New Thoughts on Besov Spaces

=

-+

�

repre sentation provide s the pos sibility of carrying over many re­ sul t s from the spe cial case Q = JRn to the case o f a general Q

•

E . g . the Den s ity theorem can be e stabl i shed in thi s fash ion .

Conve r se l y , i t i s also convenient in particular in more compl icated ins tance s , to use the q uotient repre sentation as a de finition . Q JRn =

We shall there fore in what fol lows mostly take

•

Let us however al so men tion the fol lowing rather elementary re sul t . � ( Q) i s not changed i f we make a p change o f coordinate s .

Invariance theorem . local C

00

�

Thi s provide s us with the pos sibi l ity o f de fining W (Q ) when Q i s a mani fo ld not embedde d in JRn ( at least i f Q i s compac t ) .

6

We end our s u rvey o f Sobolev sp ace s .

We now f i x attention

to the problem of de fin ing space s analogo u s to in tege r k i s re pl ace d by any re a l n umbe r s space s of " fractional o rde r " ) .

when the

( a k ind o f Sob o l e v

Seve ral reasons fo r why thi s

i s o f impo rtance w i l l appe ar l at e r o n . that there i s no uni q ue

wfp

( n atural )

I t t urn s out howe ve r

way to achieve this .

The

fo l lowing p o s s ib i l i t ie s are ava i l ab l e : 1

°

Po ten t i a l o r Lio uvi l le sp ace s

l � p � oo .

Le t

J

( 1 - /::, )

1/ 2

�

P , where s re a l ,

where

2 2 a a -2 + + i s the Lap l a c i an . Such a " symboli c " 2 CJx ax l n ope rator we alway s de fine using the Fo urie r tran s fo rm . I . e . /::,

=

•

•

•

deno t ing the Fo urie r tran s fo rm by

A

we req u i re that

Jf ( O In the s ame way fractional powe r s o f J are de fine d by the formul a

where

s•

time s we also nee d the " homo geneo u s " ope rato r powe r s I s

=

s

( Some -

de note s the space o f tempe re d di s t rib ut i on s . I

=

,;-:::.:-;;:-

are the gene ra l i z e d po ten t i a l o f M . Rie s z .

- 2 we ge t the Newton pote n t i a l .

The operato r s J

occa s iona l ly cal l e d the Be s se l potential s . )

The

If s

a re

We then de fine

7

which space is e q uipped with the norm

It i s po s s ible to show that ( use Mikhl in ' s mul tip lier theorem) wpk if

s

=k

intege r..:_ 0 , 1 < p < oo

so at least for 1 < p < oo P ps i s a true generalization of Wpk 2 ° Be sov or Lip s chitz space Bps q where s real , }

1 2P1

q � oo .

Be fore giving the de finition let us right away

remark that

Bps q is a true generali z ation of � p only i f p Let us also notice that so

= Lip s

if

O < s < l.

On the o ther hand

Bl oo 00

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

q

2.

8

(5)

+

s oo B p

The p roblem o f demons trating the e q uivalence i s in general a non-trivi a l one .

The s i tuation i s comp l i cated by

the fact that there are in the l i terature a mul titude o f d i f fe re n t but e q uivalent de finition s .

Mo s t o f the de f in i t ions

are goo d only in ce rt ain interva l s o f s .

Le t u s try to make

a s urve y : 0 < s < 1.

a.

We set q

sq B p with

6

h

=

{ f I fEL

f (x)

=

p

&

( J n :rn.

f ( x+h )

- f (x) .

If q

= oo

the interpre tation o f

the de f in ing e xpre s s ion i s

s up

The no r.m i s given by

Be l ow

( b . - j . ) we do no t wri te down the e xpre s s ion fo r t he norm

because it can be formed in exac tly the s ame manner . b.

1 < s < 2.

We s e t

9

Bps q

=

c.

0 < s 1- intege r .

{ f j f�:: Wp1 & D. f�:: Bs-l , q (j=l, p J

• • •

,n) }

Extension o f the procedure initiated

in b . ( k=integral part o f s )

d.

0 < s < 2.

We can now set q )

with

�� e.

dh ) 1/q < n I hj

00

}

f ( x ) = f ( x+2 h ) - 2 f ( x+h ) + f ( x ) . Procedure analogous to the one in b . and c .

We use

kth order di fference s k

\)

2.:

=0

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

I t i s plain that the de finition indicated under the headings a . - c . al l are somewhat re late d .

Now we indicated a

somewhat different approach first deve loped systemati cal ly in the thes i s 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 ( tempere d ) solution

of the boundary problem

10

2 a u 2 at u

;::

;::

-

if t > 0

b. u

0,

if t

f

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

t

u ( x, t )

f (y)

dy .

Then we have q 1/ q dt ) < T o < s < 2.

g.

Now holds

=

sq B p

h.

{ f \ fE L p 0 < s.

}

00

&

( f;

a2u l \ 2 1 \ t 2 L Cl t p { s t

q 1/ q dt < ) T

00

}

Exten s ion o f the proce dure be gun in f .

For a l l the s e case s we have a t least

s > 0.

and g .

Howeve r it

is e a sy to mo di fy the above appro ach so as to cove r the case of negative s i.

s < l.

( an d s = 0 )

too .

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

o f the boundary p rob lem

11

if t > 0 if t =

v= f

0

Then holds {

j.

f l f E:

LI &

s real .

( -6

00

I It

�� ts

I I .L p

q dt ) 1/ q < t

00

}

Analogous .

We are now faced with the problem o f see ing what i s common in all the se case s .

First let us cons ide r a smal l variant o f

a . , the case s b . -e . being analogous : a' .

O < s < l. &

Bps q = { f I fs Lp where

ej

=

(O,

•

•

One can show that

•

(

,l,

I l li te . fj I L q p ) dt 1/q < oo ( . =l, J J t) ts

!0

00

.

•

•

• • •

,n) }

, O ) i s the j th b as i s ve cto r o f En .

I f we compare a ' . with f . say , we see that the integral s are bui l t up in same fashion .

We have thus to con front the integrands only , i . e . the expres sion s li te . f and t� at J re spectively . I t i s now readily seen that they both are the e ffe ct of a trans lation invariant operator depending on t acting on f , i . e . o f the form ¢ t * f where ¢ t are " te s t functions " depending on t . The dependence on t i s now particularly simple :

12

or , expre s sed in terms of Fourier trans forms ,

¢ ( t� ) where

<jJ

1

i s a given te st function . /"-

!-,

te . J

f

(e


i tt: '

Indeed we find A

J-l) f ( �)

and t l � l e -t l � l f { � ) re spe ctive l y .

( 6)

B sq= { f I P

We are thus le ad to try the fol lowing de finition -s { fooo ( t I I

*

fI IL

under suitab le re strictions on

p

t �

)q

) 1/q

<

oo

}

What are the re s trictior

Le t us here devise every crude •

In view of

(5)

we have in any case

Thus we must have

13

Re strict ing attention to the case

p

q and using P l anchere l ' s

formula we get

Replace now � by t - l �

and let

t

-+

0:

i� . If = e J _ l ( 7) � ( � ) = I � J e - J � J , s..2_1 . Thus ( 7)

which thus i s a nece ssary condition . implie s

0

<s<1

and i f

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

Of course we canno t expe ct to get the comp lete

answer with such crude weapons .

Next we ob serve that

( 7) ,

A

on

the othe r hand , certainly i s ful filled i f � van i shes in a ne ighborhood o f

0

and oo .

Moreove r we f i x attention to the

case when we can do wi th j ust one � in

(6)

- obviously

cannot vani sh for all t at some point

�.

We are thus lead

to impose the fol lowing condition o f Tauberian characte r ( analogous to Wiener ' s ) : (8)

{t � I t

>

o} n

{ ¢ "�

o}

'I

p' for each �

'I

o

�t ( � )

14

I n fact it w i l l b e enough t o work with a s tronge r fo rm o f i t

s upp

(8 I )

{ b - l < I E;. I

¢

< b } w i th

b >11

where we o ften fo r conve n ience choo s e to work " in b ase 2 " tak in g thus

b

=

2.

1

To te l l the who l e truth we have a l so to

add a te rm o f the type

I 1 �*

t= o }

{;

I

fl

= { I t;. l

where � s a t i s fie s

L p < 1}

•

We have a l so ove rlooke d the re gularity conditions to be imp o s e d on ¢

( and � ) .

due course

( Chap .

But a l l th i s wi l l be made more pre c i se in

3) .

In con c l usi on we i n s e rt he re two s imp l e i l l u s trative e xamp l e s where the e s sen ce o f the te chnique b a s e d o n the Taube r i an c ondition

( 8 ' ) wil l be app arent .

Le t us howeve r f i r s t po i n t o ut that the re are a l s o o the r more con s t ruct ive de finit ion s o f Be sov space s . a.

App rox ima tion theory

(s > 0) .

Let us con s i de r the

be s t approximat ion of f in L by e xpone n t i a l fun c t ion of type p < r

(9)

:

E (t 1 f)

Then holds

=

inf

iI

f-g l

lL

p

w he re

s upp g c {

I t;. l

<

r }

15

Bps q = { f!

:fE:

L

&

p

00 (!0 ( rsE (r, f) ) q

dr ) 1/ q r

<

oo

•

I f n = 1 and p = q = oo thi s contains the non-periodic analogue o f the classical re sults o f Bern ste in and Jack son for approximation by trigonometric pol ynomial s . b.

Inte rpo lation ( s real ) .

For real interpolation holds

Bps q with s e . g . by de finition ( c f . Chap . 2 ) 00 { f I uo

dt ) 1 /q

T

<

00

}

with (10 ) K(t , f ) = K ( t , f ; Notice the formal analog between ( 9 ) and ( 1 0 ) .

U s ing complex

interpo lation we get inste ad s0 p

[P

I

s1 p ) 8 with s

P

Now to the e xample s that were p romi sed. Example 1 .

We ierstrass non-diffe rentiable function .

Weie rstrass showed in 1 872 that the function

16

00

f (x)

( 11 )

\)

=1

a\) cos ( b\) x ) where

a
was not di ffe rentiable at any point provided I t i s needl e s s t o point out here the pro found influence that thi s counter-example has exe rc i sed in the devel opment o f In 1 9 1 6 Hardy e xamined Weierstras s '

analysis as a whole .

function and he demonstrated the s ame re sult unde r the weake r assumption re sul t .

ab �1 .

We sha l l now give a simple p roo f o f Hardy ' s

In place o f ( 1 1 ) we conside r the more general function f (x ) =

(12)

00

I

v=O

cv e ib\!X

where { cv } i s any se q uence with I I cv I < oo I claim that the fol lowing holds true . •

Propos i tion .

Let f be given and assume that for some

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

(13) Then holds

c\) = O (b -v s ) and

holds with 0 replaced by Proof .

f E: B soo 00

•

An analogous statement

o.

Let us take Fourier tran s forms in ( 12 ) :

We get

17

where

8

i s the de lta function .

¢

(1) Cv 8

Using ( 8 ' ) it now fol lows

( � -b } where one take s t \!

With no los s o f general ity we may assume that

¢ ( 1}

b=

\!

1.

There fore taking the inverse Fourier trans form we end up with Cv

\)

e ib

X

In particular hol ds thus

On the o ther hand , s ince

J cp t (-y) ( Note that n

=

1! )

-we

1

f (y) dy

t

!(- z) f (y ) dy t

obtain using ( 13)

The p roof o f f E Bsoo is s imilar. 00

Having e s tabl i shed the propo sition it i s easy to prove the non-differentiabil ity o f the Weie rstrass function .

Take

18 thus c = av with a < 1 and ab � 1 and assume f is differentiable at some point x 0 . With no loss of generality we may assume that x 0 = 0 (by translation , if necessary) and that f ( O ) = f ' ( O ) = 0 (by subtracting a finite number of terms , if necessary ) . Thus ( 1 3 ) holds with s = 1 and in place of 0 . We conclude that av = (b - v) . But this clearly implie s ab < 1 , thus contradicting our hypothesis . Example 2 . Riemann ' s first theorem on trigonometric series . In his famous memoir on trigonometric serie s from 185 9 Riemann considered functions or , better, distributions of the form \)

o

0

f (x)

00

n=-oo

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

00

l:

m=- oo

(Notice that -F ' = f ( in distributional sense , of course ! ) . ) The " first theorem" referred 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.

19

Notes For a modern treatment of the variational approach to Dirichlet ' s problem see Lions [ l ] or Lion s-Magene s [ 2 ] .

;

In

partial d i f fe rential e q uations the space w {Q ) is al so o ften den oted H 1 {Q ) . One o f the c lass ical papers by Sobolev i s [3] .

See a l so his book [ 4 ] .

The first systematic treatment

o f Bpsq (Q ) of s > 0 with de fin it ions o f the type a . -e . using finite d i f fe rence s is Be sov [ 5 ] . The space s Bps q (Q ) , s f

intege r are o ften denoted by w; {Q ) , known a s Slobode cki j space s. The space s Bps oo (Q ) are o ften denote d by Hps {Q ) , known as Niko l sk space s . For other work s o f the Sovie t ( = Nikolski j ) School ( Nikol ski j , S lobodecki j , I lin , Kudrj avcev, Lizorkin , Besov,

Burenko v , etc . ) see the book by Niko l ski j [ 6 ] and also the survey articles [ 7 ] and [ 8 ] .

Somewhat outdated but s t i l l read-

able are f urther the survey article s by Magenes-Stampachia [ 9 ] and Magenes [ 1 0 ] where a lso the applications to partial dif ferential e q uations are given .

In the case p = q = 2 see Peetre

[ ll ] , Hormander [ 12 ] , Vo leviv-Pane j ah [ 1 3 ] .

The t �e atment of

nLipschitz space s " in Stein [ 14 ] , Chap . 5 i s based on Taibleson ' s approach [ 1 5 ] .

All o f the relevant works of Hardy and Littlewood

ca n be found in vol . 3 o f Hardy ' s collected works [ 1 6 ] .

In thi s

context see a lso the relevant portions of Zygmund ' s treatise [17] .

The se authors are concerned with the periodic 1-dimensional ca se ( T 1 rather than llin ) . The first systematic treatment of Be so space s using the definition with general

¢

was given in [ 18 ]

20

( c f . also [ 19 ] ) .

But the spe cial case p

=

q

=

2 appears

al re ady in Hormander ' s book [ 1 2 ] whe re a l so 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 app l i cations of it to approximation theory ( see his lecture note s [ 2 0 ] , [ 2 1 ] ) .

The constructive charac­

terization via approximation theory is uti l i zed in Niko l skij ' s book [ 7 ] .

( C f . a l so 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 riz ation via

interpolation originate s from Lions ( see e . g . Lion s-Pee tre [25 ] ) .

The t re atment o f the We ierstrass non-di ffe rentiable

function given here goe s back to a p aper by Freud [ 2 6 ] ( see also Kahane [ 2 7 ] ) .

Riemann ' s theory o f trigonometric seri e s

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

Quotation :

Le s auteurs ont et e soutenus par In te rpo l . J.

Chapter

2.

L . Lions and

J.

Peetre

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

Thi s chapter i s e s sentially a digre s s ion .

We want to

give a rapid survey o f those portions o f the theory o f inte r­ polation space s which wil l be use d in the se q ue l . F i rs t we review howeve r some notions connected with topological vector space s . The mos t important class of topo logical vector space s are the locally convex space s .

In a locally convex space

E

there

exi s ts a base o f ne ighborhoods o f 0 consi sting o f symmetric , balance d , convex sets I , i . e . ( 1- T) U

( 1)

+

T

UC U

if

a UCU

i f J a/ .2_ 1 . and

0< T

.2_ 1 .

A subclass o f the locally conve x space s are the normed space s .

In a normed space E the topo logy come s from a norm ,

i . e . a re alvalue d functional l l x / I de fined on E such that

( 2)

// x + Y 11.2. I I x I I + I I Y I I

( triangle ine q ual ity)

( homogeneous ) / I ex I I = l c I 1 /x / I // x / I > 0 i f x 1- 0 , //OJ / = 0 ( po s itive de finite ) A complete normed space i s t .e rmed a Banach space . In the type o f analysis we are heading for , howeve r , a somewhat larger class of topo logical space s i s neede d , namely 21

22

Thi s me an s that we rep l ace

the locally q uas i - convex one s .

(1) (1

I

by

( 1-T ) U + T UC A U

)

where

A

i s a con s tant 2_

1

if 0 < ' 2 1

whi ch may depend on

U.

I n the s ame

way we arr i ve at the concept o f qu a s i -n orme d sp ace and q ua s i norm i f we rep l ace

(2 1 )

I I x + Yl l

Note that

(2 " )

(2 1 )

l l x + Yl l

<

( 2 ) by

A ( I I xl l +I I Yl l

) ( q ua s i - triangle inequal i t y )

ce rtainly ho lds true i f

<

1 < l l xl l p +I I YI I P)P

( p - t r i angle inequa l i t y )

!

-1

2 1) .

A comp l e te

q u a s i -norme d space we c a l l a q uasi-Banach space .

The q u a s i -

where

A

and p are re l ated by

A

=

2P

(0 < p

no rme d space s can a l so be chara cte r i ze d be ing local l y bounded . The dual o f a topo log i c a l ve c to r space E i s de note d by E 1 • I t always carri e s a local ly convex topo logy wh i ch i s comp a t ib le wi th the dua l i ty , fo r i n s tance the we ak topo logy or the s trong . In p articular i f E i s a q u a s i -Banach space then E '

i s a Banach

space in the strong t opo logy . We pause to give some e xamp l e s o f q uasi -Banach space s . Example l ( Lebe sgue space s ) . e q uipped with a me asure

w

.

If

Le t � be any me a sure space 0 < p<

oo

we de fine L =L W ) p p

!

