Preface Contents ESTIMATES FOR TRANSLATION INVAR1ANT OPERATORS IN L p SPACES

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ESTIMATES FOR TRANSLATION INVAR1ANT OPERATORS IN L p SPACES

BY

L A R S H O R M A N D E R Stockholm

Preface

T h e t h e o r y of b o u n d e d t r a n s l a t i o n i n v a r i a n t o p e r a t o r s b e t w e e n L v s p a c e s i n s e v e r a l v a r i a b l e s h a s a t t r a c t e d m u c h i n t e r e s t i n t h e ] i t e r a t u r e d u r i n g t h e p a s t d e c a d e , p a r t l y d u e t o i t s a p p l i c a t i o n s i n s o m e f i e l d s s u c h a s t h e t h e o r y o f p a r t i a l d i f f e r e n t i a l e q u a - t i o n s . T h r o u g h t h e w o r k of C a l d e r 6 n , Z y g m u n d a n d o t h e r s r e a l v a r i a b l e m e t h o d s h a v e b e e n i n t r o d u c e d w h i c h h a v e p e r m i t t e d t h e e x t e n s i o n t o s e v e r a l v a r i a b l e s of r e - s u l t s o r i g i n a l l y b a s e d o n c o m p l e x m e t h o d s i n t h e c a s e of a s i n g l e v a r i a b l e . F u r t h e r ,

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94 LARS ~SRMANDER

a suitable framework for a general theory is given by the theory of distributions, for translation invariant operators are essentially convolutions with distributions (see section 1.1).

The purpose of this paper is thus the study of the spaces Lp q Of tempered distributions T in R" such t h a t with a constant C

IIT~ul[a<~ClluUp

for all infinitely differentiable u with compact support, the norms being i q and L p norms. I n section 1.2 we discuss those properties of these spaces which follow from M. Riesz' convexity theorem and the theory of the Fourier transformation in L p spaces.

Some of these results are taken over from Schwartz [13], and others have been used implicitly in various papers on convolution transforms. I n section 1.3 we study homomorphisms of the Fourier transform M," of Lp p induced by a mapping in R ~. I t turns out t h a t if the mapping is twice continuously differentiable and p * 2 , it must be linear. This improves a result of Schwartz [13], but for p = 1 it is weaker than known resu]ts concerning the algebra

M11

Of Fonrier-Stieltjes transforms. I n section 1.4 we prove t h a t the Wiener-L~vy theorem is valid in a certain subalgebra of M~ p whose relation to M~ p is studied. The proof is rather trivial but we have included it because of its similarity with a result in Chapter I I which is essential in Chapter I I I . (Closely related results concerning sequence spaces are due to Devinatz and ttirschman, Amer.

J. Math. 80 (1958), 829-842.)

Chapters I I and I I I are devoted to the numerous estimates which originate from Riesz' theorem on conjugate functions (Riesz [10]). I n Chapter I I we discuss the real variable method introduced b y CalderSn and Z y g m u n d [2] in the study of conjugate functions in several variables. I t has also been used later b y Z y g m u n d [18] to prove the Hardy-Littlewood-Sobolev estimates of potentials and also b y Stein [15] in studying estimates of the kind which we discuss in Chapter I I I . The main theorem in section 2.1 describes the general situation in which such arguments apply. I n section 2.2 we show first t h a t our theorem contains the results of CalderSn and Z y g m u n d [2]

and Z y g m u n d [18] mentioned above. We then show t h a t it also gives a short proof and a slight improvement of a theorem of Mihlin [8], [9]. (The proof given by Mihlin depends on a paper of Marcinkiewicz [7] which is based on the Littlewood-Paley theory (see Chapter I I I ) and on the properties of Rademacher functions.) We end the section by proving a theorem of the Wiener-Ldvy type for a certain algebra of homogeneous functions of degree 0 contained in Mp ". Closely related results are due to CalderSn and Z y g m u n d [3] but are not sufficient for the applications in Chapter I I I .

