c- Bsq · a c. f l . e . p

Notes ( for the appendix )

2 8 8

A . Some general references perta ining to the trace problem

can be found in [ 5 6 ] . The re ference to Li zorkin is

[ 7 4 ] •

B . For the extension theorem for domains sati sfying a kind

of cone condition see Stein [ 1 4 ] , chap . 7 . Regarding

"extension by 0 " see Arkeryd [ 7 6 ] and the re ferences

given there . The interpolation technique used i s the one

of [ 5 5 ] ( c f . also [ 5 6 ] ) .

C . The problem discussed here , and the result have their

origin in the work of Boman [ 1 6 4 ] .

D . For an over al l introduction to pseudo di fferential

operators see Hormander [ 1 6 5 ] .

2 8 9

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3 05

The final version of the notes was corrected by my

student Bj orn Jawerth . It i s a pleasure to express my sincere

gratitude to him . It i s perhaps intere sting to point out that

Jawerth has indeed been able to solve several of the questions

left open in chap . l l (on the case 0 < p < l ) . In particular

he has extended the structure theory ( chap . 9 ) to the full

range 0 < p < oo • Another result of his i s that ( B Oq ) 1:;: Bsq 1

p i f 0 < p < l , 1 ..2. q < oo . Thi s is a complement to th. 1 3 . The

most signi ficant achievement is however a description of the

trace of Bsq when 0 l , but there appears tlf: new restriction

s > 1/p + (n - l ) ( 1/p - l ) . A s imilar result is valid for Ps p

( and for the Triebel-Lizorkin spaces Fsq as well ) . The essential p idea of his proof depends on an approach to the Polya-Plancherel

theorem based on the rea l variable techniques of �efferman-Stein

/ 36/ (a possibility which was already briefly mentioned in

Bergh-Peetre /154/ ) . None of Jawerth 1 s results have yet appeared .

( regarding the problems on p . 2 4 5 some progress has been made

by me , see my ( hopefully ) forthcoming paper "Hardy classes on

manifolds " . )

J . P .

(Mathematics Series Volume 1) Peetre, Jaak-New Thoughts on Besov Spaces-Duke University (1976)

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