23

to be the space o f

J.l

I I fl l L p

-measurable fun ct ions such that

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

(with the usual interpretation i f p = 00 ) . space i f

Thi s i s a Banach

l � p � oo but only a quasi -Banach space i f

0

< p< 1. 1 = 1

Al so note that , as is wel l-known , Lp' :::Lp ' where .!. p + p' ( conj ugate e xponent ) , in the forme r case ( excluding p = while , by a theorem by Day , Lp' = 0 in ]J has atoms ) . Thus , the Hahn-Banach we see that q uasi-normed space s may

h

t e latter case theorem

00)

I

{unless

being violated,

behave quite

differently

from Banach space s . 0

Example 2 ( Lorentz space s ) . Let Q and

JJ

be as be fore .

< p , q � oo we de fine Lp = Lp (Q ) to be the space o f q q ]J-measurable fun ct ions such that If

I I fll L pq

(f;

1 1 p t * q ( t f (t) ) � )q

He re f * denote s the decreasing re arrangement o f I fl the formal analogy with the de fini tion of Be sov-space s .

Notice We

The space L i s also known as weak see that Lpp = Lp poo Lebe sgue o r Marcinkiewic z space and i s sometime s denoted by •

Lp * (or Mp ) On 1 y i f 1 < p � oo , 1 � q � oo or p = q = 1 is Lp q a Banach space . In al l o ther case s it i s a q uasi-Banach space . •

One can show that Lp' :::: Lp , , i f 1 < p < oo , l � q < oo o r P = q = 1 . q q E xa mple 3 ( Hardy space s ) . If 0 < p < oo we de fine Hp =Hp ( D ) _

_

24

to be the space o f fun c tion s D

di sc

{

=

z

I IzI

< l } C C s u c h that

u;

s up 0 < r< l If

l� p �

space .

oo

holomo rphic in the un i t

l f ( re

ie

)

l

p

d 8)

l p

H i s a Banach sp ace , o therwi se a q u a s i -Banach P

By the cl a s s i c al theo rem of M . Rie s z on con j ugate

functions we have H ' ::: H , p p

l< p <

if

oo

•

The dual o f H 1 has

re ce n t ly been iden t i fied by Fe f fe rman - S te in . 0

The dual o f

< p < 1 on the o ther h and was previou s ly de te rmined by

Duren Romberg and Shie l d s . l - l, oo L ip s ch i t z ) space B

E

I t i s e s sential ly the Be s o v ( o r •

The theory o f H

p

sp ace s has

( to

some e x ten t ) been e xtende d to seve ra l variab le s by S tein and We i s s .

I n the non-pe riodic c a se , whi ch i s the one o f inte re s t

:

n l ) t o u s , they de fine the sp ace H ( JR p

usin g a s u i t ab le

gene ra l i z ation o f the Cauchy- Riemann eq uation s . to the se space s l a te r on

( Ch ap .

11

) .

That much fo r q uas i - Ban ach sp ace s . turn to inte rpo lation sp ace s .

We re turn

We are re ady to

Ro ughly spe aking i t i s an

a ttempt to tre at var ious fami l ie s o f concre te sp ace s Be sov , Lebe sgue , Loren t z , Hardy , et c . ) view .

( p o tent i al ,

from a common point o f

To be mo re p recise l e t there be given two q ua s i -Banach

space s A0 and A1 and a Hausdo r f f topo logical ve c to r sp ace

A

and a s s ume that both A0 and A1 a re continuo u s l y embe dde d in A -+

The entity A co up l e .

=

{ A0 , A1 } w 1 l l then be te rme d a q u a s i -Banach •

We sha l l now indi cate seve ra l proce dure s wh i ch to a

•

25

given q uasi - Banach couple F(

-+A

-+

-+

A as sociate a q uasi-Banach space

) continuously embedded in

-+

A.

The dependence o f F ( A

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

F ( A ) i s an inte rpo lation functor o r , by abuse o f language , inte rpo lation space . 1°

Complex space s ( Calde r on ) .

Here we have to re strict Let o < 8 < 1 .

ourselve s to the Banach case only . -+

We say

a E[ A 1 8 = [ A0 , A 1 J 8 i f and only i f the re i s an f = f ( z ) , z= x + iy , such that

that

(a)

f ( z ) i s holomorphic and bounded i n the s trip

0<x<1

with value s in A 0 + A1 with continuous boundary value s on the boundary l ine s x = 0 and x = 1 , (b) (c)

f ( iy) i s continuous and bounded with val ue s in A 0 , f ( l + i y ) i s continuous and bounded with value s in A1 ,

(d)

a = f (8 ) . -+

We e q uip [ A

le

with the norm

I I al l E xample 1 .

r

A. 1

8

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

We have 1 p

1 -e

Po

+

(0 < 8 < 1 ) .

Thi s i s e ssentially just a re statement o f the classical interpolation theorem o f

M.

Rie s z -Thorin ( 1 9 3 9 ) .

The main s tep

26

i n the proo f i s always the construction o f an "optima l " f . Here the fo llowing function wil l do : p ( 1- z ) f (z) Example

2.

=

Po

IaI

sgn a .

We have Pps

if s

This was al ready stated in Chap . 1 and a proo f will be given in Chap .

3.

We shall not di scus s any o f the deepe r prope rtie s of + A ] e but content ourselve s to state the following : + + Interpol ation theorem. Let A = { A O , Al } and B = { B O , B l } + + be two Banach c ouples and let T : A + B be a morphism o f couple s ( i . e . a linear mapping such that T : A0 +s 0 T : A1 +B 1 + con tinuously ) . Then T : [A] e+ [ B ] e continuous ly . Moreove r +

holds fo r the operator norms the convexity ine q ual ity

�

sup I I Ta I I I a I I Here generally spe aking I I T I I A , B A a�O + e . that [Al e i s an in terpolation space o f e xponent

We say

This is j ust a mere res tatement o f the functorial character + o f [A] e ( modulo the verification o f the convexity ine q uality ! ) .

27

We leave the details to the re ade r . 2°

Real s pa ce s (Lion s , Gagliardo ) .

the general q uasi-convex situation .

Now we can con sider

First we introduce two

auxil iary functional s , terme d the K- and the J- functional re spectively , as fol lows :

0 < t < oo, a E A 0

If

+

A1 ( l inear hul l ,

j oin ) we put +

K ( t , a ;A )

If

0 < t < oo

K ( t , a ; A0 A 1 ) =

inf a=a 0 + a 1

a E A 0 n A 1 ( in tersection , meet , pul lback ) we put

+

max ( I I a I I A , t I I a I I A 1 0

J ( t , a ; A)

They are in a sense dual to each othe r . 0
,::;,

00

Let

)

•

0 < 8 < 1,

Now we can de f ine +

a E (A) S q < =>

K( t , a ) ) q te

dt

T )

1/ q <

00

or < = > ( Banach case only) 1) or

3

u = u ( t ) ( 0 < t < oo ) :

dt ) 1/ q < oo and 2) t

dt a = fooo u ( t ) T

28

<=>

u

u \) ( \)::: 0 , � 1 ,

and

2)

s =

We e q uip

-+

(A }

00

2::

v =-

oo

+

2 , • • • ) : 1)

00

2::

\) = -

00

\) J (2 , u v ) ve 2

<

00

uv

8 q with the q uas i-norm I I a i l (A)

u;

8q

( K ( t 8, t

a) ) q

dt ) 1/q

T

::: (Banach case on ly) inf f; ( J (t U ( t ) ) ) q dt ) 1 /q t u t \) J ( 2 , a) ::: inf ( 2:: ) q ) 1/ q • \) 8 u \) = 2 1

00

00

Thus there are several de finition s :

in the Banach case three ,

in the general q uas i - Banach case only two. we s tate this as a theo rem.

For re fe rence s

These three ( tw o) de finitions are

Eq uivalence theorem. e q uivalen t .

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 fun ctions wi th value s in a space which is not ne ce s s arily locally convex. Example 1 .

We have

L

pq

if

1 p

1 -8

Po

+

8 < pl ( 0 8

Notice that no condition is imposed on q 0 and q 1 •

<

1) .

Thus in

29

particular we have 1 p

Lp i f

1- 8 + Po

--

8 pl ( 0 < 8 < 1 ) .

This i s e ssentiall y a res tatement o f the clas si cal interThe key to those

polation theorem of Marcinkiewicz ( 19 39 ) .

re sults are certain e xplicit e xpre s sions for the K-fun ctional , the s implest of whi ch is the fo l lowing : K ( t , a ; Ll , � ) Example 2 .

Re cently Fe ffe rman , Rivie re and Sagher have

extended the above re s ults to the case o f Hp space s , and even In particular the i r Lorentz analogue s H p q 1 p

Hp i f q

8 pl ( 0 < 8 < 1)

holds . Example 3 .

I t was mentioned in Chap . 1 that Bps q

1.

f

s

=

( 1 -8 ) s + 8 s o l

(0 < 8 < 1)

I t i s also possib le t o show that Bpsq i f s

(0 < 8 < 1) .

30

The proo fs wil l be found in Chap . 3 . Th is example i s e ssentially a simpler special 0 case of the pre ce ding one . Let c be the space o f continuous Example

4.

bounded functions in

I I al l and let

C

lR

=

(

--oo , oo

)

, with the norm

sup I a ( x ) I

co

1

be the space o f function s whose first derivative 0 exists and belongs to c , with

I I al l

c

sup I a ' (x) I

1

( He re we are cheating a l ittle bit , since this become s a true norm only after identi fication of functions which diffe r by a constant. )

Then (C

0

1

c

1

)

eoo

(o < e < 1)

holds , where Lip

i s the space o f functions satis fying a 6 Lips chitz (Holde r ) condition of exponent e , with the norm sup l a ( x ) - a ( y ) I I x - Yl 6 Thi s fol lows at once from the fol lowing expre ssion for the K - funct ional :

31

K ( t , a ; c 0 c 1 ) ::::

w

( t , a) = sup I a ( x+h ) -a ( x ) l I h l 2t

( modulus of continuity)

Because i t i s 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 di splay a detailed We have the fo l lowing two obvious

Eas y side .

Proo f : e stimate s

If

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

a = a 0 + a1 then fo l lows

Taking the

inf

we thus get w

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

a = a 0 +a1 whi ch i s , i f not optimal , at least "approximate l y " optimal . Hard (er) s ide .

We mus t find a decompos i ti on

We choose a 0 (x)

•

l(X)

...

t

.!_ t J0 a ( x+y ) dy ,

32

We then have

� f:

( a (x ) - a (x a(x

a1' ( x )

+

+

y) ) dy

t ) - a (x) t

whi ch clearly yields I I a0 I I 0 c

I I al I I 1 c

<

w

< t, a>

< W( t , a)

t

Thus we end up with K(t,a)

<

-

I I a0 I I 0 c

+

t I I a1 I I 1 c

.2. 2

w

( t , a) .

The proo f i s complete . Remark .

The above can be general ized to the fo l lowing

more general situation A0 A1

= =

E =

D ( A)

any Banach space , =

the domain of the in finite simal ge ne rator

33

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

E;

i . e . we have

G ( s ) , G ( t ) -rid

G ( t+a ) = G ( t )

s trongly as t-rO ,

G ( t ) a-a i f a € D (A ) . ! I G ( t ) I I < C , Aa = l im t t+O 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 re q uirement t i l G ( t ) Aa l l � C , which in particular impl ie s that G ( t ) i s a ho lomorphic semigro up , we can al so prove that K ( t , a ) :::

sup 0 < s �t

s I IG ( s ) A a I I

The detail s are le ft fo� the reader .

Thi s perhaps helps the

reade r to under s tand why the various de finition s of Be sov space s in Chap . 1 , unde r the headings a . - j . , are e quivalen t . I n Chap . 8 we shal l howeve r , give a di ffe rent ( equivalence )

34

proo f. We no·w list some auxil iary properti.t! s of the space s +

(A t

q

•

First we have an

Interpolation theorem.

Analogous to the inte rpolation

theorem in the complex case . We al so compare the re al and the complex space s . Comparison theorem. +

C A) e 1

C

(Banach case only)

+

+

[Al e c (A) e oo

CO <

e

<

•

We have

1) .

A particular instance o f it is the re l ation ( see Chap . 1 ) :

1)

Proof (out line ) : the

repre sentation

+

Let

a E (A) 8 1

•

Then

a

admits

u ( t) dt . with a sui table u . We T a = f ( e ) wi th a holomorphic f s imply

a =

!00 0

obtain a repre sentation by taking

+

It fo llows that a E [Al e

whi ch i s a k ind of Mel lin transform. 2)

Let

+

a E [Al e

Then

•

a = f ( e ) with f holomorphi c . K ( t , f ( iy) )

<

a

admits a repre sentation

Obviously we have

l l f ( iy) I I A0,:s

c

•

35

K ( t , f ( l+i y) )

�

t iif ( l +iy) II A � C t 1

Thus the three l ine theore m , usually name d afte r Doetsch , but real ly due to Linde lo f , I have been to ld , yields K (t, f (8) )

K ( t , a) and we have

-+

a E (A) 8 00

�c

t

8

The proof i s complete .

We al so mention anothe r Comparis on theorem. -+

( A) 8 q

1

C

We have

-+

(A) 8 q

if

2


In parti cular we see that

The mos t important re sult of real interpo lation is howe ve r the fol lowing : Re iteration theorem.

Let

-+

X

{ X0 , x 1 } be any q uasi-

Banach couple such that -+

-+

( A) e . q . cx i C ( A) e . oo l

l

for some q 0 and q 1 > 0 but that

l

( i= O , l) Then it fo llows

36

+

(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 q 1 . Finally we mention the following powerful Duality theorem. (Banach case only) Assume that A0 (') A1 is dense in both A0 and A1 • Then •

holds. This contains in particular the result concerning the dual o f Lorentz space . We also get ( Bpsq )

I

�

Bp-si , q if 1 � p < I

oo

1�q <

oo

Indeed we use the duality theorem along with the fact that ( Pps

)

1

::::

p

-s p if 1

1

�p <

00

(One can also determine the dual when 0 < q < 1 . In the Besov case one finds : ( Bpsq )

I

� �

Bp-si , oo i f 1 � p <

oo

, 0
37

We conclude thi s chapter b y giving some applications whi ch are intended to di splay the power of the technique o f inte rpo lation space s .

a

I - where we res trict our-

We cons i de r the potentials 0 <

selve s to the case

a


In Chap . 1 they were de fined

using the Fourie r tran s fo rm .

Howeve r , i t is al so po s sible

to de fine them as ce rtain convo lution s . I where

c

-a

I x-y I

f (x) = c

a-n f (y ) dy

i s a cons tant e xpre ssible in

fol lowing holds true . Thea rem 1 .

We have

I -a

Inde e d

Lp+Lq

r

factors .

i f q1 = p1

I t implies the fo llowing

�P

The a

n'

1 < p < na

i f .!_ .!_ - � ' 1 < p < !!, n q = P k Indeed thi s i s preci sely the Sobolev Embedding theorem

Coro llary .

We have

+L q

( Chap . 1 ) , e xcept that we cannot capture the case

p = 1 in

thi s way . Proo f o f coro l l ary :

For simplicity let us take k

We start wi th the fo llowing Fourie r tran sform i dentity f

n l: j=l

Taking the inve rse Fourie r tran s fo rm we ge t f

n l: j =l

a.* D . f J

J

1.

·

38

- i � j 1 1 � 1 2 • The important point is that one < C l x l 1 -n There fore one gets

where has

l f l < C I -l l grad f l

=

c l x l l -n * l grad f l

Let now f e: Wp1 Then j grad f I e: Lp by definition and f e:Lq by th. 1 where � = p1 - n1 . The proof i s complete . Proof o f th. 1 : The proof goes via O ' Neil ' s inequality stated below. Indeed it is readily checked that l x I a-n E:L p oo if (n- a) p = n . There fore lx I a-n * f ELq if 1 1 a 1 -1 + -l - l or, after elimination o f if q p p q p n There remains O ' Neil ' s inequality. a e:L p oo' f E:Lp => a * f ELq i f 1 -1 -l -1 , 1 < p < 1 < < oo . q p p This more recent result should be compared with the classical Young' s inequality. aE:L p , fE:Lp=> a * f E: Lp i f 1 q p.!:.. + .!:..p - 1 , l =< p =< p ' , l =< p < oo For the proof of O ' Neil ' s ineq uality ( the proof of Young ' s inequality is similar using Riesz-Thorin) we consider Then we have the mapping T : f->a * f where a e: L p oo •

P,

=

+

T: L1 -+ L p oo T : L p' l -+L oo

P

' ,

P

(by Minkowsky ' s inequality) ,

(by the fact that L 'p 1 L p oo By interpolation we then get �

)

•

39

T : ( L1 , L , l ) r p 8

-+

( L oo , L ) r • 8 p 00

But Lpr i f 1p

( Ll , L ' l ) r 8 p ( L oo ' Loo ) 8 r p

=

L

1 qr i f q

Elimination o f 8 give s pre ci sely

1- - 8 8 1 + PT , 1-8 p 1 p

1 q

+

-8

00

1 p - l.

+

Thus we have T : Lpr -+ L qr =

r and notice that Lpp

Final ly we take p

Lp ' L CL q whi ch qp

yields

We are through . Next we conside r , takin g n Hf ( x )

=

1

X

*

f (x)

=

=

1 , the Hilbert trans form

f ( �-y) dy p.v. �

=

s

l im -+o

f

IY 1 2:E:

f (x-y) dy y

where the inte gral thus i s a principal value ( p . v. ) in the sense o f Cauchy.

Notice that

40

i sgn t;, f ( t;,) "'

In the case o f T1 ( the periodic case ) thi s i s the operation which to a function , given by the boundary value s o f a harmonic function in the unit disc D, assign s the conj ugate fun ction . The fo llowing cla s s i cal re sul t holds true . H : Lip s -+ Lip s , O < s < l . Later on we shal l prove much more general resul t s { for Theorem 2 .