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TRANSLATION INVARIAlgT OPERATORS 95 I n Chapter I I I we study estimates of convolution transforms involving parameters. We do not make a systematic theory similar to t h a t in Chapter I for such families of transforms but restrict ourselves to results parallel to those of Chapter I I . I n sections 3.1.-3.3 estimates involving L 2 norms with respect to the parameters are proved. I n section 3.4 they are shown to contain the known results concerning the functions of Littlewood-Paley, Lusin and Marcinkiewicz as well as other estimates which m a y be of interest in the theory of partial differential equations. The proofs are similar to those in Section 2.1. Real variable methods have previously been used by Stein [15] in studying the Marcinkiewicz function in several variables but our method differs considerably from his. B y studying the adjoint transformations which map functions in the product space of R" and the parameter space on functions in R ~, we obtain estimates also when 2 < p < ~ . I n t h a t case the results known previously are rather incomplete when n > 1 and the proofs when n = 1 seem difficult. We also obtain simple proofs of general "inverse" estimates. The results concerning Mp p which follow from the Littlewood-Paley estimates are not studied here so we refer to Little- wood and Paley [5] and Marcinkiewicz [7].

This paper is essentially self-contained, which m a y be an advantage to the non specialist in view of the extensive literature in the field. Necessary prerequisites are elements of distribution theory, including the Fourier transformation (see [12]); Riesz' convexity theorem (see [ l l ] and [16]), Marcinkiewicz' interpolation theorem (see [18]), and also basic facts concerning bounded operators in Banach spaces. The bibliography is very incomplete so a reader interested in studying the literature closely should consult the references given in the quoted papers also.

C H A P T E R T

General t h e o r y

1.1. Translation invariant operators as convolutions

We denote b y L p, 1 ~<p~< o~, the space of measurable functions in R '~ with integrable p t h power, and write

(1) It is convenient to se~ formally I[u[[~=oo if u~.L ~.

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96 LXaS Xt()RMANDER

W h e n p = ~ this shall be u n d e r s t o o d as t h e essential s u p r e m u m of I/I- B y L ~ we d e n o t e t h e space of functions in L ~ which t e n d to 0 at ~ a n d b y C t h e space of continuous functions. M will d e n o t e the space of b o u n d e d measures d # n o r m e d b y

fld•l.

W h e n 1 4 p 4 ~ we shall use t h e n o t a t i o n p ' for t h e c o n j u g a t e e x p o n e n t defined b y l i p + 1 / p ' = 1.

If h E R ~ we d e n o t e b y ra t h e operator defined b y

(r~u) (x)=u(x-h).

D E F I N I T I O N 1.1. A bounded linear operator A ]rom L ~ to L q is said to be transla- tion invariant i /

l:h A = A vh, h e R ~.

Such operators which are n o n trivial do n o t exist for all p, q.

T ~ E O R E M 1.1. I / A is a bounded translation invariant operator /rom L ~ to L q and p > q we have A = O i/ p < ~ and i / p = ~ the restriction o/ A to L ~ is O.

Proo/. First note t h a t if p <

IIn+~ull~-->2"~llull~, ueL~; h-->~; (1.1.1)

t h e same is t r u e for p = ~ provided t h a t u E L ~ . I n fact, we can write u = v + w where v has c o m p a c t s u p p o r t a n d II w I1~ < ~. F o r sufficiently large I hl the s u p p o r t s of v a n d ~hv do n o t meet, hence

IIv+~vll~=21'~llvll~.

Since I I I v l l ~ - I I ~ l l ~ l < ~ a n d I I I v + ~ v l l ~ - I l u + ~ u l l ~ l < 2 ~ a n d s is a r b i t r a r y , we o b t a i n (1.1.1).

:Now assume t h a t

I I A ~ I I ~ < C l l u l I ~ , u E L p, (1.1.2) with q < p < ~ . The linearity a n d t r a n s l a t i o n invariance of A give

W h e n h - + ~ it follows f r o m (1.1.1) t h a t

IIA ~11~< 21~-~'~ c II ~ I1~, (1.1.3)

which improves (1.1.2) since t h e e x p o n e n t is negative. If C denotes the smallest con- s t a n t such t h a t (1.1.2) holds we t h u s get a contradiction unless C = 0 , t h a t is, A = 0 . The same a r g u m e n t s a p p l y when p = co > q provided t h a t we replace L ~ b y L ~ . The proof is complete.

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TRAI~SLATION II~IVARIANT OPERATORS 97 A l t h o u g h s o m e w h a t incomplete for p = ~ this result will justify us to assume t h a t p ~< q in w h a t follows.

Let $ be the space of infinitely differen~iable functions u such that sup Ix~l)~ul <

for all ~ a n d •, a n d with the t o p o l o g y defined b y these seminorms. Here zt = (~1 . . . aj) a n d fl = (ill . . . ilk) are multi-indices, t h a t is, sequences of indices between 1 a n d n,

D ~ = ( - i ~ / ~ x ~ , ) ... ( - i D / ~ x ~ ) ; x~=x~, ... xzk.

We use the n o t a t i o n Ice I for the length j of t h e multi-index g. 8 is dense in L" if p < ~ , a n d its closure in L ~ is C N L f f . T h e dual space of S is d e n o t e d b y S' a n d its elements are called t e m p e r e d distributions. (See Schwartz [12].)

THEOREM 1.2. I / A is a bounded translation invariant operator/rom L" to i q, then there is a unique distribution T E S' such that

A u = T ~ u , u E $ .

F o r t h e proof we need a l e m m a which is a v e r y special case of Sobolev's lemma.

LEMMA 1.i. I / a /unction v in R ~ and its derivatives o] order ~ n are in L ~ locally, the de/inition o/ v may be changed on a set o/ measure 0 to make it continuous. Then we have with a constant C

, v ( x ) , ~ C :,~,<.( f , D ~ v , ' d y ) ~''. (1.1.4)

{y-x[41

Proo/. The assumptions concerning v are also satisfied with p = l a n d (1.1.4) follows f r o m H61der's i n e q u a l i t y for e v e r y p if it is p r o v e d for p = 1. I n t h e proof we m a y also assume t h a t x = O a n d t h a t v has c o m p a c t s u p p o r t in t h e u n i t sphere.

F o r let ~ be a f u n c t i o n in C~ with s u p p o r t in the u n i t sphere a n d which equals 1 in a n e i g h b o u r h o o d of 0. T h e n w = v ~ has c o m p a c t s u p p o r t in t h e u n i t sphere, a n d Leibniz' f o r m u l a shows that

lY[<I

H e n c e if we prove t h e s t a t e m e n t of the l e m m a for w, it follows t h a t v is continuos in a n e i g h b o u r h o o d of the origin after correction on a null set a n d t h a t (1.1.4) is valid for x = 0. The general s t a t e m e n t of t h e l c m m a t h e n follows b y its t r a n s l a t i o n invariance.

7 - 60173032- Acta mathematica. 104. Imprim6 le 21 septembre 1960

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98 LARS H()RMANDER

N o w l e t h ( x ) = H ( x l ) . . . H ( x n ) w h e r e H is t h e H e a v i s i d e f u n c t i o n w h i c h e q u a l s 1 for x > 0 a n d 0 for x < 0 . W e t h e n h a v e

~n h//~ xl ...~ x~ = 6

i n t h e d i s t r i b u t i o n sence. H e n c e , since w h a s c o m p a c t s u p p o r t , w = w ->+ (~ = w ++ ( ~ h / ~ x 1 ... ~ xn) = (~'~ w / ~ x I ... ~ xn) ++ h.

I n t h e r i g h t h a n d side we h a v e a c o n v o l u t i o n b e t w e e n a n i n t e g r a b l e a n d a b o u n d e d f u n c t i o n , h e n c e a c o n t i n u o u s f u n c t i o n , w differs f r o m t h i s c o n t i n u o u s f u n c t i o n o n l y on a null set a n d if i t s d e f i n i t i o n is c h a n g e d t h e r e we h a v e

(x) l<.II "w/ xl

... a x , I d x ,

Iw

w h i c h c o m p l e t e s t h e proof.

P r o o / o/ T h e o r e m 1.2. L e t A b e t h e o p e r a t o r in t h e t h e o r e m a n d u E S . W e c l a i m t h a t

D ~ ( A u ) = A ( D ~ u ) (1.1.5)

in t h e d i s t r i b u t i o n sense. T o p r o v e t h i s i t is c l e a r l y e n o u g h t o c o n s i d e r a d e r i v a t i v e of t h e first o r d e r . P u t v = A u a n d d e f i n e uh (x) = u (x 1 + h, x~ . . . x~) a n d vh s i m i l a r l y . Since A is i n v a r i a n t for t r a n s l a t i o n we h a v e A u~ = vh a n d h e n c e

W h e n h--~O t h e differenee q u o t i e n t ( u h - - u ) / / h c o n v e r g e s t o O u / O x 1 in L p, h e n c e ( v h - v ) / h e o n v e r g e s t o A ( O u / O x l ) in L q n o r m . H e n c e (1.1.5) f o l l o w s .