We have

arbitrary r , general Be sov space s and general convolution operators ) . Proo f of th. 2 : H \)f { x )

We e xpre s s H as a sum H = J

IV

00

L:

V = - oo

H v where

f { x-y ) d y , y

( Th i s amounts to about the s ame a s taking princip a l value s ! ) We write now y) dy H vf ( x ) = !I f ( yx\)

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

\) l y) dy - f ( x-2v + ) + f ( x-2 ) f (xDH vf ( x ) = - !I 2 \) 2 v +l \) y 2

+

v f ( x+2 ) = 2

This give s the e stimate s

Df ( x-y ) dy . y

+

41

or

v c 2 l it I I 1 c

or

c

l it I I 1 c

or in terms o f the J-functional J (2

v

1

Hv f ) <

c

I I f I I 0 or c 2 I I t I I 1 c v

c

I f we apply thi s to an arbitrary de compos ition

f = £ 0 + £1 we can al so write thi s in terms o f the K- functional : J(2

\)

1

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

\) f E Lip s , be cause Lip s = ( C 0 , c 1 ) e s ' we have K ( 2 , f) � c 2 v s v and thus J ( 2 , H v f ) � C 2 v s . Now re cal l that H f = IH v f . Using again Lip s = (C 0 , c 1 ) s and the second o f the e q uivalent 8

If

de finitions using the J- functional , we thus concl ude The p roo f i s complete .

42

Notes Concerning topological ve ctor space s see the

book by

Kothe [ 2 8 ] which also contains a bri e f treatment of locall y bounded space s .

Concerning the dual o f Lp in the q Banach case see Haaker [ 2 9 ] where a short proof o f theorem also i s indicated.

Cwikel [ 31 ] .

quasi­ Day ' s

See al so Cwike l-Sagher [ 3 0 ] ,

For Lo rentz space in general there i s the ex-

cel lent survey article by Hunt [ 3 2 ] .

The classical theory of

Hp space s can be found in Duren ' s book [ 3 3 ] .

The dual of

0

Hp ( D ) i f

was determined b y Duren-Romberg- Shields [ 34 ] . The ir re sult was extende d to the case o f Hp ( lR n+ + l ) by Wal sh [ 35 ] . The dual o f H 1 ( lR � + l ) was determined by Fe ffe rman ( The case o f H 1 ( D) i s o f course imp l i citly contained there in . ) For an introduction

and S te in in thei r fundamental work [ 36 ] .

to Hp space s of seve ral variables see Ste in-We i s s [ 3 7 ] , Chap . 3 or Stein [ 14 ] , Chap . 7 . For

a

more de tailed treatment o f interpolation space s

we re fe r to Chap . 3 of the book by Butzer-Berens [ 3 8 ] . Seve ral other books deal ing with interpol at ion 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 ourse lve s with

j ust a sketch of the hi storical development of the theory .

F ir st

-

o f a l l , a discussion of the cla s s i ca l interpolation theorems of ( M . Rie sz - ) Thor in and Marc inkiewicz can be found in Chap . 12 of Zygmund ' s treati se [ 1 7 ] .

The abstract theory of interpo lation

space s wa s created around 1 9 6 0 by Lions , Gagliardo , Calderon ,

43

Krein and others . Calde ron [ 4 1 ] .

The complex spaces are s tudied in

The real space s are s tudied in Lions-Peetre

[ 2 5 ] and in the pre sent form - with explicit men tion o f K ( t , a ) and J ( t , a ) - in Peetre [ 4 2 ] .

The extens ion to the

quas i-Banach case come s late r .

See Kree [ 4 3 ] , Holmstedt [ 4 4 ] ,

Sagher [ 4 5 ] , Pee tre -Sparr [ 4 6 ] .

In the latte r work it i s not

even as sumed that the space s are ve ctor space s , i . e . the additive structure alone enters .

Conce rn ing integration in

quasi -Banach space s , see Peetre [ 4 7 ] and the works q uoted there .

Concerning inte rpolation of Hp space s see Fe ffe rman­ S te in [ 36 ] , Riviere - Sagher [ 4 8 ] , Fe f fe rman-Riviere - Sagher [ 4 9 ] . Compare with the class i cal treatment in [ 1 7 ] , Chap . 1 2 .

More

pre cise re sults conce rning the compari son o f the complex and the real space s can be found in Pee tre [ S O ] , [ 5 1 ] .

In the

latte r paper the re i s al so mentione d a third type o f inte rpolation method which somehow lie s in between the re al and the complex. Concerning the dual o f Bpsq when 0 < q < 1 ( o r q =00 ) see Peetre [ 5 2 ] . See also Flett [ 5 3 ] . The present treatment o f O ' Ne i l ' s inequal ity [ 5 4 ] can be found in Peetre [ 5 5 ] .

See

al so Pee t re [ 5 6 ] where the same type of techni q ue is applied to general integral operators whi ch need not be translation invariant.

For Young ' s ine qual ity ( via Thorin ' s theorem) see

[ 1 7 ] , Chap . 1 2 .

The re sult on Rie s z po tential s ( th . 1 ) i s

due to Sobo lev [ 3 ] but was late r independently redis cove re d by Thorin [ 5 7 ] . In the case o f T 1 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 ] .

44

Quotation :

The sphere i s the mos t uniform of solid bodies Origen, one of the Fathers of the Church , taught that the blessed would come back to life in the form of sphere s and would enter rolling into heaven J . L . Borge s "The Book o f Imaginary Beings" De finition and basic propertie s o f Besov space s . •

•

•

•

Chapter

3.

Now we are ready to embark on a more systematic study of Besov spaces . First we collect for reference some basic facts concerning tempered distributions and Fourier transforms which we have already freely made use of in the preceding. Let S be the space o f rapidly decreasing functions , i . e . f E S <=> for all multi-indice s a, S , x s D f (x) = 0 ( 1 ) as x -+ oo If we e q uip i t with the family of semi-norms a

sup X S becomes a Frechet space . Obviously S is stable for derivation and multiplication with coordinates : for all a , S f E S = > x 13 o f E S and these are , moreover, continuous a operations . The dual space S' = S ' ( JRn) is called the space of tempered distributions . By abuse of notation the duality between s • and S i s generally written as an integral : < f , g> e . g. if

8

f n f ( X ) g ( X) dX i f f E S JR

1 1

g ES

is the "de lta function" then we have

•

I I '

45

= g ( O ) = JlRo ( x ) g ( x ) dx By duality D and

v.

extend to

if

g E:

s

S' .

In dealing with the Fourier trans form it i s often convenient to have in mind two space s lRn , one " latin " space lRn = lRnX with the gene ral e lement

x = (x 1 , . . . , xn ) and the dual " greek " space with the general element � = ( � 1 , , � n ) , the dual ity

R�

•

•

•

+ x n E;, n . Thi s i s al so natural from the point o f view of phys ics where x o ften is " time " ( se c ) and t;, " fre quency" ( se c - 1 ) so that x E;, i s " dimensionle s s " . I f f E:

s

= S X its Fourier trans form is an e lement

given by

A

That

J n e -ixE;, f ( x ) dx . ::IR X

Ff ( t;, )

f ( t;, A

f E: S t;, can be seen from the basi c formulas ( i t;, ) f . a

( D af )

(1)

(x s f )

(2)

( - iD

E;,

) S Ff .

More gene rally (1' )

F (a

(2 ' )

F (b f )

*

f)

Fa F f , 1

Fb

*

Ff,

A

f = Ff of

46

under suitable assumption on a and b. We will also need the formula "'

f ( t t;, ) where

(3)

The inverse Fourier trans form is given by

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

2 n I f ( x) I dx

JR. X

1

Since F and F -l are continuous operations F : Sx + St;, , F - 1 : S t;, + Sx , they extend the duality to tempered distributions S ' . Formulas ( 1) and ( 2 ) (or ( 1 ' ) and ( 2 ' ) remain valid for tempered distributions . Remark. Using instead duality 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 : Lp + Lp ' p if 1 < p < 2 . These are the theorems of Hausdorff-Young and Paley. -+

47

If

f

i s a function o r a ( tempe red) distribution we

denote i t s support by supp f , i . e . the smal le s t closed set such that

f

van i she s in the complement .

more to a phys ical language supp

A

f

con s i s ts o f those

fre q uenc ie s whi ch are neede d to build up b ination s o f characters e ix s

f

from l inear com-

•

:

that

Appealing once

Now let {

V cj>v(S) 'I 0 i f f E;,s int 1\ whe re 1\= {2V-1� I s I �2 +l }

(5)

( Tauberian condit ion ) v

v

1 _:_cs>o i f s;s�s={(2-s)-1 2 .::_lsiS2-€)2

(6)

l;v(O

(7)

v S I o6¢v ( 01 � c 2- I I 6

S

for e ve ry

Sometime s we shal l al so re q uire that 1

(8) Let al so

cil

be

an

(9)

cp €

(10)

�(0

( or

00

L: -oo
o

(x) )

auxil iary " te s t funct ion " such that s

'I 0

if

E;, s int K , whe re

K ={lsi �1}

( Tauberian condition )

48

( 11 )

I � I

> cs >

o if

t,; s

K

€

Example 1 . I f ¢ i s any function in s with supp = RO ' I I .:. c s > o if t,; s Ro € then we can define v by v � ( f,; /2 ) . If we in addition assume that (E,; ) > 0 v ( E,;) then we get ( 8 ) , upon replacing v (f,; ) by A

A

I

=

A

if necessary. In this case we can take l:

-1

v=

- oo

A

< U . v

This special type of test function , we encountered already in Chap . 1 , except that there we used a discrete parameter t ( roughly t ::: 2 - v ) We are now in a position to formulate our basic definitions Then Definition 1 . Let s be real , 1 � p � oo , 0 < q � oo we set •

•

{fl f

s

S' <

&

(Besov sp ace ) This space we e q uip with the ( quasi-) norm

00

}

49

Some words o f explanations are in orde r here . Compared with Chap . 1 two changes have been made . Firstly , the parameter q has 0 < q < 1 included in the range . Thi s means Bpsq is not always a Banach space . Secondly , as already noted , we used the discrete parameter V ( V = 0 , � 1 , � 2 , ) instead of the continuous one t ( O < t < oo). That we neve rtheless obtain the same space s at least if 1 � q � oo will be clear later on . It will be al so proven in due course that the definition is independent of the particular test functions {¢v} �=-oo and Finally as was already said in Chap . 1 we usual ly do the calculations "in base 2 " . It is clear that 2 can be replaced by any number b > l . Definition 2 . Let s real , 1 � P � oo Then we set (with J �) •


•

•

•

=

{ f If

s

S'

(potential space ) This space we e q uip with the norm IIf II

pps

= I IJs f I IL

p

& II

Js f I I L

p

< oo }

50

This is exactly as in Chap . 1 . Example 2 . If f o (delta function) , so that then we have n 00 ' 1 1) .!. + f s Bp p (p p' ::

"

f

:;:

1,

and this is the bes t result in the sense that f � Bp-sq if n s > - p� ' or s - p' q < 00 In fact we have

I

We have also (Use ( 2 ) and ( 7 ) to estimate n but it is not possible to make as strong a coni f s < - p' elusion as in the Besov case . Example 3 . More generally , if f ( � ) :: l � l -0 in a neighborhood of oo and i s C00 elsewhere then f s Bp 0 -n/p ' ' oo and this is again best possible in an analogous sense . Example 4 . More generally if in a neighborhood of oo and C00 elsewhere then f s Bp0 -n/p ' ,q if q > .!. T

•

It is often convenient and sometimes even nece ssary to work with "homogeneous". ( q uasi- ) norms ( i . e . homogeneous with respect to dilations , or in D) . We there fore also de fine the following modified spaces .

51 De f inition 1 .

We set

Thi s space we e q uip with the ( quas i - ) norm 00

2:

V =-co De fini t ion 2 .

We set (with

I

r-IS:)

{ f/f E S' & / /I s f //L p

<

00

}

Thi s space we e q uip with the norm

He re ari se s howeve r a certain comp li cation .

Namely they

are not true ( q uasi - ) norms since they are not pos i tive de finite ( indeed / /f// .s q= O f i s a polynomial ) . The same phenomena Bp we encoun te red alre ady in Chap . 2 in connection with the e xample with Lip s . Al so I s f cannot be de fined for al l f E S . Indeed we would l ike to have I s f ([;) =/ E; / s 1 ([;) as in the But the fact that [; = 0 i s a s ingularity i f s 0 case of Js •

<

i s an obstacle . The remedy for all thi s is to do the calcul us modulo polynomial s , of degree < d , where d is a suitab le n umber .

Let us

52

give a complete analysis of the situation . For simplicity and with no essential loss of generality we may assume that ( 8 ) i s valid. Let us consider the doubly infinite series 00

V =�oo

v * f.

It is easy to see that one half of it, namely the series 00

V=L: O

<j>

f

V*

converges weakly in S ' for any f s S' , for the Fourier transformed series

does so. Indeed we have an estimate of the type I f f ( � ) g ( O d� l


L:

l a l� m , I B I� m

_,

! l � l l o 8 g ( O i d� if gs S

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

Indeed it turns out that each term is

- VA )

0 (2

with A

>

0.

The

53

hal f o f our se rie s -1 E \) =-

00


\)

*

f

cause s much more trouble . if

s > O and

= 0 if

1 i f n = 1 and f (s ) = -

E.g.

.;

I t i s how-

s < O i t i s not conve rgent .

eve r true that the derive d seri e s 00

\)

S ' i f I a I i s sufficientl y large

converge s in I a I � d.

= --00 1

say

1

if

To see thi s we use the above e stimate in the case o f

the serie s

A

( The term factor k i l l s

A

alone give s d > B!)

B if

0 ( 2V

B) wi th

B > 0 but the extra

The conve rgence o f the de rived

serie s i s however e q uivalent to the existence of se q uence { PN } �=l

o f polynomials of de gree < d such that 00

E v =-N converge s in

S ' as N


\)

*

+ oo

verge s modulo polynomial s . from

f

f •

-

In other words the se rie s conI t i s clear that the l imit di ffers

by a distribution with

A

supp f = { O }

1

in o ther words

54

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

00

L:

v = - oo +

¢v *

f (modulo polynomial)

polynomial.

We want however to say a little bit more about how big the number d must be . First we state the following lemma that will do us great service in what follows too . Lemma l . Let f E: s• with supp f K ( r ) = { I � I � r } Then holds n (­pl •

(12)

(13) holds

Assume that supp f (13

I

R (r )

r

<

I � I <2 r}

•

Then

)

If r > l we can as well substitute J for I . Remark . I f f E: S . to say that supp f is compact is 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 ine q uality ( first stated '

55

for

T1 and trigonometric po lynomial s ) . Proo f :

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

wel l take

r = 1.

(Al so it would have been sufficient to

prove ( 12 ) for p 1 = oo , for in view o f Holder ' s ine q ual ity we have -

1-8 p

+

�

.

00

)

A

Now let ¢ be any function in S with supp ¢ compact and � ( �) = 1 holds .

if

� E: K ( l ) ( =K 0 ) .

Then the i dentity

f = ¢* f

Using Young ' s ine q ual ity ( see Chap . 2 ) we now get =

Thi s fini she s the proo f o f ( 12 ) . identity

D

a

f =

D

a

and we are through .

¢* f .

c

1 p

I I fi lL p

+

'P"

1 -1 .

To prove ( 13 ) we use the

Minkowsky ' s inequality then yie lds

The proof o f ( 1 3 ' ) goe s along similar

l ine s . Let now f b e a di s tribution which ful f i ls in de f . 1 .

Then holds in particular

by ( 12 ) (with p l see that i f s < � p

the condi tion = 0 ( 2 - vs ) or

I l ¢v * f l I L n -� ) P v ( ) . We there fore P oo ) i l ¢ v * f i i L oo= 0(2 our seri e s conve rge s i n L and s o i n S' . ;:>

00

A similar argument shows that thi s i s true al so i f

56

n s = p , q ;;; 1 .

Wi th the he lp o f

(13)

(with p = oo ) we can

extend thi s argument to the derive d serie s . We find that it converge s in S ' i f l a l ,2: d and s < d + pn or s = d + p!!. q �1 . Thus to summarize the s i tuation I l f l I s q become s a true B p ( q uas i - ) norm i f , with d as above , we agree to do the calcul ations modulo polynomia l s of degree

>

time polynomial s o f degree f

s

< d , e xcluding at the s ame

d.

We can now a l s o give a pre ci se de finition o f S ' w e de fine I s f b y the formula

Is .

If

(modulo polynomial s ) Each term i s here uni q ue ly de fine d (by the re q uirement that its Fourier trans form should be l � l s ¢v f) but the sum is determined only up to a polynomial . To s ay that fs P s i s thus interpreted so that there exists 00

v =-oo

Da I s (

cp

v *f)

-+

Da

g

g s Lp such that as N

-+

oo

•

we agree to adopt the s ame identification convention for :P s as for :B s q . p p s sq s is The connection betwee n Bp q and :Bp ( or Pp and sq sq al so apparent now. Namely i f f s Bp or f € Bp then we have

f>;)

57


Now by the above remark

* f

i s an entire fun ction of

e xponential type ( � 1 ) , thus in particular c In other words the di stributions in B s q and B s q have the s ame local p p regularity prope rtie s . In what fol lows we shal l mos tly work with B ps q but many of the proofs are valid for ( Re ade rs should check thi s point e ach time ! ) counter Bps q and not

00

In the appl i cations we will often en-

We also indicate two more generali z ations of Pps

;

B q and

First we notice that in the definition s the unde rlying space Lp = Lp ( ffin ) coul d be replaced by any trans lation invariant Banach space of fun ctions or distributions X. ( Such •

space s are sometime s termed homogeneous . ) For the new spaces we sugge s t the fol lowing notation : B s q x , P s X . We may al so introduce analogous Sobolev space s WkX . In the s ame way we use B s q x , P s X , �X . Example 5 .

If

X

L we are back in the old case . p

we have :

Example 6 .