L e m m a 1.1 n o w shows t h a t A u is a c o n t i n u o u s f u n c t i o n a f t e r c o r r e c t i o n on a n u l l set if u E $ a n d t h a t , t h i s e o r r e c t i o n being m a d e ,

I (A u) (0) l -< c y ID u[I,.

H e n c e ( A u ) ( 0 ) is a c o n t i n u o u s l i n e a r f o r m on $ so t h a t i t m a y be w r i t t e n (A u) (0) = T (4) = (T ++ u) (0),

w h e r e ~ ( x ) = u ( - x ) a n d T E $ ' . I n v i e w of t h e i n v a r i a n c e for t r a n s l a t i o n of b o t h sides we g e t

(A u) (x) = (T + u) (x)

for e v e r y x, w h i c h p r o v e s t h e t h e o r e m since t h e u n i q u e n e s s of T follows i m m e d i a t e l y f r o m t h e proof.

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T R A N S L A T I O : N I N V A R I A N T O P E R A T O R S 99 If p < ~ , the space $ is dense in L ~ and the operator A is obtained as the closure of the operator u-->T~eu. If p = ~ and q < co, we have T = 0 in virtue of Proposition 1.1; and if p = q = ~ the distribution T is obviously a bounded measure.

The case p = ~ therefore does not present great interest. Thus the study of the translation invariant operators is essentially equivalent to the study of the spaces Lp q of the following definition.

D E F I N I T I O N 1.2. The space o/ distributions T in $' such that

**IIT*ullo<Ollutl,, ueS, (1.1.6)**

where C is a constant, is denoted by Lp q. The smallest constant C which can be used in (1.1.6) will be denoted by L~q(T).

Lp q is thus isomorphic to a closed subspace of the Banach space of all bounded linear mappings of L ~ into i q, hence is also a Banach space.

Let :~ denote the Fourier transformation u-->~, [.

~ | - . ,

~(~) je 2~'<X~>u(x) dx, ue$,

extended to all T E S' by continuity or, equivalently, the formula ( u ) = T ( ~ ) , u 6 S ,

(see Schwartz [12], Chap. VII). We recall t h a t the Fourier transformation is an isomorphism of S and of S'. Then the mapping u - - > T ~ u , u E S , can also be written

u->:~ 1 ( ~ : ~ u )

and is thus via the Fourier transformation equivalent to multiplication by ~.

D E F I N I T I O N 1.3. The set o/Fourier trans/orms T of distributions T E Lp q is denoted by M , q and we write

i p q (~) = L~ q (T).

The elements in M~ q are called multipliers o/ type (p, q).

Sometimes we shall write Lpq(,~ and Mpq(n) in order to emphasize t h a t the number of independent variables is n.

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100 LARS HORMANDER

1.2. Basic properties of

Mpq

Our first theorem in this section is very well known b u t we formulate it for completeness and reference.

THEOREM 1.3. Let T be a distribution =4=0. Then the set of points ( x , y ) E R 2 such that T E Lllx 11~ is a convex subset o/ the triangle

0~<x~<l, 0~<y~<l, y<~x, (1.2.1) which is symmetric with respect to the line x + y = 1. I n this set log Lllx 11y (T) is a convex function o/ (x, y) with the corresponding symmetry property.

Proof. That the set in question satisfies (1.2.1) follows at once from Theorem 1.1.

The s y m m e t r y is proved as follows. Let x', y' be defined by x § x' = y + y ' = 1. Then if

**II T*ull,, < C Ilull,,x, ueS,**

we get from HSlder's inequality

IT u v(0)l <Cllull,=llvll,=.; u, yeS.

Since convolution products are associative and commutative we get from the converse of HSlder's inequality

II T vll,,. Cllvll, ..

Hence (x,y) and

(y', x')

belong to the set in the theorem at the same time, and

L1/~: 11~ (T)=LI/~,

1I=' (T) since the role of (x, y) and

(y', x')

may be interchanged in the above argument.

Finally, the convexity follows from Riesz-Thorin's convexity theorem (see Riesz [11], Thorin [16]). The proof is complete.

We next list some cases where Lp q is easy to describe precisely.

T H E O R E ~ 1.4. We have

L p : C = L I ~ ' = L p', p < ~ ; L ~ = = L l l = M , (1.2.2) with equality also o/ the norms.

Proof of the theorem. I n view of Theorem 1.3 it is enough to prove that Lp = = L "' a n d t h a t L:r = = M. The last fact is essentially the definition of bounded measures a n d was already observed after Theorem 1.2. The first follows from the fact t h a t

L ~' is the dual space of L" when p < ~ .

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TRANSLATION INVARIAbIT OPERATORS 1 0 1

COROLLARY 1.1. TEL1/x 1/y /or all (x,y) in the triangle (1.2.1) i/ and only i/

T ~ L 1 N L ~.

Proof. I n view of T h e o r e m 1.4 we have T E i11x lly for all three corners (x, y) of t h e triangle if a n d o n l y if T E L1N L ~0. B u t t h e c o n v e x i t y p r o p e r t y in T h e o r e m 1.3 shows t h a t T is t h e n in L1/x I/y for every (x, y) in t h e triangle.

COROZLARY 1.2. Let p<~q and set 1 / p - l / q = 1 - 1/a. Then we have i a ~ i p q and

/ e i a . (1.2.3)

I / a = 1, one may replace L 1 by M .

Proof. I n virtue of T h e o r e m 1.4 t h e corollary is t r u e for p = a ' , q = oo a n d for p - 1 , q = a . Hence it follows in general from the c o n v e x i t y properties in T h e o r e m 1.3.

N o t e t h a t (1.2.3) means e x a c t l y t h e well-known i n e q u a l i t y

Ill <- llo<ll/llollull,,, fcL ~

u s.

We n e x t t u r n to some results which are best expressed in terms of the Mp q spaces.

THEOREM 1.5. With equality also o/ the norms we have

M2 2 ⁼ L ~176 (1.2.4)

Proo/. L e t T E L2 2 so t h a t ~ E M~ 2. T h e n T ~- u E L e for all u E S, hence the Fourier t r a n s f o r m T d E L 2, a n d

for all ~ E S. This proves first t h a t ~ is a locally square integrable f u n c t i o n a n d t h e n t h a t ]T(~)] ~ < i 2 2(~) a l m o s t everywhere. On the other hand, if I ~(~)1 ~< C almost everywhere t h e same a r g u m e n t proves t h a t M2 2 (T) ~< C. Hence Me 2 (T) is t h e essential s u p r e m u m of T, which proves the theorem.

COROLLARY 1.3. For every p we have

M~ v ^cL ~, (1.2.5)

II !

<~ M , ~ (1), / e M , v. (1.2.6) Proo/. The c o n v e x i t y a n d s y m m e t r y s t a t e d in T h e o r e m 1.3 show t h a t M ~ ' c M ~ 2 a n d t h a t

Me 2 (l) ~< M . ~ (1).

Hence t h e corollary follows f r o m T h e o r e m 1.5.

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102 LARS HORMANDER

For p 4 q we can only prove a weaker regularity result. B y LPloo we shall mean the set of functions which belong to L p on every compact set.

THEOREM 1.6. The /ollowing inclusions are valid:

Mp q ~ L~lo~ if p ~> 2; M p q ~ Lq'1or i f q ~< 2.

(1.2.7)

Proof. Since M ~ q= Mq2" according to Theorem 1.2, it is sufficient to prove the latter half of (I.2.7). Let T ELp q, q ~ 2 . For every u E S we then have T ~ u E L q and in view of the Hausdorff-Young theorem on Fourier transforms of functions in L q, q ~ 2, (Zygmund [17]), we obtain ~ E L q" for every u E S, hence for every ~ E S. This proves the theorem.

If p>~2 or q~<2 we thus have i p q c L 2 1 o c . Elements in two such M p q classes m a y thus be multiplied together pointwise, giving a locally integrable product,. This gives a sense to the statement in the following theorem.

TttEOREM 1.7. Let 2<~p<~q<~r or p < ~ q ~ r ~ 2 . T h e n i/ / E M , q and g E M q ~ we have / g E M p ~ and

M , ~ (/g) <. M p q (/) i q ~ (g). (1.2.8) The translation invariant operator corresponding t o / g is the product o/those corresponding to g and to ].

COROLLARY 1.4. M p p is /or every p a normed ring with the operations o/ point.

wise multiplication and addition.

This result is partly given by Schwartz [13]. Note t h a t Theorem 1.4 shows t h a t M11 = M ~ ~ is the algebra of Fourier-Stieltjes transforms.

Proo/ o/ the theorem. I n view of Theorem 1.3 it is enough to consider the Case p ~< q ~< r ~ 2. Denote by A I the closure of the mapping

L p D S ~ u-->:~ -I (]~) E L q.