If X

L

pr ( Lorentz space ) we also write =

P s Lpr

Thus

58 The se spaces we may call Lorentz Besov, Lorentz potential , Lorentz - Sovolev spaces . Example 7 . Another important case i s ( i . e . f E: F-l Lp <= > f E: Lp ) • The space s are s q related to certain space s Kp 1' ntro d uce d b y Beurl1ng and Herz . The precise relation is ·

F -1 Kps q or Secondly we notice that if � is any q uasi - Banach space of se q uences then we may replace the defining condition by

We then obtain space which might be denoted by B �X ( and analogously B � in the homogeneous case ) . Clearly we get B � = B sq x if � = £sq where a = { a v } � =O E: iff � (2\! s I a ) ) 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 B sq x even in a more general , abstract form will be discussed in Chap . 10 . Now, all definitions being made , we can start our study of Besov spaces . We begin with a completeness result. Theorem l. If l ,�: q � oo B� q is a Banach space , if sq . 0
59

First we prove a use ful technical lemma .

g E V ' F;, ( the space o f all di stributions (not tempe re d ) in lRn F;, ) De fine fv and F by fv = ¢v g and F = g . As sume that Lemma 2.

Let

A

•

+

( 14 )

g E S F;, )

Proo f :

l:

v=O A

f = g . ( In particular thi s

•

It i s clear

We may assume that ( 8 ) i s valid .

F ES' .

that

00

fs Bps q such that

Then the re e xists holds for

(

00

We shal l prove that the se rie s

conve rge s in

S'.

I f the sum i s denoted by f ' and i f we put

f = F + f ' 1 then i t fol lows that

* f = F so that f E Bps q I t is also clear that f = g 1 because o f condition l: in S ' on the f To e stablish the conve rgence o f (5 ) V = l \! Z I -0 f other hand i t suffice s to prove the conve rgence of \) =1 for cr sufficiently large . To do thi s we use (12) and in L A

•

00

•

00

( 13 ' ) o f Lemma 1 to conclude that <

<

c

2

n p

- cr

)

I I f \) I l L � p

c

2

v(

I t i s now clear that the serie s converge s in n - s.

cr> P

n

P - cr L

00

s)

if

We al so take thi s opportun i ty to mention the fol lowing

use£ul characteri zation of

60

Le1Tli\1a 2 ' Let { fv } � =l be a se q uence and F a member of s • such that supp f C R , and supp F C K and assume that (14 ) v v holds true . Then the series L:=l fv converges in Let f ' be its sum and define f = F + f ' . Then f Bpsq Conversely every ft.: Bpsq can be obtained in such a manner . 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 =

00

\)

E:

S• . •

iP

Then

i s complete , thus q uasi-Banach, i ff every series is convergent in E . The L: 1 such that i=l proof is the same as for the normed space , in which case we of course can allow ourselve s to take p = l . Now finally to the Proof of Th . l . Bps q is p -normable if P = min ( q , l ) . Let L: f 1. be a series in Bpsq such that i=l 00

E

x .

00

00

L:

i=l

j j f 1. j j p

Bpsq

< oo

61

We shal l show that i t converge s in f E: Bps q and that (15 )

II f I I s q � B

p

<

S • to some e lement

11 IIf.1 I I P q ) P 'I � i=l B

•

()()

E f; and letting i=M -'M -+ oo we see that the se rie s in fact conve rge s to f in Bps q

Applying the same e s timate to the " tai l "

•

To establ i sh ( 1 5 ) we first observe that for e ach v the se ries ()()

()()

E

.

•

A

"'

supp FC K .

Also i t i s e a sy to see that +

(

()()

2:

V =Q

Using Lemma 2 we see that

fE: Bps q and ( 15 ) fol l ows .

Next we cons ider various compari son (embedding) theorems . First we compare B ps q with S and S' which i s a rather trivial matter . Theorem 2. We have a ( continuous ) embedding S -+ B�q · Al so S i s dense in Bps q i f p , q < oo. Consider f v = � v * f . Then by ( 7 ) Proo f : Le t f E: S for any o , IDS f v ( .;) 12.c i � I - I SI- o holds .

o r , for any k ,

Using (2) we find

62

i

f v (x) i .:S.-

C

2v (n-o ) / ( 1

+

( 2v l x l ) k

It follows that I I fv I I L � p

C

1 2v (n ( l- p-) -O )

Taking sufficiently large we see that f E: 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 of those f in Bps q such that supp f i s compact certainly is dense in Bps q , i . e . the exponential functions . I f suffice s now to invoke the classical fact that if p < co the exponential functions are dense in Lp (non-periodic analogue of the Weierstrass approxi­ mation theorem) . The proof is complete . Theorem 3 . We have an embedding Bps q S ' . Proof: Only the continuity has to be verified. To this end it suffice s to remark that if f EBps q then is sufficiently large (more precise result will be given in a moment ! ) and that this corre spondence is a continuous one . For the embedding L S ' i s apparently a continuous one . Next we compare Besov spaces with the same p . Theorem 4 . We have the embedding o

A

-r

co

-r

if s 1 < s or

63

Bpsl -+ pps -+ Bpsao . I f s k integer > 0 then oo Bpk l -+ �p -+ Bpk Moreove r �p = Ppk i f 1 < p < oo (or k =

Al so

Proo f :

=

0) .

As was already stated in Chap . 2 , thi s can be

prove d using interpolation and the theorem below. dire ct proof re sul ts e a s i l y i f we notice that nSq N

Howeve r a s q C n l l N

under the said condition s relating the parameters . The proof o f the s tatement invo lving Pps is left to the reade r . The proo f o f the last s tatement concerning Wpk wil l be po stponed to Chap . 4 .

I t i s based on the Mihklin mul tipl ie r theorem.

Much more interesting i s the fol lowing Theorem 5 (Be sov embedding theorem) . We have the ems q bedding Bps q -+ Bp 1 provided l Proo f : After al l the se preparations , the proof can almost be reduced to a trivial ity .

Then by

Let

( 12 ) o f Lemma 1

in the said conditions on the parameters .

for any

p1� p

Since clearly s q The proof i s f�::Bp l l

we see that

•

For comparison we write down the corre sponding result for potential space s . Theorem 6 (po tent ial embedding theorem) . We have the s p s 5 s 1 = pn - s , B p l provide d � embe dding P p -+ Pp l p l l l -

64 p 1 > p , s 1 < s and 1 < p < oo . It admits the following immediate Corollary ( Sobolev embedding theorem) . We have �p + Lp np - k , p 2:, p , k integer 2:,0 and l < p < oo . l provided 1 Remark . As we know the corollary remains true for p = 1 too but this calls for a special proof (c. f . Chap . 1 ) . Before proving Thm. 6 we first settle the q uestion of real interpolating Besov and potential spaces , for the proof re quires interpolation . The result is already known to us from Chap . 2 . Theorem 7 . We have Bpsq i f s

=

It has several important corollaries Corollary 1 . We have

Proo f : use the reiteration theorem (Chap. 2 ) . Corollary 2 . Bps q does not depend on {

-

•



65 00

Bps q i s invariant for a local c coordinates ( in JRnX ) Corollary 4 .

change o f

•

He re come s the

Proo f o f th . 7 :

Let

By ( 1 31) o f Lemma 1 (with

J in place o f I ) and Minkowsky ' s ine q uality we have (16)

I l L � C 2 - V s iiJ s (

I I

( The e stimate ! I

1L � p s £1 I I p l p

Taking the in f ove r al l such de compo s ition s we get

In a s im ilar way we obtain

By

66 Hence + (

00

\)

l:

=1

Since , generally speaking , K ( t , a) < max ( 1 , ts ) K ( s , a ) , -

because K (t , a ) is a concave function of a , we see that the last expression e ffective ly can be estimated by 00

dt ) 1 /q = c l l f l l s C ( 0f ( t - e K (t , f ) ) q T (Pp 0 s s Thus we have proven (Pp 0 ' Pp 1 ) 8 q C Bps q . With no loss of For the converse inclusion let generali ty assume that ( 7 ) holds true . Using again ( 16 ) we obtain W@ may

We have likewise the estimate

s J ( l cp * f ; pp o I

67

sl p _:s_ c I I cp * f I I L p )p

I

00

E ¢ * f the e q uivalen ce theorem yie lds + Since V =l V s f E: ( Pp 0, The proo f is comple te . thus proving ::l We still have to prove th . 6 . I t i s howeve r convenient •

to in sert he re the fo llowing re sult which has been more or l e s s imp l i ci t in the pre ce ding discussion . Theorem 8 . For any n we h ave an i somorphi sm J n : Bps q - + Bps- n , q and an i somorphi sm J n·. Pps -+ Pps - n Thi s means also that in what fol lows we can o ften take s = 0 . 0

Proof :

The s tatement for potential space s results at

once from de f . 2 .

For Be sov space s i t fol lows by interpol ation

( th . 7 ) or by a s imple dire ct argume n t .

The i somorphi sm cla s s of B; q depends only on q ,p . The i somorphism c lass o f Pps depends only on p . Proo f : Obvious . Pps ) 8p Coro llary 2 . ( Pps 1 - e + � ( o < e0 < l ) l if 1 Po P1 P Proo f : Use Coro l lary l .

I

0

For completene s s ' s ake let us also mention the fol lowing e lementary re sul t , the proo f of whi ch i s le ft to the re ade r . Theo rem 9 . Fo r any multi-in dex a we h ave con tinuous map s 0 D : B s q ->-B s - 1 a � q and Ps Ps- I a I p P a· P a p Now final ly to the .

Proo f o f th . 6 : s1

__,.

•

For the s ake of s implicity let us take

0 which by th . 8 is no re striction . By th . 5 and th . 4 n - s . By inte rpolation where p

68 ( th .

7) '

keeping

p

fixe d but varying

s , p 1 we get

P + s; + Lp oo • A new interpol ation ( us ing cor . 2 o f th . 8 ) P 1 . s Ps + L now with s fixed and varying g1ve p p l p + Ip 1 . s Passing back to a gene ral s we have Pps + p 1 A final pl interpolation ( th . 7 and cor . 2 of th . 8) , thi s time with s , p 1 s p The p roo f i s fixed and p s 1 varying , leads to Pps + Bp 1 l complete . 1

The above proo f actual ly yie l ds more than the s s p theo rem says , namely Pps + Pp l p ,... Bp l p ( He re we used the l l Lorentz-potential and Lorentz-Be sov space s . ) Remark .

1

•

•

The re al inte rpolation o f potential space s was settled in th .

7.

As we said , one get s Be sov space s as the re sult

o f the interpo l ation .

Now we write down the corre sponding

re sul t for complex inte rpolation . Theorem 1 0 .

We have

s s [ p 0 ' pP 1 ] e P

P; i f

=

( l-8 ) s 0 + e s 1 ( o < e < 1 ) and l < p < oo s s Then there exists Proof (outline ) : Let fo: [ P o ' PP 1 l e P a ve ctor val ue d hol omo rphic function h ( z ) such that s

•

f , s up 1 1 h < i Y ) 1 1 s 0 � c pp

h <e )

=

( and

s < oo ) . sup l l h ( z) l l s0 O < Re z < l pp + pp 1

Conside r the function

sup l l h ( l+iy ) l l p s l � c p

69

s ( l- z ) J o

+

slz

h (z)

I t can be proven ( e . g . using the Mikhlin with value s in Lp mul tip l ie r theorem , c f . Chap . 4 ) that the ope rators J iy ( y real ) •

are bounded in Lp and that (17) for some

A > o. hl ( 8 )

Using ( 1 7 ) we see that J s f , sup l l h l ( iy ) I l L ,;S c ( 1 p sup I I h 1 ( 1 + i y ) I I L ,;S C ( l p

+

+

A IY I ) ,

I Y I ) A.

I t i s pos s ib le to show that the three l ine theorem ( see Chap . 2) is s ti l l applicab le . Thus we conclude J sf s Lp and s s s f sP; . This prove s [ Pp o ' Pp 1 ] Pp · For the conve r se we have to f ind an at leas t approximatel y optimal repre sentation f

=

h ( 8) .

A natural choi ce is h (z)

Using ( 1 7 ) one gets then the estimate s

70

so i t i s not pre ci se l y what one wants to but the difficul ty can be easily overcome by replacing

h ( z ) by m ( z ) h ( z ) where

m ( z ) i s a scalar value d holomorphic function , with m ( 6 ) = 1 and whi ch 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 Coro l l ary 1 . [Wp 0 , l < p < oo . Corol lary 2 .

The space s

change of coordinate s i f

PROBLEM. case

p

=

s

1
Pps

are invariant for a local

< oo .

To e xtend th . 1 0 and its coro ll arie s to the

1 or oo

Remark .

()() C

The proo f o f th . 1 0 bre ak s down for

p

=

1

or

oo since ( 1 7 ) is not val id in this case . I t is not even known k th at the space s P s1 ln Wl are s table under any interpo lation .

proce s s .

Remark .

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

re fined re sult s involving the inte rpol ation of Besov and potential space s when al l parameters vary s imultaneously . The characte ri z at ion o f Be sov space s as inte rpo lation space s obtained in th . 7 is in some sen se a constructive characterizat ion .

Another constructive characteri z ation can

be obtained vi a approximation theory . to Lemma 2 ' above . )

( I t i s in fact re lated

Be fore s tating the e xact re sul t 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 prove d here are den sity theorems .

A typical such re sul t - in fact the prototype - is the In the case of llin

Weie rstrass approximation theorem ( 1 8 75 ) .

it says that. the e xponential functions are den se in Lp i f l < p < oo. On T 1 we have to take the trigonome tric polynomial s . O f course We ierstrass himse l f did not have Lp but Cu the space o f bounded uni formly continuous functions . By the way both ,

uni form continuity and uni form conve rgence make the i r appearance in thi s context .

On a compact topological group it i s

the Peter-We y l theorem. Returning to We ierstrass and :JRn let us set E (r , f)

=

in � jjf - g j j L supp gCK ( r ) p

which q uantity i s called the be st approximation o f

f

in Lp The n the We ie rstrass

by functions of e xponential type � r . theorem c an be rephrased as E (r , f)

..., ( l ) , r -+ oo for any

f

s

Lp

•

Another type o f theorems are now conce rne d with the degree o f accuracy o f the approximation .

What ( smoothne s s ) propertie s

have to be impo sed on f in order to as sure that E (r , f)

=

O ( r - 5 ) , r -+

oo

72

s

whe re p

= oo

i s a given number > 0 ?

In the case o f T1 and

the p roblem was complete l y solve d by Berns tein and

Jackson in the 1 9 1 0 ' s , e xcept for the case

s

=

inte ge r which

was f i l led in much late r on ( 19 4 5 ) by Zygmund . The answe r i s that f must be in B s oo The case o f seve ral variab le s 00

•

was probab ly first conside re d by Niko l ski j ( around 1 9 5 0 ) .

Now

we give a general treatment o f thi s prob lem within our general framework . Theorem 1 1 .

Let

s > 0 , 0 < q .::_

oo

Then we have dr ) l/q < r

Proof : 0 < q <1 f s Bps q •

For simp l i city we take

1 2_ q

:::._ oo .

00

The case

re q uire s some s l ight change s in the argument . De fine
g Then we get

< Hence ( floo ( r s E

(

dr ) 1 / q r , f) ) q r

Let

73

00

L:

V =O

<

C(

�c

00

2 v+l ( r s E ( r , f) !2

L:

( 2VS

00

00

V=O

00

L:

A=O

)

q

dr --

1/q � r )

q I I
1 /q

( 2\) s I I
L:

L:

A =0

<

p

1/ q

2 - As ( �c I If I I s q

Bp

<

oo .

Thi s prove s half o f the theo�em . The proo f in the othe r dire ct ion i s pos s ibly e ven s imp l e r . I f 2 v - 1 .:_ r , v* f doe s not change i f f i s repl ace d by f-g . Thus we have I I
v r� 2 -l .

I t now fol lows readi ly dr r

and thus upon summation ( c(

00

L:

v=O

00

L:

v=O

74

The proo f i s complete . Final ly we determine al so the dual o f our space s . Let

Theorem 12 .

s

real , 1 � p < oo

1 � q < oo

Then

holds

Proo f :

The statement for potential space s fo llows

8

�

and the fact that Lp' Lp , i f 1 � p < oo Indeed let SE ( Pps ) ' and de fine T by putting T ( f ) = S ( J- s f ) . Then TELp' • Hence , by the above the re e xi sts hELp , such There fore we find that T ( f ) = ff h re adily from th .

s

( f)

Jf g

where we have set

g = Js h .

I t i s plain that now e as ily seen that the mapping S + g : ( Pps ) ' + is an i somorphi sm .

g E P� � · I t i s Pp- � actua l ly

The statement for Be sov space s fol lows from the one for potential space s , whi ch we j ust prove d , using inte rpo lation ( th . 7 ) .

We pre fe r howeve r to give a direct proo f . If E i s any Banach space we denote by £ s q ( E ) the space o f se q uences 00 a = {a \) } \)=0 with 00

2:

v =O

( 2 vs

I

I av

I

q

I E) )

1 /q

< oo •

75

We use the fact that (Q, s q ( E ) ) ' � Q, - s q ' ( E ' ) i f 1 � q

<

oo .

Al so we recal l that ( E1E9 E2 ) ' �E1 ' E9E2' for any Banach space s sq E1, E2• Let now s t:: ( Bp ) ' • We de fine T by setting cp v * f

S (f) if F

and extend it by Hahn-Banach ' s theorem to the who le of the space Lp ® Q, sq ( Lp ) . By the above the re e xi s t G ELp , and -s {g v } 00v-_0 t:: Q, q ( LP ) such that

fF

G

dx +

00 Z:: v =O

f fv

gv dx

I t i s no e s sential re s triction to assume that supp

A

GC

2 I<

,

supp g vc 2 Rv

De fine now

g = G +

By Lemma 1' ( conveniently modi fied ) we see that Also we can see that S ( f ) = J f g . to fi l l in the re st o f the detail s .

00

g v =z:: O -s v' g E Bp ' q

•

We l eave i t to the reade r

76

No te s Di stributions ( or generali zed function s ) have a long hi story .

They can be traced back to Sobolev ( 19 3 4 ) but h i s

work went unnotice d for a long t ime . and popularized b y L . Schwartz ( 19 4 5 ) .

They were rediscove re d An introduction to

distributions can be found in many modern texts on functional analys i s ( e . g . Yoshida [ 5 8 ] ) or partial di f fe rential e q uations ( e . g . Hormander [ 1 2 ] ) . Conce rning the " cl a s s ical " Fourie r transform the be s t source i s perhaps S te in-We i s s [ 3 7 ] . As was said already in Chap . 1 o ur tre atment o f Be sov space s is the one o f Peetre [ 1 8 ] . p

=

q

=

In the spe cial case

2 thi s can be found alre ady in HBrmander [ 1 2 ] .

The

basic underl ying ideas can a l so 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 space s has been used in seve ral pape r s .

Let us mention Lo fstrom ' s the s i s [ 5 9 ] where

problems pertaining to theore ti cal numerical analy s i s are treated.

The point of view of Shapiro [ 2 0 ] , [ 2 1 ] and in

particul ar that o f Boman- Shap i ro [ 6 0 ] come s close to ours . ( These are dire cted towards q ue stions o f approximation theory and the Be sov space s do not enter exp l i ci t l y , c f . Chap .

8. )

Concern ing Bernste in ' s ineq ual ity ( Lemma 1 ) see books in approximation theo ry ( e . g . [ 2 3 ] , [ 2 4 ] , or [ 6 ] ) . Gene ral i z ations 5 of the type B q x or more generally EP x have been conside red

77

by q uite a few authors :

Golovkin [ 6 1 ] , Calderon [ 4 1 ] ,

Torchin sky [ 6 2 ] to mention j ust a few .

The fun ctorial point

of view appears e . g . in Gri svard [ 6 3 ] and in Donaldson [ 6 4 ] . The Lorentz-Be sov space s are mentioned in Pee tre [ 1 9 ] . The space s Kps q make the ir appearance in the early work o f Beurl ing ' s on spe ctral synthe s i s in the middle 4 0 ' s ( see e . g . [ 6 5 ] ) but in ful l generality they were introduced only in Herz ' s paper The connection with interpol ation space s was cl ari fied

[66 ] .

by Peetre [ 6 7 ] and Gilbe rt [ 6 8 ] .

See al so Johnson [ 6 9 ] , [ 70 ] . Conce rning the " homogeneous " space s Bps q and Pps , the ir origin i s o f a more dubious nature . Anyhow they are in Peetre [ 5 5 ] and also , pe rhaps in a more clear form , in Herz [ 6 6 ] .

The

work of Shamir [ 71 ] , [ 7 2 ] ought to be mentioned here too . some problems they simply are a must .

In

Conce rn ing the Aoki -

Rolewicz lemma see Pee tre - Sparr [ 4 6 ] o r Saghe r [ 4 5 ] .

A s was

mentioned in the notes to Chap . 1 the characte riz ation of Be sov space s using interpol ation space s goe s back to the work o f Lions ( see e . g . [ 2 5 ] ) .

Th . 10 on the other hand probab ly

come s from Calde ron ( see [ 4 1 ] ) .

From the hi s torical point

o f view i t i s inte re sting to note that Lions himsel f was lead to introduce inte rpo lation spaces in proving a spe cial case (p

=

q

=

2 ) o f Cor . 4 of th . 6 .

Regarding the characte rization

using the be st approximation see Nikolski j [ 6 ] .

Regarding

Lemma 1 ' see l ikewise Niko l sk i j [ 7 ] and Triebel [ 2 2 ]

•

The proof o f the duality theorem given here i s similar to the one o f Triebel [ 73 ] .

It is not clear where Be sov space s with

78

s�0

really appear.

Regarding the case

Li zorkin -Niko l sk i j [ 74 ] .

s

=

In the special case

0

see Lions p

=

q

=

2

space s with "negative norms " were systematically used in partial d i f fe rential e q uations alre ady in the middle S O ' s ( c f . e . g . Hormande r [ 1 2 ] 1 Peetre [ 1 1 ] ) .

0
<1

Concerning the case

1

see Pee tre [ 5 2 ] Fle t t [ 5 3 ] . The re sul t i s that Bp- �oo if 1 � p < oo 1 0 < q < 1 . I f general ly speaking ( Bps q ) X being any homogeneous Banach space and x 0 standing for the a fact closure of S in X one can show that ( ( B; oo ) 0 ) ' B� � l '

�

�

1

belonging to folklore , that has been o ften rediscovered ( see

[ 5 �) .

The proof is the same as for th . 12 .

Quotation :

Exe rcise for the reade r .

Chapter 4 .

Compari son of Be sov and potential space s.

In thi s chapter we will pre sent a more detailed compari son of Besov and potential space s .

Thi s wil l provide an

occasion to introduce some basic Calde ron -Zygmund and PaleyLittlewood theory , which wil l be needed in what fol lows .

We

know already that ( Chap . 3 , th . 4 ) 1� p <

if

00

With the aid o f Plan chere l ' s formula i t i s also e asy to show that

We now show that the l atter re sult can be e xtended at leas t to the range

1 < p < oo

Theorem 1 .

We have

s2 Bpsp -+ Pps -+ Bp

if

1 < p� 2

Bpsp

if

2�p<

Bps2 -+ P ps Proof :

-+

00

In view o f th . 8 o f Chap . 3 it suffice s to do

the proof in the spe cial case are going to prove that 79

s

=

0.

In othe r words we

80

if

1< p � 2

if

2 � p < ()()

To achieve this we shal l invoke the following ( o f Paley- Littlewood type ) .

Let { ¢ v } and be a s in Chap . 3 , so that ( 4 ) - ( 7 ) and ( 8 ) - ( 9 ) o f that Theorem 2 .

chapter are ful f i l led. (3)

I I£ I lL p

*

I I

Then

()()

2 E I ¢ v* £ 1 l � * f I lL + I I ( v=O p

1/2

�

I IL p

ho lds . I t is a l s o true that ()() 2 1/2 ( 4 ) I I f I I L "' I I E I ¢ v* f I ) I IL v = - oo p p ( The latter fact i s nee de d for the analogue o f th . 1 for B s q and P s , of course ! ) p p We po s tpone the discus s ion o f th . 2 for a momen t and complete the proof o f th . 1 .

We prove ( 1 ) ; ( 2 ) i s prove d in

exactly the same manner ( or using duality) Let fsB p (where 1 < p < 2 ) . Then

�

•

()()

E

v = 0 holds .

Hen ce , using the fact that

Q,

p

+

Q,

2

if

p < 2 (where

81

J Thus

=

tP

we have written

t 0 P ) we get 2 P/ 2

00

( L: \) =

0

fE: Lp by th . 2 . Conve rsely let

I \) * fl )

dx < oo

This prove s the l e ft hand part o f ( 1 ) . f E: Lp .

Then by th . 2 and using Je ssen ' s

ine q ual ity we obtain 2 1 2 � I I <1> v * f I 1L ) 1 V=O p

<

00

f E:Bp0 2 and we have e stab l i shed the right hand side too .

Thus

The proof i s complete (modulo the proo f o f th . 2 ) . Remark .

The above sugge sts the introduction o f certain

new space s Fps q

de fined a s fo llows :

;

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 wi l l be based on the

fol lowing Theo rem 3 .

( o f Calderon- Zygmund type ) .

Let E 1 and E 2

be two Hilbert space s and consider the convolution operator Tf ( x ) whe re

a

a * f (x)

J a ( x-y) f ( y ) dy

i s an ope rator val ued function and

f

a vector

82

valued one .

As sume that

( 5 ) l x l f2: 2 t l l ( a ( x-y ) -a ( x ) ) e i i l l i i l E 2 dx � c e E 1 , y l � t for eve ry

e s E 1 and every

t

> 0,

( i .e . , J J � ( O I I E , E � C ) 1 2 Then

holds . I f ( 5 ) holds with indi ce s 1 and 2 inte rchanged then al so with

1 < p �2

replaced by

2 �p <

oo .

(8)

holds

Thi s i s , in

part i cular , the case i f

( 9 ) I I D a a U) I I E , E 1 2

� c 1 �; 1 J a l i f -

l a l� n

( Mikh l in condition ) Since there are many exce l lent treatments in the l ite rature we are not going to give the full proo f o f th. 3 .

Seve ral

83

comments are , however, in orde r . I f E i s any given Banach space we denote by

Lp ( E) the space o f measurable fun ction s ( on a give n mea sure space � with measure

�

)

wi th value s in E such that

In an analogous way we de fine the space Lp (�) or all other q space s we nee d . I f E 1 and E 2 are two Banach space and A : E 1 -+ E 2 norm by

is a continuous linear operator we denote its

sup I I Ae I I E 2 efO

I

I Ie I IE . 1

I f E i s a Hilbert spa ce , Planchere l ' s formula i s still val id in the space

L2 ( E ) and ( 8 ) for

conse q uence o f i t .

For

p

=

2 i s a dire ct

1 < p < 2 ( 8 ) re sults from ( 7 ) using

inte rpolation ( ve ctor value d analogue o f Marcinkiewicz ) . e xtend i t to

2 < p
,;;;p ,;;; oo and

E

(L

L p (E ' ) i f p (E) ) ' is a reflexive Banach space . ) The hard part

have to use dual i ty . 1

To

( I t i s known that

:::

,

It depends on a certain decompos it ion of L 1 func­ ns (the Calder6n-Zygmund decompo sition) which again i s obta ined

is thus ( 7 ) .

�

from one covering lemma or the othe r . the detail s .

We sha l l not enter into

84

We shall have to use the general ve ctor value d form o f th . 3 .

Vector valued function s will also have to be con side re d For the bene fit o f the reade r let us write down

later on .

condi t ions ( 5 ) , ( 6 ) , and ( 9 ) in the s calar value d case ( E1= E 2 = C (5 ' )

)

lxl

•

They read as follows :

{ 2 r 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 l ater on ( see Chap . 6 where it belongs logica lly ) in thi s special case , i . e . , the s tatement ( 9 ' ) = > ( 7 ' ) , in fact , even a s tronger form

( The s tatement ( 9 ' ) => T : Lp + LP i f 1 < p < oo i s the Mikhlin or Marcinkiewicz mul tip l ie r theorem . ) Here we shal l of it.

re s trict ourselve s to some s impler comments .

First o f al l

( 5 ' ) i s ful fi l le d 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 c 1 on the un i t sphere S ) . I f moreove r !8

a ( x ) ds = 0 (6 ' ) i s true too .

These are the operators

con side re d o riginal ly by Calde ron and Zygmund ( 19 5 2 ) .

They

85

are character i ze d by be ing not only trans lation invariant but also dilation invari an t .

If

n =

1

the re i s , up to a cons tant

mul tiple , only one such operator - the Hilbert transform 1 ) . Th . 3 special i zed to thi s case i s no thing but (a (x) = x the cla s s i cal theorem of M. Ries z on con j ugate fun ctions ( 19 2 7 ) . We shal l al so - be fore re turn ing to the main road - amuse ourselve s by s tating a condi tion emanating from Cotlar whi ch guarantee s ( 6 ' ) , i . e . ( 8 ) with

p = 2.

The n i ce thing i s that

we then have condition s on Lp boundedne s s where the Fourie r tran sform doe s not ente r e xp l icitly ( i f this i s so n i ce ) . We state it as Theo rem 4 .

{

Let a satis fy ( 5 ' ) and

I

I a ( x ) I dx < C r

( 1 1)

K r)

Ix

(12)

J K (r)

a ( x) dx

0

( He re K ( r ) = { l i; l � r }

i s the bal l wi th radius r and cente r 0 . ) 0 00 and in particular a"' E.: L Then we may conclude that a E.: B 1 2 Note that both I

•

( 1 1)

•

and ( 1 2 ) are ful fi l led in the Calde ron-

Zygmund case . The proo f again will be postponed to Chap . 8 where i t be longs logic al ly . However , we can now comp lete a point left open in Chap . 3 . Proo f o f th . 4 o f Chap . 3 ( completed) : (k

inte ger

> o,

l < p < oo )

.

Let

fE.:P

If l a l � k we can write

�

86 2 D f = a * J f with � ( !;) = ( i l;: ) / ( 1 + j �;; j ) s/2 • I t i s a readily ver i f ie d that the Mikhlin condition ( 9 ' ) i s ful f i l le d

D f s Lp • Thus f E J< i t follows that p a k k L: Conve rsely , let f s Wp . Then we may wri te J f - i a i � k with suitab le a a to which ( 9 ' ) again i s appl icab le . We concl ude that Jkf s Lp and g:: J

th . 3 , let us give the We begin with the following general

Proo f o f th . 2 :

In some o f the previous work we have imposed on

observation .

{ rl' v } ,

c}

conditions ( 8 ) o f Chap . 3 .

We let

{ lj;v } ,

set ful fil l ing ( 4 ) - ( 7 ) and ( 9 -1 0 ) of Chap . 3 .

1J'

be another

We shall inste ad

re q ui re that

It is not hard to see that thi s can be achieved :

It can even be arrange d that

lj;v= cpv ' IJ' =

Conside r now the mappings T: and

f -+ (


*f

s imply take

• S and T formal ly de fined by

, { ¢v * f } �= O

)


87

S : (F, { � }� , v

v =-00

)

+

'

* F +

00

E � v * fv v =0

In view of ( 1 3 ) we have S s

so that

o

T

id ( identity)

i s a pro j e ction ( retraction o f T )

I f we now apply th . 3 with E 1 (14 ) Indeed

(9)

T : Lp + Lp

Lp ( £2) i f

EB

=

c , E2

=

•

_Q,2

we find

1 < p� 2 .

i s ful fi l led for we ce rtainly have

2 ( I D a

z v=O

In the s ame way tak ing

I

1/2 D a � v ( E;, ) 12 ) <

E1

C

C I E;. �- I a I we find

(15 ) Thi s completes the p roo f for the range 2 � p<

oo

l < p � 2.

The case

now fol lows at once by duality .

Remark 1 .

A simple proo f o f one hal f o f th . 2 using only

the s calar value d form o f Mikhlin ' s theorem and a standard probabi l i s ti c argument run s as fol lows .

For a random se q uen ce

88

w= { + 1 , + 1, + 1 ,

•

•

}

set 00

v l:=O -+

Tw Again we find

with a bound uniform in w <

C

I If I I L p

Take now the mathematical expectation of the p-th power ( 16 ) But ( 2. ( 1 1 T wf l l i, ) ) l /p= ( f f.. ( I * f + v � O +

A

and any numerical

p ) 1 /p :::: ( A I 2 + 'f I a I 2 ) 1 / 2 , 0 < p < oo a + + ( A I I E I 0 v v �0 v (Khintchine ' s ineq uality) holds . ( The more "mathematically oriented" reader could here 00

V'='

89

have used Rademacher series instead in which case we use Littlewood instead of Khintchine ! ) Inserting this in ( 16 ) we get ( 1 4 ) anew. It would be desirable to have a similar proof of ( 1 5 ) too . Remark 2. The above proof of th. 2 shows real ly more : That Lp is a retract of Lp Lp ( l) (or what is the same Lp ( £ 2 ) - the two spaces are clearly isomorphic) . We can express this in terms of a commutative diagram. ED

1

id Since Lp and P; are isomorphic we have l ikewise id

Pps

t

p

;

T

·�

�

It is also easy to see that id

Bps q

�· s

Lp

ED

Lp ( £ 2 )

1

90

Notice that the operators T and S are indepe ndent 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 be s t pos s ible . To fix the ideas again let

s =

0.

Then we shal l prove

�

(17)

B q

( 18 )

Lp

+

+

Lp = > q

<

min (p , 2 )

�

B �> q > max (p , 2 )

By a variation o f the arguments below one can al so show that for the l imiting case s

p = 1 and p = oo th . 6 of Chap . 3 i s

alre ady the bes t possib le . 2

�

To fix the ide as let us take

p

<

1

<

p� 2.

The case

oo can be handled in analogous manner (or mos t simply

by duality ) . The proo f o f ( 1 8 ) i s then particul arly simple so we s tart with i t .

Let g be a fixed function i n

S

such that

supp g is con tained in a sufficiently smal l ne ighborhood o f (1,0, , 0 ) and let a = { a, ) v = O be any seq uence . Let us choose f as fol lows •

A

•

00

•

f (U

00

00

L:

\)§;,0

v=O

or f (x)

00

\)

L:

=0

a v

91

About our se q uence o f te st functions { \) }00\) = 0 we may assume that ¢ v( � ) = 1 near 2\) e 1 . There fore we have 1"2 \) x 1 v * f = av e g I t fo llows that •

I I f I I O ::: I I a I I B q Q,q

•

p

On the o ther hand using bas ic facts on l acunary Fourie r se ries we readily see that

Lp � BpO q we mus t have q > 2 = max (p , 2 ) There fore i f

£2

�

Q,q which implie s

•

For the proof o f ( 1 7 ) we con sider an

f

s uch that

n f ( �) = I � � p' ( log i � I > - T in a neighborhood o f C

()()

oo ,

e l sewhere

I t i s pos s ib le to demonstrate the asymptotic deve lopment f ( x ) - C l xl

T >

1/q.

p'

3

o f Chap .

There fore i f

Remark.

1 T ( log -jxl )

There fore

with a suitab le C . other hand by th.

- n

3

sp0q� Lp

, x

�o

f E: Lp i f f T > 1/p. (with o = n/p ' ) we mus t have

q ,;S p

On the

0

S imilar techniq ue s c an be used to show that

92

th . 5 o f Chap . 3 cannot b e improved upon .

93

Note s . Th . 1 i s e xp l i ci t ly s tated in Be sov [ 5 ] and Taibleson [ 1 5 ] but its roots lie much deepe r ( I f n 14 . )

=

1 c f . [ 1 7 ] , Chap .

Th . 3 goe s b ack to Calderon - Zygmund [ 7 5 ] ( scalar value d

case , dilation invariant ope rators ) .

They thereby e xtende d

M . Rie s z theorem - whi ch was first proved by complex variable te chni q ue s - to the case o f several vari able s .

The ir result

has important applications to e l liptic p artial di f fe rential e q uations ( c f . e . g . Arkeryd [ 76 ] ) .

A conside r ab le simp l i fi­

cation and c lari fication of the proof in [ 75 ] was obt.ained by Hormander [ 7 7 ]

who also e xp l i citly s tated condition

(5' ) .