A x is a bounded operator from L p to L q. Similarly we define a bounded operator A 0 from L q to i r. Then we have

: ~ ( A i u ) = / ~ , u E L ' ; : ~ ( A g v ) = g ~ , v E L q. (1.2.9) I n fact, these identities arc valid by definition when u and v are in S. To prove the second identity, for example, we note that the Hausdorff-Young inequality shows that the mapping L q ~ v - - > : ~ ( A g v ) E L r" is continuous. Since gEL~loo the mapping

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T R A ~ S L A T I O / ~ I I ~ V A I ~ I A ~ T O P E R A T O R S 103

Lq~v--->g~ELlloo

is also continuous. S being dense in

L q,

the second formula (1.2.9) follows. Taking

v = A z u we

thus have

: ~ ( A g A 1 u ) = / g ~ , u E L p,

hence in particular this is true when u E $. This proves t h a t

/g

is the multiplier corresponding to

A~Ar,

hence (1.2.8) is valid. The proof is complete.

Without the restriction given on the exponents, Theorem 1.7 would not always have a sense (see Theorem 1.9}. However, we can always prove a m u c h weaker statement showing t h a t the local smoothness of the elements in M~ q increases with p and decreases with q. Combining this with Theorem 1.4 we could also get another proof of Theorem 1.6.

T ~ E O R ~ M 1.8.

I / / E M p q and g E $ we have

g / E M ~ q

if

r ~ p ; g / E M ~ ~

if

s>~q.

Proo/.

Since

Mp q= MaY

it is sufficient to prove the first statement. L e t f = T I , g = T g . Then T g E S a n d for u E S we get

]] ( T I ~ T g ) ~ u l i q = ]] Tf)t-(T~ ~u)Itq<ipq(/) ]] Tg~ullp <~ipq(/) i / (g) IiulIr.

Hence

T I ~ Tg E L~ q

so t h a t the Fourier transform

g / E Mr q

and i ~ q (g/) <~ i ~ ~ (g) M~ q (/).

Theorem 1.6 does not give a n y information when p < 2 < q. We shall now study t h a t case starting with the following lemma.

L E M ~ A 1.2. I / U E $ a n d us E $ is de/ined by

~t (~) = ~ (~) d ~ t~t~,

we have/or p > 2 with a constant C~

Ilut II, ~< c~

Itl

t ER.

Proo/.

First note t h a t Parseval's formula gives

To estimate the m a x i m u m of

ut,

we introduce polar coordinates,

o I~1=1

d r deo.

(1.2.10)

We shall integrate b y parts with respect to r. Note t h a t

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104 L A R S H O R M A N D E R R

0

(1.2.11)

for all real R, a a n d t. I n fact, a change of variables gives t h a t

R c

f et(2ar+tr~) dr=t- 89 e ia'/t f eir2dr

0 b

if t > 0, where b = a / ~ t a n d c = R 1/[+ a / ~ . Since f + : e ~r' d r is c o n v e r g e n t as a general- ized R i e m a n n integral, (1.2.11) follows for t > 0 , hence b y complex c o n j u g a t i o n for t < 0 . I n t e g r a t i n g b y p a r t s in (1.2.10) a n d using (1.2.11), we o b t a i n

]us(x) l <,< Cltl-89 f lOCt/Or + (n-1)r-l ~]rn-l drd~o=Cl ltl-89

F r o m this estimate a n d t h e fact t h a t ]]us]ls is c o n s t a n t we o b t a i n

flu, l, dx<.(c, ltl 89 l dx=Vo ltll 89

which proves t h e lemma.

THEOREM 1.9. I / p < 2 < q there exist elements in Mp a which are distributions o/

positive order, that is, which are not measures.

Proo]. Assume t h a t t h e s t a t e m e n t were false, so t h a t e v e r y / e M p a is a measure.