The ve ctor value d c a se was first clearly conce ive d by

J . Schwartz [ 7 8 ] who used it pre c i se ly for proving theorems of the Paley-Littlewood t ype .

Le t us further mention Benedek­

Calde r6n-Panzone [ 79 ] , Littman-McCarthy-Rivie re [ 80 ] , Riviere [ 81 ] and for a general introduction S te in [ 14 ] .

The

Paley-Littlewood theory arose from the work o f the se authors in the 3 0 ' s .

Again original ly complex variab le techni q ue s ,

notoriously comp l icated by the way , we re use d . Chap . 1 3 .

See [ 1 7 ] ,

For the Paley and Littlewood theory in a rathe r

general abstract situation ( di f fus ion semi-group s ) see Ste in [ 82 ] .

I t i s inte re s ting to note that the Mikhlin or

Marcinkiewicz theorem historical ly was p rove d using Paley­ Littlewood theory . First by Marcinkiewic z ( 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 . [ 14 ] ) .

Th . 4 goe s back to the work o f Cotlar [ 85 ] ( c f . Concerning lacunary Fourier serie s see [ 1 7 ] .

Quotation :

S ame as for Chap . 2 .

Chapter 5 .

More on interpolation .

We know already several re sults on interpo lation 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

re sul t s the e xponent p was fixed all the time (except in the cor . to th . 8 where p varied but the o ther parameter s was kept fixed ) .

Now we wish to see what happen s i f al l para-

meters are varied at the s ame time . Taking into account remark 2 in Chap . 4 we see that the interpol ation of Be sov and potential space s can be reduced to the interpolation of the space s t s q (A ) and Lp ( A ) , i . e . , vector value d se q uence and function space s . We there fore begin by reviewing what i s known to be true about thi s . Let us recal l the de finitions o f the above space s . Let A be any q uas i -Banach space . We denote by t s q (A) , where s rea l o < q .::_ oo , the space o f se q uences a = { a v Soo= O with value s in A such that

S]

being any mea sure space c arrying the positive mea sure

ll ,

we denote by Lp (A) , where 0 < p � oo , the space of l-1-mea surable a = a (x ) ( x E S] ) with val ue s in A such that fun ctions

95

96

I I a I I Lp (A)

( J � < I I a ( x ) l l i ) d ]J (x) ) l /p

< oo .

In an analogous way we introduce the Lorentz space Lpr (A) where 0 < p , r � oo More generally , w being a positive ].l -measurable function in � , we define the space of ].l -measurable functions a = a (x) such that •

The space £ s q (A) is really a special instance of Lp (A,w) . Indeed take : �

=

{ o ,l , 2 , ( {v } ) = 1 w (v ) = 2v s q=p

J.l

•

.

•

}

(discrete measure)

We there fore start with Lp (A,w) . The following results are wel l-known and completely understood . For the proofs we refer to the literature . We separate the complex and the real case . Theorem 1 . (vector valued analogue of Thorin ) . Let A = { A ,A1 } be any Banach couple . Let 1 � p , . p 1� oo Then 0 0 holds

97

l w) provided

1 p

W

1

-+

Let

Theorem 2 .

A

(0 < 8 < 1) .

be a q ua s i-Banach couples .

valued analogue o f M. Rie s z ) .

Let

0 < p0

= 1 p

p rovide d

W

I

1

p1 �

-+ Lp ( (A) 8p

oo

1

•

( i ) vector Then holds

w)

w o 1- 8 w l 8 ( 0 < 8 < 1 )

More gene rally we have

and the reve rsed embedding i f analogue o f Marcink iewi cz ) . 0 < Po

again

1

p1 �

(L

Pa r a

1 p

provided Remark .

=

=

=

1- 8 Po

oo

•

(A , w) , + _j_

r � P·

( i i ) ( ve ctor value d

Let A be any B an ach space .

Let

Then hol ds L

plrl

(A , w) ) r 8

pl ( 0 < 8 < 1 )

Lpr ( A 1 w )

•

Notice that in part ( i i ) o f th. 2 we take

and w 0 = w 1 = w . Thus we do not have a ful l analogue o f the Marcinkiewicz theorem in the s calar case .

A0

A1

A

98

Let us now turn our attention to the space s 3.

Theorem

+

Let

A

=

£ 5 q (A) .

{ A 0 , AJ! be any Banach couple .

Le t

Then holds

provide d

, s

=

( 1 - 8 ) s 0 + es 1 ( o < e < 1 )

In view o f the above ob servation that £ s q (A)

Proo f :

i s but a spe cial case of

Lp ( A , w ) thi s is j us t a re cast o f

th . 1 . ( i ) Let A = { A 0 1 A1 } be any q uasi-Ban ach 0 < q 0 1 q 1 � oo then holds :

Theorem 4 . coup le .

Let

(1)

5 q £ o o ( Ao )

provide d

1 q

I

=

5 q £ 1 1 (A1 ) ) eq I

5

< 1 - 8) s 0 + e s 1 < o < e < 1 )

•

More generally holds : (2) The e xponents ( ii )

Le t

min ( q , r ) and A

max ( q 1 r) are the bes t possible .

be any q uasi - Banach space .

Let

Then holds

99

Jl.

(3)

s0q

provide d ( ii i )

0 (A) ,

Jl.

s 1q 1

(A) ) r = 8

Jl.

sr (A)

s = ( 1- 8) s 0 + 8s 1 ( O < 8 < 1 ) .

Let

A

Take further

Let

be any q uasi-Banach space . s0 = s1 = s .

0 < q 0 , q 1 � oo

Then holds

1 q

provide d

Proo f :

( i ) Again

Jl.

s q (A) being a spe c i al instan ce o f

Lp ( A , w ) , ( 1 ) i s a s traight forward con se q uence o f p art ( i ) o f th . 2. Let us next fix attention to the first -+ in (2) . If

we can again make appe al to part ( i ) o f th. 2 .

r �q

r �q.

us there fore assume

For any se q uen ce

Let

a = { a v � =O

let us write a =

00

L:

\) =0

a

E

\) \)

whe re

E

\)

= ( 0 , • • • , 0 , 1 , 0 , • • • ) (with the 1 in the v -th position )

ain that for e ach v v I I Jl. s J.. q J.. (A . ) < 2 ].

\) S .

].

From thi s we obtain by inte rpolation

i

0 ,1)

•

100

whe re we have wri tten Q,

T = Ass ume now that

r

s 1 q1

(A1 ) ) r e

i s s o small that

T

•

is

r -normable .

We

know that thi s i s po ssible in view o f the Aoki-Ro lewicz lemma . There fore tak ing

r-th powers and forming the sum we ge t

v <

C(

00

I

=0

I= 0

I I a v E: v I

l � I rT ) /r -

l l a.v I I (A. )

8r

) r)

oo

1/r

= c I I a I I sr + £ (A ) er

We con tend that the first + in the fol lowing two case s : 2 ° r sufficiently smal l . To obtain the same contention for gene ral

r ( :;; g ) we have to use inte rpol ation .

General ly speaking , let us wri te s q £ o o (A ) , o

Q,

s 1q 1

(A l ) ) 8 r .

+ T.

i

1

(i

=

and assume that

Q,

sr . + 1 (A sr i )

1

0 'l)

•

0

I

1)

101

Inte rpo lating this give s (5 )

sr

�v n

O (A+ e r

o

)

,

1 -

�v n

sr

1 ( A+

er1

) ) , r + ( T 0 , T 1 ) , r where 1\

1\

A

+ r ( 0 < A < 1) 1

r

By ( 1 ) , which we have already proven , we have 9,

sr o

( A+ er ) , 0

9,

s rl

(i\ er

> > 1 Ar

Howeve r , b y a ce rtain complement to the re iteration theorem we have

Thus the le ft hand side of ( 5 ) i s e f fe ctive l y

9,

sr ( A ) 8r

•

Also

by the s ame token

�

T. right hand side i s T . 9,

Altogethe r we have shown

sr (A+ ) T , er +

unde r the said as sumptions .

Thi s fin ishe s the proo f for the

102

first

-+

in ( 2 ) .

the hard case i s 2

\)

Let us turn to the second -+ in r ,2, q .

(2) .

Now

For e ach v we have the ine q uality

s.

2 I I a ) I A . � c I Ia I I s . q . ( i= e , 1 ) l. l. (A ) � l. i

By interpolation we obtain v < 2 s l l a l l A-+ er with

T

c l la. I I T

having the s ame meaning as above , i . e . , the middle

space appearing in ( 2 ) .

r

If

=

oo

this fin i she s the proof .

The general case i s obtained b y interpolating between the cases

r = q

and

r = oo

•

Having thus e stab lished ( 2 ) let us indicate a counte re xample which shows that the e xponents

min ( q , r ) and

max ( q , r) in ( 2 ) c annot be improved on , in genera l . pre c i se let us assume that

P

To be

is a numbe r such that

s q s qo -+ ( � o (A o ) ' � 1 1 (A l ) ) e r We want to prove that

p � min ( r , q ) . Taking s 1 = 0 we obtain to the right by part ( i i ) o f th . 2 a Lorentz se q uence space . I t i s easy to see that this p = q , q < r . Taking again A o = A 1 = A but re q uiring now s o =I s l part ( ii ) ( which we have sr not yet prove n ) give s the space � (A) . This clearly shows entai l s by nece ss ity

p < q

or

103

p

� r.

The proof that

min ( q , r ) i s be st po s s ible .

Thus

max ( q , r ) is be s t po s s ib le is similar . We s tart from the estimate

(ii ) 2

\) s .

1

I I a v I IA � I I a I I s . q . R.

�

1.

(i

=

0 , 1)

(A)

Applied to a gene ral decompo sition

a

=

a 0 + a 1 thi s leads

to R.

Tak ing

s 0qO

s q (A) , £ 1 l ( A ) )

rth powers and forming the sum we get

dt ) l/q

T

< c I Ia I I

s q ( R. 0 0 (A)

1

s q £ 1 1 (A) ) 8 r

by the de fini tion o f the interpolation space s via K ( see Thi s prove s one hal f o f ( 3 ) .

Chap . 2 ) .

For the conve rse let

us write 00

L:

\) =-

u

00

\)

with

u \)

=

a \) E: \) if \) � 0 ,

We have

£

s.q. l.

1.

2 (A)

-v s 1.

l la 1 1 A

104

Thi s give s

Thus we ge t I I al

<

l

(£

s 0q 0

C I I al

£

(A ) , l

£sr

s lql

<

(A) ) 8 r

C(

oo z::

\) = 0

(2

v

( s -s 0 )

J(2

v

( s0-s1)

(A)

by the ( di s crete ) de fin ition o f the inte rpolation space s via J ( see Chap . 2 ) .

Thi s completes the proof of ( 3 ) .

( i i i ) Immediate conse q uence of part ( i i ) o f th . 2 . PROBLEM. (2

s 0 qO

( AO ) '

To find a precise de s cription of s £ lql (A 1 ) ) er i f r � q .

We are now re ady to proceed to the app l i cations to Be sov and potential space s . Contrary to our hab i t we shal l s tart s wi th Pp , because thi s i s here that much s imple r . Then hol ds Theorem 5 . ( i ) Let l < p 0 , p 1 < oo •

s0 [ PP o

( ii )

if

s =

We also have ( in the same condition s )

l ti S

Proo f :

( i ) By remark 2 in Chap . 4 we have the commuta-

ti ve diagram p

id

s. p.

l

l

1s . �

p

l

p.

l

�

Lp . Ell Lp . l

l

(Q,

s.2 l

)

0, 1)

(i

By interpolation we obtain ( recall the functorial characte r o f our inte rpolation " space s " ! ) s [P 0

p

'

Po

1

id s [ Pp 0 ' 0

p

sl pl

sl pl

T

Lp Ell Lp (Q, s 2 )

� ��

He re we have used th. 1 .

From this diagram, can now be re ad

off: f E[P

so P0

,

P

s

1 ]8

P1

<=> I I * f i l L

vs + I I ( I (2 O = v P

I ¢ v * fl

But the latter condi tion means pre ci sely that

2 ) 1/2

I l L < P

uc

Thi s

f ' i she s the proo f o f part ( i ) . p 0 = p 1 = p , part ( i ) i s j ust a re statement o f Chap . 3 . We thus obtain a new proo f o f the latter

Remark . of th.

9

resul t . th .

8

If

If

s 0 = s 1 both parts are contained i n cor . 2 o f of Chap . 3 .

106

To state the full re sul t for Bps q we need also the sq = B sq L Lo rentz - Be sov space s Bpr pr ( see Chap . 3 . ) . Notice that Then

Theorem 6 . holds s q [B o o Po if

1 p

(ii)

1- 8 + 8 -pl Po Le t

s q (B o o Po 1 p

Bps q

1- 8 + 8 I s qo ql

1 q

1 ,;;;, p 0 1 p 1 ,;;;,

(6) if

s q Bp l l ]8 l

co

I 0 < q0 1 q 1 ,;;;,

s q B 1 1 )8 q P1 1- 8 qo

1 1- 8 + 8 I P1 q Po

+

( l- 8 ) s 0 + 8s 1 ( O < 8 < 1 ) co .

Then holds

Bps q q 8 I ql

( l - 8) s 0 + 8 s 1 ( 0 < 8 < 1 }

S

-

(7)

More general ly we have

if

s 1 rnax ( q 1 r) Bpr

s , rnin ( q , r) Bpr

(8 ) 1 p

1- 8 + Po

--

1 q

1- 8 qo

--

+

I

S

107

In parti cular holds ( r (9 )

B s , m1 n ( q , p ) P ·

-+

p)

s q s q ( B o o , Bp l l ) p Po 1 8

B s ,max ( q , p ) P

�

�

Al so the e xponents min ( q , r ) and max (p , r) i n po s s ib le , at least if (10)

so - sl < l n= n - - n P o pl

l � Po , p1

< oo ,

( 8)

are best

s 0 1- s 1 and

Then holds

(iii)

( iv ) Let

l � p < oo

0 < q0 , q1 �

oo .

Then hol ds

1 q

8 ql ( 0

<

8 < l)

•

Part ( i i i ) i s of course j us t a re statement of

Remark .

th . 7 o f Chap . 3 o f which we thus get a new proo f . ( i ) By Remark 2 in Chap . 4 we now have the

Proo f :

cornm tat ive di agram : s.q. Bp 1. 1 id

1

ls . q .

B

1 1

P 1·

T

� �

Lp .

1

E9

.Q,

s 1. q 1.

( Lp ) . 1

(i

0 ,1)

108

By interpo l a t i on thi s yie lds s q o o [B p id

s q l l B ' le P

T

�

l

s q s q l l o o [B B ' p Je P

He re we have used th .

L tB p

Q, s q ( L ) p

� 3.

That

f

s

sq s q s q l l o o [B , B ] <= > f t: B e P Po Pl

fo l lows exactly as in the proo f o f th . 5 . (ii)

The proof o f th i s part goe s along s im i l a r l ine s s tarting

with the same diagram .

Fo r

(6) we use

(1)

of th .

course i s obtained j us t b e spe c i a l i z ati on f rom we use

( 2 ) o f th . 4 .

spe c i a l i z at ion .

Again

(9)

(6) .

Chap .

For

(8)

fo l lows from i t j us t be

The only thing that remain s - and that i s the

hard p o i n t in fact - i s to see that the e xponents and ma x ( q , r )

( 7 ) of

4.

are b e s t po s s ib le .

By du a l i ty

( th .

min ( q , r ) 10 of

3 ) we can re strict attent i on to the l atter case .

thus that we have for some

A s s ume

P

(11)

As in Chap . 4 l e t s upp g and l e t by

g

be a f i xed funct ion in

i s contained in a sma l l ne i ghborhood o f 00 a : {a } V = O v

be any s calar sequence .

such that e = 1

(1 , 0 ,

We de f ine

•

•

f

•

,0)

109

00

A

f ( s) =

00

\)

g ( s-2 e 1 ) or f ( x ) = I: av g v ( x ) v=O

where thus gv (x)

. \) e �x 2 g ( x ) . A

Since I Ig I l L doe s not depend on v and s ince s upp g v R v pr s q i f f a E: Q.s q . obvious ly , it i s readily seen that f E: B pr From ( l l ) follows We thus get an embedding

where we have used th . 4 , part ( i i ) , incidental l y .

But the

inve rse mapping ( de fined for al l fun ct ions of the form I: a v g v ) i s continuous . Thus we get Q. sr -+ Q. s p which entail s p > r . For the proo f of p � q we take inste ad 00

A

f (s ) wher

00

= I: \) =0

thi s time g (x) \)

v v v 2 - s 2 - n n e ix2 g ( 2 - v nx) .

We re q uire that 1 ° supp . g v C. R v and with c pr independent o f v , with p

cpr 2 - vs 2 0 I 19v I I L pr and s re l ated as in ( 8)

.

llO

1

Th i s leave s us with the con di t i on s °

!!. p i condi t ion

2

t;+

n

s. l

(10 )

(i

0,1) .

=

0

n

�

1

and

Upon e l iminat ing , we find p re ci se l y

I f th i s i s s o we see that

•

( 11)

Oq q a E: Q, = £ . In view o f q q l o -+ £ p • ) By p art ( £ , £ 6 r

f E: B

sq pr

i ff

again this le ads to ( i i i ) o f th .

4 we mus t the re fore

P2: q .

ne ce s s ar i l y have ( ii i )

Use part

(ii)

of th .

( i v)

Use part

(iii)

4.

o f th . 4 .

PROBLEM . To find a pre ci se de s c ription o f s q s q l l ' o o lf r � q. ( I t i s thus n o t a Be sov space . ) ) B (B 6 r P0 P1

1

A fte r thus having te rmin a ted our di scus s ion o f inte rpola -

tion o f potential and Be s ov sp ace s le t us indi cate a few app l i cations of the re sul ts obt aine d .

( Other app l i cation s

wi l l be give n l ate r . ) We be gin wi th the fo l lowing impo rtant co ro l l arie s . Coro l l ary =

5

( 1- 6 ) s

o

1

+

1.