Mapping / on t h e restriction to t h e set {~;1~1 ~< l} we get a closed everywhere defined m a p p i n g f r o m M , q to the space of b o u n d e d measures in t h e u n i t sphere, with t h e n o r m defined as t h e total variation. I n virtue of t h e t h e o r e m on t h e closed g r a p h t h e m a p p i n g m u s t be continuous. I n particular

j IIId~<CM,r /eS. (1.2.11)

T a k e a f u n c t i o n u in S so t h a t 4 ( 0 ) 4 0 a n d define us as in t h e lemma. W i t h ] replaced b y fit t h e left h a n d side of (1.2.11) is i n d e p e n d e n t of t a n d 4 0 . I n virtue of T h e o r e m 1.4 a n d L e m m a 1.2 we o b t a i n Lv. ~176 (us)=L~ v (us)= I[us[lv-+o if p > 2, t h a t is

My, ~~ (~t)=MlV(~s)-->O when t-->oo if p > 2 . F u r t h e r , it follows from T h e o r e m 1.5 t h a t

= II a

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T R A N S L A T I O ~ I ~ T V A R I A N T O P E R A T O R S 105 which is i n d e p e n d e n t of t. T h e logarithmic c o n v e x i t y of the Mp q n o r m as a function of

1/p

a n d

1/q

which is c o n t a i n e d in T h e o r e m 1.3 n o w i m m e d i a t e l y shows t h a t

Mpq(dt)-->0 if

p < 2 <q, t-->~.

H e n c e we get a contradiction if f is replaced b y ~t in (1.2.11), a n d t - + ~ . proves the theorem.

I n particular we get t h e following familiar result.

This

COROLLARY 1.5.

I/

p > 2

there exist functions u E L p such that ~ is a distribu- tion of positive order.

Proof.

E v e r y element in M1 ~ is the Fourier t r a n s f o r m of a function in L p (Theorem 1.4).

W h e n 1 < p ~< 2 ~< q < ~ , an i m p o r t a n t subclass of

Mp q

is given b y t)aley's inequality:

THEOREM 1.10.

Let q~>~O be a measurable function such that

With a constant Cp depending on p and on C we then have when l < p ~ 2

(1.2.13)

~)l/p

(fla/~[~w~d <~.lluli., u e L ' . (,.2.14)

N o t e t h a t t h e i n t e g r a n d m a y be written ]~[P~2-P so t h a t it is n a t u r a l t h a t we define it to be 0 when ~ = 0 .

Proof.

F o r the sake of completeness we recall the proof, following Z y g m u n d [18].

W h e n p - 2 the i n e q u a l i t y (1.2.14) follows with C 2 = 1 from P a r s e v a l ' s equality. W r i t e d # ( ~ ) = ( ~ ( ~ ) ) 2 d ~ a n d

Tu=d/q:.

N o t e t h a t it follows f r o m (1.2.13) t h a t

I n fact, writing m (s) = m {}; ~ (}) >~ s} we have

/~{~; q)(~)~a}= f s 2 d ( - m ( s ) ) ~ 2 f m(s) s d s + lim

o 0

since

s m ( s ) ~ C

in virtue of (1.2.13). We n o w o b t a i n

/~ {~;

](Tu) (~)]>~ s} <~ 2 C I]

u ]]1/8, u e L 1.

1.2.15)

(1.2.16)

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106 LARS HORMANDER

In fact, since I(T u) (~)J < II u lli/~ (~}, the set in question is contained i n the set where

~(~)~<l[ulll/s. F r o m the validity of (1.2.14) when p = 2 it also follows t h a t

ff {~; I(Tu)(~) [ > s} < (11 u ll~/s) ~,

u e L ~. (1.2.17) We now only have to invoke Marcinkiewicz' interpolation theorem (Zygmund [18], Theorem 1) in order to conclude from (1.2.16) and (1.2.17) t h a t (1.2.14) is valid.

I f we combine Theorem 1.10 with the Hausdorff-Young inequality

II~ll~,~<llull~, 1~<p~<2, (1.2.18) and use H(ilder's inequality, we obtain the following

COROLLARY 1.6. I / q~ satis/ies (1.2.13) and l <p<r<~p" < ~ , we have

I'~(p(l"-"~'"l'd$ ~<c, llull,,

u e L ' . (1.2.19) This reduces to (1.2.18) when r = p ' and to (1.2.14) when r = p .

THEOREM 1.11. Let / be a measurable /unction such that, with 1 < b < r we have ]or some constant C

m {~; It (~)1 ~> s} < c / s b.

(1.2.20)

Then / e M ~ q i/ 1 <p-~<2 ~<q< o~, 1 / p - 1/q= 1/b. (1.2.21) Proo/. Since M~q=Mq2" we m a y assume t h a t p<.q', for otherwise we have q'<~ (p')' =p. With ~ = I]1 b and r=q', the assumptions of Corollary 1.6 are then satisfied and since 1 / q ' - 1 / p ' = 1 / p - 1/q= 1/b we obtain

II/~llq. ~< c,, II ~11,,, ueZp.