Le t

6s ' l

1 p

1 < P 1 p < ()() an d l e t l o 6 1- 6 + (0 < 6 < p Po l

--

I I£ I I

c s < p p

( 2)

( 13)

I If I I

Proo f :

B sp p

< c

1)

-

I I£

1 1 1 s- o6 p

I If I I

1 6 1 �0 p

p

p

I If I I

l

6 p

Then ho lds

51

p

Po

I If1

6

•

s

p

l

l l

We make use o f the fo l lowing ge ne ral re s u l t for

interpolat ion sp ace s :

(*)

Let A

=

{

A0 1 A 1 }

be any q u a s i - Banach

111

1i. 8q

couple and let A be a space such that q

>

0.

Then holds :

+

A for some

I f one applies ( * ) , ( 1 2 ) and ( 1 3 ) readily fol low , mak ing also appeal t o part ( i ) and ( i i ) o f th . 5 re spe ctive l y . s

=

Corol lary 2 .

Le t

( 1 -8 ) s o + 8 s 1 , p1

=

I I f I

<

=

1- 8

Po

Then holds :

In part i cula r ( p 0

1

p +

0

I

8

pl �

pl

I

00

1

q

I

o

<

q0 , ql �

-e + e -1

qo

ql

oo

and let

(0 < 8

<

1) .

8 c 1 1£ 1 1 1 � q I I£ 1 1 8s q • B 0 0 B 1 1 Po P1 q0 , p1 q 1 ) holds

IB sp � c p

1- 8 8 I I£ 1 1 s p 1 1£ 1 1 s p • B a o B 1 1

Po

pl

Use again ( * ) but now in con j un ction with ( 6 )

Proo f : in th . 6 .

P o or p 1 = 1 . We give a l so an appl ic ation where inte rpolation i s used To e xtend Cor . 1 to the case

P OBLEM.

in a m re e s sential way . come/ l ater on . )

Let

n

=

( Seve ral s imil ar appl ication s will 1 and assume that 1 � p �

oo

•

Then

i s the space o f function s of boun ded pth variation in the P sense of Wiener , i . e . , f s Vp i f f for e ve r y famil y of d i s j oint

V

inte rval s

Ik

=

[ ak , bk ] c: JR holds

112

(14 )

with C depen ding on f on l y .

I t i s re adi l y seen that th i s i s

I � � lv

a Banach space , taking

= in f C , a t l e a s t i f we coun t p modulo polynomi a l s o f de gree 0 , i . e . , mo dulo con stants .

We h ave

Theorem 7 . 1

B•p '

(14)

whe re

1

p

1 p

1- 8 Po

+

(V ) , V Pl 8 P Po

+

V P

(0 < 8 < 1} .

(v

( 15 )

+

+

+

p

00

Al so in the s ame condi tions

1 P s " p oo -

I Vp ) 8 oo Po l

1 pI

-

•

B

00

He re we use for the f i r s t time the homogeneous Remark . space s B. We alway s le ave to the reade r to ve r i fy in e ach •

case t h at a re s ul t prove d fo r B i s va l i d a l s o fo r B . P roo f :

1)

We f i r s t prove the m iddle + in

Con s i de r a f i xe d fami ly o f di s j oint inte rva l s L k Wi th i t we a s s oc i a te the mapp i n g :

It

i s c le ar that

U: V p

+ l

P· £ 1

(i

0 , 1)

( 14 ) . =

[a , b ] C �. k k

11 3

By inte rpolation we obtain p p -+ ( £ 0 1 £ 1 ) e p

( 16 )

It follows that i f f EV . 2) P embeddings

f E (V

Po

, V

P1

=

Q, P

) then ( 1 6 ) ho lds true , i . e . ,

For the first embedding we noti ce the obvious

L -+ V 00

00

By interpolation we obtain o 31

11

I

• B ool oo ) e P -+
But by ( 9 ) o f th . 6 tak ing 1

-

• Bp p Al so

I

1

1

p

1- e -1- +

e

00

-

-+ o 3 i 1 , Boo l ) eP 00

y what j u s t was proven

so it suffice s to invoke the re iteration theorem. indeed

We get

11 4

+

3)

I n the p roo f o f the third + see Chap . 8 .

(We have inc luded

i t for the s ake o f completene s s ) . 4)

fol lows from ( 1 4 )

( 15 )

Remark . space

V a

( II

b -a l k k

l-

!

P

=

1 w p

e xten d th.

-a p

I

( 14 )

by

f (b ) - f ( a ) k k

I

p l /p < ) = C

<

oo

One can a l s o show that

Obvio u s ly v p

6 .

One can a l s o con s i de r s l i ght ly more gene ral

de fined by re p l acing

p

( 14 I )

using aga in ( 9 ) o f th .

if

1 < p < oo .

7 to the space s

The re a de r i s in vi te d to try to

115

Note s Concerning th . 1 see Calde ron [ 4 1 ] and conce rning parts of th . 2 (wh i ch goe s back to Gagl iardo ) Lion s-Peetre [ 25 ] 1 the e xten sion to the ful l range ( 0 100 ] i s o f l ate r origin ( see Holmstedt [ 44 ] 1 Peetre [ 86 ] ) .

A counte r-example for a

general ve ctor value d analogue o f l'-1arcinkiewicz theorem was I t should al so be mentioned that the

found by Kree [ 8 7 ] .

interpol ation in the case of weights first was considere d by Ste in-We i s s [ 8 8 ] . ( c f . Triebe l [ 7 3 ]

1

Th . 4 was f i r s t s tated in Peetre [ 1 9 ]

[22 ] ) .

Part ( i ) o f th . 5 and al l o f

th. 6 except for ( 8 ) i s due to Gri svard [ 6 3 ] . from Peetre [ 5 5 ] 1 [ 1 9 ] .

The res t come s

In [ 5 5 ] the re i s al so an applic at ion

to the in terpo l ation theorem of S tampacchia [ 89 ] . Vp eminate from Wiener [ 9 2 ] . to Marcinkiewicz ( see [ 8 3 ] ) . Kruger-Solomj ak [ 9 3 ] .

The spaces

The l ast + in ( 14 ) goe s back Regarding inte rpol ation see

Ine q ual itie s o f the type appe aring in

Cor . 1 and Cor . 2 to th . 6 were first obtained by Nirenberg [ 9 0 ] and Gagliardo [ 9 1 ] .

116

Chapte r 6 .

The Fo urie r tran s fo rm .

Th i s sho rte r chapte r i s re a l l y an i n troduc tion o r prep aration fo r Ch apte r 7 . Le t u s begin b y re s tat ing what we a l re ady know re garding the a c t i on of the Fo uri e r trans fo rm in var ious c l a s s e s of functions

( see Chap .

3) .

Th i s is e s sential ly the Hausdo r f f -

Youn g theorem :

1)

( 1)

an d i t s re finement usual l y a s s o c i ated with Pale y .

F : Lp + Lp I p

(2)

if 1 < P < 2

c!.

_

p

+

1

1)

p'

Now we t urn to various vari an t s and gene r a l i z ation o f the se re s ul t s . We re cal l that

(1)

and

from the endpo int re s ul t s of the

( 2 ) we re obtained by inte rpo lation p

=

1 and p

(quan t i tat i ve ve r s i on o f )

=

2 , i . e . , by the use

Riemann- Lebe s gue lemma and

Plan che re l ' s fo rmula re spe ct ive ly .

We now obse rve that

(3)

wh i ch i s a k in d o f imp roveme n t o f the Riemann - Le be sgue lemma . Inde e d i f

• Q oo

f s B

1

then fo r e a ch

the re fore by the s a i d lemma

A

"'

'1'\)

v

A

we have

f s L 00

I t fo l lows that

A

f

is

11 7

R VE

bounded in the sets uni formly in v i f

v v ( 2 - E ) - l 2 ..::_ I t;. I .::_ ( 2 - E ) 2 1

=

But the se sets obvious ly

E > 0 i s fixe d .

ove rl ap i f E i s sufficiently sma l l .

f E L oo

There fore

•

�ve

Noticing that

now interpol ate between ( 3 ) and P l anchere l . B• 02 2 = L 2 1 thi s yields :

F: But by Thm. 6 1

( 8)

J3 0 1min (p ' 1 r)

pr

Also ( L

o f Ch . 5 -+

00 ( B lo

I

Lp , r .

00

:8 202 ) r i f e

l p

1- e 1

+

2

e

There fore we have proven the

fol lowing Theorem 1 .

We have

(4)

F : BOprmin (p ' , r)

Taking

r

=

p ' we find in particular

If r

=

p ( 4 ) leads to a weaker conclus ion

-+

Lp ' r

if

1< p< 2

One can also • formulate result s in terms of the Beurl ing-He rz spaces Kps q ( see Chap . 3 for de fini tion ) and more general ly the Lorentz­ • sq Beurl ing -Herz space Kpr ( the obvious de fin i ti on ) .

t han ( 2 ) , thi s in view of th .

7

of Chap . 4 .

118

We have

Theorem 2 .

0q f : :B p

(6)

We al s o have

if

l < p�2,

O < q�

( genera l i z in g ( 3 ) ) .

if

0 < q

<

00

We invi te the re ade r to provide the obvi ous pro o f o f th .

2 and a l s o o f the fo l lowing Coro l l ary .

We have

if

l < p �2.

Note that a we aker fo rm o f

(5}

re s u l t s f rom { 6 ) by taking

p' .

q

\ve now go on s tudying a s l ightly di f fe re n t p rob lem .

For

certain re a son s i t i s conve n i e n t to inte rchange the role of x

and

�.

In o th e r words we con s ide r the inve r se Fourier F

trans form

-l

rathe r than F

•

S ince we have perfect

symme try we have o f co urse

-1

(1I )

F

(2 I )

F

-1

-r L

:

L

:

L -r L p p'p

p

p'

l�p� 2

if

if

(p .!. +

l < p� 2

(p .!.

1

1)

p' +

1

p'

1)

119

Notice that sin ce

p�

2

we have

condition s have to be imposed on

p'

� 2.

f

We now ask what

in orde r that

f

should

be long to a Lebe sgue or Lorentz space wi th an e xponent < 2,

f E Lq . He re i s a first re sult in this sense . Theorem 3 ( Be rn s te in ) . We have

e . g.

( 7) Proo f :

By Schwarz ' s ine q ual ity and

Let

Planche re l we have

Taking the sum it fol lows that

l lfl l Ll�

C

00

\! �-""

dx � C

00

\)

l:

= -00

I If I I

... and

( 8)

f E L1 • Theorem 4 ( Sz a s z ) . F

.

-1 : B 2

n ( p1

We have

2) , p 1

-+

L

p

if

0 < P < 2.

120

Proo f :

Adap t i on o f the ab ove p ro o f fo r th .

Remark . with

1.

Us ing in te rpo l a t ion one c an al so prove

l � p � 2 s tarting with ( 7 )

F - l . L -+ L 2 2

and

(8)

(i.e. ,

P lanche re l 1 s the o rem) . Re turn ing to t h . l we see that , in view o f th . 5 o f chap .

3,

(7 I )

F

I f in Fo r

( 7 ) en tai l s a l s o -1

if

:

n � . p ' 00 • p'l B by B (7 I ) p p

( 7 1 ) we rep l a ce

(7 1 )

l� p� 2 .

F : Ll

together w i th

L

-+

oo

i s n o longe r true .

imp l ie s

l •� p ' -+ L oo and we know that such an e s timate i s be s t p o s s ib le • B p Howeve r , the re is the fol low ing s ub s t i tute : Theore m 5 :

F

-1

:

\'Ve have

co o , l• � p , ,.. . B I I B 00 p

In particular hol ds

p 2

-+

if

l)

(p

l o, 2 l . + F () :B 00 . � l

Proo f :

Use Co r .

2

o f th .

6

o f Chap . 5 .

proof i s e q ual ly s imp le ! ) Coro l l a ry .

Take

n

=

l.

Then holds :

(A d i re ct

121

Proo f :

Use th . 7 o f Chap . 5 .

The next rather natura l deve lopment i s to substi tute for L1 in th . 3 the space We argue as fo llows . e stimate I I ¢ * f I l L v l

Now

A

cp

•

Q oo

Bl

oo l , we have to But in view of th . 3 we have

•

To show that

R

v

(Sl )

=

where the in f is taken over al l Then i t i s easy to see that

We have proven : Theorem 6 :

A

f

to

Let us the re fore for any open set Q

p

Then

:8°

f o f course depends on ly on the re stri ction o f

I I g I I · sq B

(9 )

E

A

v

the net

g.

f

Assume that sup v

in f

set

l l hl l B sq ( rl) •

p

h whose re s triction to

Sl

is

122

Corollary.

In part icular this i s so i f

(Mikhlin condi tion ) whe re

i s the smal l e s t intege r > n2 .

h

We are now in a position to fil l in a gap le ft open in connection with th . 3 o f Chap . 4 (name ly that ( l ' ) entail s (5

I

)

)

o

Theorem 7 .

( "Mikhlin " ) .

As sume that for some

s

>

n/2

holds < ()()

sup v

( 10 )

Then condition s ( 5 ' ) o f Chap . 4 i s ful fi l le d , i . e . , we have f I f ( x-y) - f ( x ) I dx JRn "K ( 2 r )

( ll )

Proo f : (i.e. '

f

v

we wri te

¢v

f ).

�

C , y E: K ( r ) for eve ry

f where f v = ¢ v * f L: v =- ()() v I t clearly suffice s to e stimate for f

()()

e ach f I f ( x- y ) - f v ( x ) l dx , y E: K ( r ) JRn " K ( 2 r ) v

Choose E: so that

n 0 < E: < s- 2 .

Then we get

r> 0

123

A \) <

f I fv ( x -y ) I dx JRn "- K ( 2 r )

< 2 - 2 r- 2

f

+

n

lR

"'-

f

I f v ( x ) I dx

K ( 2r)

I x - y I 2 I f v ( x- y ) I dx + 2 2 r - 2 f I x I 2 I fv ( x ) I dx

where we at the last step have used th . 3 .

We al so ge t by

the s ame theorem :

< I Yl <

c

f

I grad

� ( x - y) I dx � C r c

r 2v 1 1 f 1 1 s � v p 2 •

r 2v

l l t::f' ) l

B2 •

Al together we have thus proven A < C min ( r 2 \) If

A

\)

,

\)

( r 2 ) -2 )

•

i s the e xp re s sion to the l e ft o f ( 1 1 ) we get A�

00

A < C < oo E v = - oo \)

n

. 2'

l

124

and the proo f i s complete . Remark .

f E: :B 0 00

F

P;

Condition .

A

( 10 )

says e s sentially that

s imilar remark appl ie s to condition

(9 )

•

125

No te s The first re sult o f thi s chapter , notab ly th . l and th . 2 together with i t s coroll ary , be long to the fo lklore . C f . e . g . Riviere-Sagher [ 9 4 ] . Th .

3

and th . 4 which indeed

go back to Be rn ste in and S zasz re spe ctive ly were first con­ side red in the context of T 1 • See the re fe rence s l i s te d in Peetre [ 9 5 ] , [ 6 7 ] .

Th .

3

goe s back to Zygmund and its

corol lary to I z umi - I z umi [ 9 6 ] ( c f . Peetre [ 5 2 ] ) . o f th . 7

i s e s sentially the one o f Peetre [ 9 5 ] .

The proo f Regarding

the interpo l ation of the Mikhlin or ( bette r ) Hormande r con di tion ( 10 )

see also John son [ 6 9 ] .

Quotation :

Your student ' s l i fe i s not entirely without value ( though I suppose he will never understand why ) . G . H . Hardy ( in a lette r to M . Rie s z )

Chapter 7 .

Multip l iers .

The general s i tuation cons idere d in thi s chapter may be Let X be any q uasi - Banach space o f functions o f distributions i n En Often one assume s that X i s invariant for translations ( i . e . fEX = > f ( x + y ) X , y E Rn )

de scribe d a s fol lows .

•

X

and - to make it symmetric - invariant for multiplic ation by characters ( i . e . fEX > e ixn f E X , n E Rn ) , but this is of cour se =

not at a l l neces sary . invariant .

We than a l so assume that the norm i s

We ask what conditions have to be imposed on a and

b in order to a s sure that

(1)

fEX => a*f

(2)

f E X = > b f E X.

X,

I f thi s i s so we s ay that

a

an (ordinary) mul tipl i e r .

The reason for such a terminol ogy

i s a Fourier multiplier and b

is of course that the convolution a* become s an ordinary A

multiplier a (= F a ) after taking Fourier tran s forms , by the formula ( see Chap . 3 ) F ( a * f) = F a F£. 126

127

I f X sat i s fie s the above invariance propertie s trivial examp le s o f Fourier multip l ie rs are Fourie r tran sforms of bounded measure s ( i . e .

a E FA or

a E A where A i s the space of bounded

measure s ) and o f o rdinary multipliers inve r se Fourier trans forms o f bounded measure s ( i . e .

or

More generally one can con side r mul tipl ie r s from one space X in to another one Y . We di scus s the problem o f Fourier mul tipl iers firs t .

Of

course with the after al l rather s imple minded too l s we are e q uipped with ( inte rpolation space s , e tc . )

we cannot hope

to settle al l problems , but neve rthel e s s a certain insight in the se matters can be gained . Generally spe aking given X let us denote the space o f all inverse Fourie r tran s forms of Fourier multipl iers by CX o r C ( X ) ( i . e . , a E C X i f f ( 1 ) holds ) . Banach algebra .

The q uas i - norm i s given by

= /. I I a * f I I and Y we w� i te C X , Y . I I a! I

Clearly C X i s a q uas i -

cx

x

-1 I I fl I x

I f we have two space s X

=

Lp ( 1 � p � oo ) and let CL . us write , o rrupting the notation j us t introduced , Cp P Let us now l i st some clas s i cal e lementary facts about the Let ljt s first consider the case

\

space s

c

X

p:

(3)

F -l L

(4)

A (bounde d measure s )

00

( inverse Fourier tran s forms of L)

=

128

(5 )

cP

(6)

A

+

cp'

if

c

c

p

+

1 1 p + p' -

pl

+

F

-1

1 L

00

if

1 � p � pl � 2

In view o f ( 5 ) one can o ften re strict one se l f to e i ther the case

1 �p � 2

or

2 �p �

We have also the fo llowing

oo

use ful Lemma 1 .

We have

(7) (8 ) 1 = -

provide d

p

1-e

--

Po

+

+

c

+

c

e pl

p p

( 0 � e �1 ) .

To see the ide a let us prove ( 7 ) on ly.

Proo f :

The

proo f o f ( 8 ) i s s imil ar but re q uire s the three l ine theorem .

a If

c

a s(

Le t

f

p1 ) 8 1 •

l: aV

with

00

=

,

c

Po

V = -oo

Then by de finition we may write 00

l:

v = -oo

sL

p i t fo·l lows t.h at a * f

00

=

l:

v =-

00

Us ing the conve xity ine q uality

av * f .

129

which is an immedi ate conse q uence o f Thorin ' s theorem , we get I I a* f I I L

p

<

=

< <

=

The re fore

00

V'J_oo I I aV * f I I L

p

<

00

I I ) I c � - 8 1 1 a) I � I I f I I L p pl 0 00 2 -v 8 J ( 2 v, a c , Cl l l l f / / L < c / . 1 f / / L L: v 0 oo = v p p

L:

V = - oo

=

a * f EL and , by de finition , P

a E Cp

•

We can now easily prove Theorem 1 ( Hi r s chman ) . (9)

F'

n 1

( •

-+

p

l

c

P

We have

if

1 p

I� -

1

2

�

In particu ar holds ( 1 0)

\�

F : B

p

Proo f :

P

,1

-+

c

P

if

1 p

> I�

- iI

By ( 4 ) and th . 3 o f Chap . 6 we have F

Also ( 3 ) give s

:

B. 1

n ,1 2

-+

130

F

:

L 00

-+

c

2

Thus by appl ication o f ( 7 ) o f Lemma 1 we get -+

c

p if

1 p

1- e + 1-

2.

e

On the otherhand by th . 6 o f Chap . 5 1

l5

Thus ( 9 ) fol lows .

==

e

00

•

To get ( 1 2 ) we have to invoke the ( obvious )

Lorentz analogue o f th. 5 o f Chap . 3 . Remark .

1- e

=r-+

The p roof i s complete .

Th . 1 shoul d be compared with the JIIJ.ikhlin

mul tip l ie r theorem.

If

p

==

1 ( Be rn s te in ' s case ) the condi tions

on a is almost the s ame but not rea l ly .

In fact th . 1 doe s

not apply to the Hilbe rt trans form o r the Calde r6n- Zygmund ope rators and of course the concl usion i s not true e ithe r in

On i f 1 < p < oo ) P the other hand i t i s possible to prove analogous interpolation

thi s c ase .

( Mikhlin says on ly

as C

•

analogue s of Mikhlin . Vve now give a simple ne ce s s ary condi tion for in Cp Theorem 2 .

We have c

p

-+

nn • -p- I 00 •B - p 00 Bp (i. p '

a

to be

131

Proof :

Let

Then i n particular w e mus t have

<

I I a l ie

< C I I ¢v I I L p

p

But - v np­ l I I ¢v I I L � C 2 p There fore we al so get - v np­ l I I ¢v * a I l L � c 2 p and

a

E:

o

Bp

n pI -

00

•

n .- p a E: Bp i

- ,

In view of ( 5 ) now also follows

00

The proof is complete . Corollary . and

-1 0 .

Th n

so to speak

b

a

E:

Lp Obvious .

Proo f : Since

Le t

a

E:

cP .

Lp I

Assume that

A

supp a

o

• say , B Ql l -+ c p we have thus enclosed c p tween two Be sov spaces . The bounds are the (6)

1

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 "mode l le d " on Theorem 3 . As sume that ( ll )

00

v � - 00

i s compact

l/ I I ¢v * a I I P ) p < cP

oo

where

C p i tsel f .

132

Then l ,:;, p ,:;, oo then

l < p < ()()

Conversely if

{ 12 )

sup

aI

I I ct> v *

\)

I

p

a t: C p where

< ()()

In other words holds :B 0 P c

(13)

Q oo l p -+ C p -+ B Cp where p

I� - l I · 2

=

In view of that we at once get ( using also th l) the following Corollary . Assume that ()()

where l = l pl p

v= -ao

- 2l 1

.

Then a t: Cp Be fore giving the proof of th. 3 it is convenient to It is remarkable settle the analogous q uestion for that we now have a much sharper result. Theorem 4 We have •

•

a t:

C

Bpsq < => sup \)

I I ¢ \) *

aI

IC p

<

oo

In particular C s;q depends In other words : only on p . (That i t does not depend on s is obvious ) .

133

Proo f :

Assume that

a

must in particular have

E: *

a

holds

C B•ps q Then i f f E: f E B•pS oo There fore •

s B q we

p for e ach v

a * fl I L � C I I f I I sq p Bp Apply thi s to lj; v * f where { ''''�' v } 00v oo i s another se q uence o f te st functions chosen in such a way that lj; v = l on R v = -

A

•

Then fol lows 2v s I I cj>: a

( 14 )

f l l L � C 2v s I I f I I L p p

*

which clearly imp l i e s

( 15 )

�;

Conve rse ly i f the e s timate

(14 ) a

E:

( 15 )

and i f f E: Bps q , a * f E: B q and

holds for

fol lows nd taking the sum we get C •Bps q • Th s complete s the proo f o f th . C •B ls q = •B lO oo e h ave Coro l lary .

a

=

4.

•

Proo f : Indeed i f c 8 � q = s 0 00 C l = B Q oo Now to the

p

' = 11

P roof o f th. f i lled with some o f th.

3

3: p

•

we show that

=

l we get J! O oo L = B• 0 00 l l

Assume that the e st imate ( l l ) i s fulThen by an easy adaptation o f the proof

134

sq a E C BpS q ' Bp l i f

(16 )

l q

(Use Holde r ' s ine q uality ) .

+

l p

Then assuming f E Lp l � p � 2 which i s no loss of generality f E B• p0 2 , so by ( 16 ) a * f E B• pOp and a fortiori a * f E Lp . There fore a E C p · That again a E C p entail s ( 1 2 ) we le ave to the re ade r to Now let

•

ver i fy . We now proceed to give some e xample s . First we remark that the cor . of th . 4 contains in particular e x . 2 o f Chap . 2 ( the Hilbert transform in Lip s ) . Indeed we have by now seve ral criteria for e specially th . 4 of Chap . 4 and th . 5 of Chap . 6 .

Both of

l the l atte r evidently apply in thi s case ( i . e . a ( x ) = p . v. x '

; ( s)

=

i sgn

s)

•

Next we cons ider the fol lowing Example l .

a < s)

=

I sI

Suppose nY

and C oo e lsewhere .

I I ¢v av I I An

in a neighborhood o f oo , with n

> 0, >

Then it i s e asy to see that

v m ( n-p - Y )

< c 2 n 1. Bp p

appl ication o f the cor. to th . 3 now yields

a E Cp

y

0

135

provide d

n P>p

p < nand p

On the other hand i f

c

m 1 1

p . It i s also possib le to show that a s c p in the l imi ting case y = n but this much harde r . Thi s shows a ¢

then

-

p

howeve r that our re sults obtained in thi s rather s traight forward way in a sense neve rthel e s s are almo s t sharp .

We

return to thi s point in a moment . and oo be st po s s ib le . • c At least it i s known that c p f B 0 p i f p ,. 2 . I t c i s not even true that B 0l p i f ,. 2 . Now we give a q uick survey o f some re sul ts for e x , Y Are the exponents

PROBLEM.

p

00

00

-+

extending the above once obtained for

ex.

By a repeated

C Lp , Lp We l l 0 assume p � p 1 for i t is e asy to see that indeed C p , p l i f p > p 1 • We omit mo st o f the proo fs or j ust give brief corruption of notation we write

Cp,p

=

hints . We have

Theorem 5 .

Proo f :

• s -s B 1

s q B• p l l

S C Bp T

Theorem 6 .

e

00

c

P, P1

same as for th . 5 . we have provided

In particular holds

1 max (p 1 , 2 )

<

1 + 1p min ( 0 , 2 )

,

1

<

p , pl <

136

B• 0 00 C p, p l Proo f :

The s ame as for th . 6 .

Theorem 7 .

We have

Bpsq -+ C Remark . p p

p*p* l i f

1 p-l*

l

p*

Thi s should be contrasted with the case

As we have remarked not even

00

-+

c p i s true un le s s

2. Proo f :

By the Lorentz analogue o f the Be sov embedding

theorem ( th . 5 o f Chap . 3 ) we ge t

Inte r -

polation doe s the re s t . Corol lary .

We have

B -s , oo -+

l p

if

l

This may be conside re d as a ve ry general form o f the - s ( see e x . l Hardy-Littlewood theorem for the potential s I of Chap . 2 ) . Theorem 8 .

We have n

- P'

-+ B Pl Proo f :

,

oo

The s ame a s for th . 2 .

137

Now we return to case o f C X , more spe ci fi cally

c

•

p We want to give a brie f discussion of the ce lebrated ( Fourie r ) multip l ier p roblem f o r the bal l . Le t us adopt for a moment the s t andpoint o f classical Fourier analy s i s .

There the problem wa s to give a mean ing

(other than the distributional ) to Fourie r ' s inve r s ion formula (2

f (x) for , say ,

f

s

Tr

) -n

A

trouble i s that

f

s

Lp ( l � p �2) . The Le t H be a i s not ve ry small at oo

L1 o r more general ly

f

given homogeneous positive function and function with

u(O)

=

l.

u

some given

Then one is lead to con side r as

approximations to the integral , " means " o f the type (2 Tr) -n A

=

I

J

A

f (� ) d �

n

m.

i s a Fourier multip l ie r in Lp ' fol lows then by routine a guments If

a (t_; )

u ( H ( t_;

I I fr - f I I L p

-+

0

as

A particularly important ca se i s when function o f the interva l [ O , l l so that fun ction o f the set have

E

= {

H (t_; )

_s_

r

-+

u

oo

•

i s the characte r i s t i c

a (U i s the characte ri s t i £C

l } ....:: nP .

In thi s case we

138

fr ( x )

( 2 'IT) -n H (

f r ( O ) ( x)

=

r,/� r

e i x E, f( E,) d E, (partial Fourier inte gral ) u (A)

A somewhat more general case i s t+

=

( 1- A) � (with

max ( t , O ) ) in which case f; a ) ( x )

=

�

( 2 'IT ) -n f e ix t, ( 1- H E, ) ) � f ( E, ) d E,, a2: 0 ( Ries z means ) .

� ( E, ) = ( 1 - H ( E, ) ) a (wi th the obvious

We agree to wri te

r + a interpretation in the limiting case a = 0 ) .

Two important special cases are : a)

E

b)

E

i s a cube o r more genera lly a conve x polyhedon . i s the uni t ball (with e . g . H ( E, ) J t, J 2 ) or more =

generally any s trictly conve x set . I t turns out that these two cases behave q uite d i f fe rently and that i t is the diffe rential geometry ( curvature ) o f the boundary a E o f

E

that cause s that di ffe rence .

The case a ) i s e asily handled. a

thi s case

=

We j ust notice that in

ao i s the product o f the characteristic fun ctions

of a finite member o f hal f spaces conside r the special case :

C::.

JRn E,

Thus i t suffice s to

a ' ) E i s a hal fspace With no los s o f generality we may then assume that E

={ E,

1

=

0

}

so that

and

=

1 - i p . v. xl

•

139

Thus i t simpl y suffice s to apply M . Rie s z ' theorem . conclude that The case

a

s

c p i ff

co

b ) i s much more compl icate d .

a 0 ( x ) ( a s wel l as function s .

1< p<

We

One can expre s s

a a ( x ) ) e xpl icitly i n te rms of Be s se l

U s ing the class ical asymptotic formula for Be ssel

function s i t then fol lows C cos ( l nx I + + 1 8 ) 2

1

+ 0(

n+3

l x I -2-

( One can also give a direct computation whi ch perhaps bette r 3E

reveals how the geometry of

come s in . )

I f we now invoke

the cor . o f th . 2 we ge t as a nece s sary condition for a o s cp 0 > �p - 21 whe re 1p = - l . The s ame applie s to We now find that i f a a s C then a > nP aa 21 " We next P ask if this is a sufficient condi tion . ( This i s (was ) the 1 ( Ste in) . mul tipl i e r problem) . Th . 1 re adi ly give s a > n -P-

I�

:

I

-

n-1- > n P p

. S lnce

-

2

p < .:!:.

or

2

we are l e ft wi th a gap .

has begun to be f ' lled up on ly re cently .

F i r s t Fe ffe rman

displayed a counte -example showing that

a 0 cannot b e a

He al so showed a positive re sul t , 1 1 + 1 and in addition > 4n then inde e d 4 2 p p L ater Carle son-S j 6lin and S j � lin i n the spe cial

a

case

n

dition .

=

=

p

multiplie r unle ss 1 that i f > n

It

2

and

2.

--

n

=

3

In particular i f

ple tely settled.

re spective ly re laxed the l atte r conn

=

2

the problem has been com-

The fol lowing figure s i l l ustrate the case :

140

We now give a simple proof o f the original re sul t by Fe ffe rman pre sumably due to Stein .

141

The k e y to the p roo f i s provided b y the fo llowing : Lemma ( Ste in ) .

Let

be the uni t sphere in

S

R1 .

Then

we have Lp -+ L 2 (S ) provide d

F

1 <

2

L2 ( S ) o f co urse denotes L2 wi th respect to the area e lement dS � on s .

where

Proo f :

a_ 1

De fine

by

a_ 1 a simi lar asymptotic formul a as for a 0 n-1-. It follows that a .:_ 0 ) , on ly the exponent is now 2 . n-1 n we h ave to To prove our a ssert1on l" ff 2 >p

Then holds for ( and a E

a

'

--

Lp e stimate a- 1

fI

But i f

I

2 a_ 1

=

C < f * f , a_ 1 >

1 � p - 1 , by Young ' s in­ q Hence by Holde r the above integral i s finite pro-

f �:: Lp then

e q uality.

A

f *

�:: Lq i f

vided 2 p - 1 +

1 1- n > 1 or -2 -

.!. > �

p

4

+

1

4n

142 The proo f i s complete . Now i t i s easy to e s tablish Theorem (Fe f fe rman ) . l p

I�

�

-

I ).

Then

aa

Proo f ( Ste in ) :

Assume

E

C . P

a

>

�

p

-

1

2'

! >! + p

4

1 4 n (where

In view o f the asymptotic formula

(explicitly given only when a = 0 ) we may a s we l l suppo se n + ! > 0 and that a = a is de fined by s= a2 a p cos l x l n l x l ;l + a

aa ( x ) =

a (x)

( in a neighborhood of

oo

)

•

Wri te A =

q,

a,

av = <jJ v a

with q, , { <jl v } �=O as in chap . 3 but for a change de fined in lRn = lR� (not lR� ) . I t now suf fice s to show that (1 7 )

with as sume assume

s

> 0.

S ince

supp f C Kv p < 2.

•

supp a v e Rv we may in conside ring a \) * f (A * f cause s no trouble ! ) We furthe r

From Holde r ' s ine q ua l i ty and Pl ancherel we get n

c 2 n

c 2

\) p

\) p

I I av * f I I L = 2

143 We next obse rve with Fe ffe rman that the main contribution to �v ( � ) come s from the set R0 = { 2 - 1 < 1 � 1 � 2 } . The re fore we have v� 1/2 1/2 2 s up ( ! I f ( 0 1 2 ds � ) I I av * f I I L � C 2 P ( ! I �v ( �) 1 d � ) p 2-1
invoking the lemma and again P lanche re l we get

I I av * f I I L p <

c

v n� C 2 p

v � - v ( n 2+ l + a ) v � 2 p 2 2 2

=

( modulo Fe ffe rman ) The re fore ( 1 6 ) fo l lows and Remark .

he proo f i s complete .

s from Fe f fe rman ' s theorem that for

I t fo

radial function s th . 1 can be imp roved somewhat . I f 1 an d 1' f u u ( x) i s a function with compact when a > n- 2 p u ( l � l > i t follows th at support <: ( O , oo ) then setting � ( � ) =

-

u

E:

p

n 1

p

1 p +

E:

( E: > 0 ) = >

whi l e a s th . 1 give s on ly u

E: p

Notice again that

n-1 + p 1

E:

n-1 > n p p

(

E: >

1

2

0)

=

>

(If

a ao

E:

cp . E:

c p we re true we

144 would have u

E

A

a

=>

but thi s i s so only i f

E

C

p

l or p

n

=

2.)

See more on the se

matte rs in Chap . 1 0 . In orde r not to con fuse the reade r we s tate explicitly that the pre ce ding discus s ion o f the multipl ier theorem for the bal l , through important by i tse l f , was on ly a digre s sion wh ich had no t much to do with our main theme , the study of Be sov space s .

That i s why the previous theorem has not eve n

got a number . We le ave now our discus sion o f Fourier mul tipliers and turn to our se cond ob j ect , that of ( o rdinary ) multiplie rs . Generally speak ing the set o f all mul tip l iers in X will be deno ted by M X.

(We only cons ide r the case o f j ust one

space . ) I f X i s a Banach space o f fun ctions or distributions in lRn it is easy to see that M X ' = MX whe re X' denote s the dua l space .

There fo re in discus s ing Be sov or po tential

space s we may re strict atte ntion to the case case

s <0

thus being captured by dua l i ty .

discuss ion o f the intermedi ate case

s

=

ought to be rather inte re s ting by itse l f . that

M LP

=

s > 0 , the We omi t the

0 , which however Le t u s also notice

L00 , l � p � oo , as i s easy to ve r i fy .

Our first re sult i s : Theorem 9 .

For any

s

>0

( and

l � p � oo

0 < q � oo ) holds :

145

Proo f :

Let

{