L e t T be the distribution with ~ = 1 . When u E $ the Hausdorff-Young inequality gives since q' ~ 2

II

T ~ u

I1o ~< II]~llq. ~ c~ II u I1,,,

which proves t h a t T E L~ q a n d hence t h a t ] E Mp q. T h e proof is complete.

When p~< 2 ~<q we can thus give bounds on the absolute value of a function / which ensure t h a t the function is in M~ q. T h a t this is not possible for other values of p and q is shown b y the following result.

THEOREM 1.12. Suppose that there exists a measurable /unction F>~O which is not 0 almost everywhere, such that every measurable /unction / satis/ying the condition

I/I <

F belongs to M~ q. Then we have p < 2 <. q.

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TI~A~SLATIO2~ I2~VAI~IA:NT O:PEI~ATORS 107 Proo 1.

if g E L ~.

We m a y assume t h a t F is bounded. The assumption means t h a t F g E M y q Thus the mapping

L ~176 9 g---->Fg E M v q

is defined everywhere in L ~ and it is obviously closed since it is continuous for the topology of L ~ on the right hand side. Hence the closed graph theorem shows t h a t the mapping is continuous, t h a t is,

M,'(Fg)<CIIglI~.

I n view of the definition of M y q this means t h a t for all u and v E S

I(Fg~;~d~l<~ M,~ (Fg)Ilu I1~ II v I1o, ~< c Ngll~ II ~ II, II v I1o,.

I J I

Hence

fF]a6la~<~cI]ull~llv]}o,.

^(1.2.22)

More generally, we get for a n y

f

F ( ~ - ~ ) l

a(~)~(~)ldX_ <~Cllull, llvll~,,

^(1.2.22)'

if in (1.2.22) we replace u ( x ) b y u(x)e-2~<x"7>and make a similar substitution for v.

11 g is a continuous positive function with f g d~ = 1 and G = F-)eg, we get b y multi- plying (1.2.22)' with g (~) and integrating

This inequality has the advantage over ( 1 . 2 . 2 2 ) t h a t G is continuous und positive everywhere. Now take v fixed with v#.0 when l~l~< 1. I t then follows from

(1.2.23)

t h a t

f l~ld#<C'llull,, ~eS.

I~1<1

Thus, if we replace u by the function ut defined in Lemma 1.2 a contradiction results unless p ~< 2. Similarly, taking u fixed we obtMn q' ~< 2, that is, 2 ~< q, which completes the proof.

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108 LARS HORMANDER

1.3. Homomorphisms of M ~ v

W e shall o n l y s t u d y h o m o m o r p h i s m s of Mp v w h i c h are i n d u c e d b y a m a p p i n g

~-->a(~) of R n i n t o R m. If / is a f u n c t i o n i n R m a f u n c t i o n a * / i n R n is defined b y ( a * / ) ( ~ ) = / ( a ( ~ ) ) , ~ E R ~.

W e first consider t h e case where a is a n affine m a p p i n g ( a ( ~ ) ) j = a j 0 + ~ ajk~k, ? ' = 1 . . . m.

k = l

THEOREM 1.13. I / a is an a/fine mapping o/ R ~ onto R m, the mapping a* is an isometric mapping o/ MrS(m) into MpV(~), /or every p. I / m = n, the mapping is onto.

Proo/. T h e d e f i n i t i o n of t h e L p spaces a n d hence of L , ~ was i n d e p e n d e n t of t h e s y s t e m of c o o r d i n a t e s e x c e p t t h a t it used a p a r t i c u l a r Lebesgue measure. However, t h e n o r m of a n e l e m e n t i n Lp p is o b v i o u s l y i n d e p e n d e n t of t h e Lebesgue m e a s u r e chosen. F u r t h e r , if T is a d i s t r i b u t i o n whose F o u r i e r t r a n s f o r m has a d e n s i t y T (~), this d e n s i t y is i n d e p e n d e n t of t h e choice of Lebesgue measure. (This is m o s t easily seen w h e n T is a m e a s u r e a n d ~ (~) t h e F o u r i e r - S t i e l t j e s t r a n s f o r m . ) H e n c e M , p is i n v a r i a n t for e v e r y change of coordinates.

C h a n g i n g c o o r d i n a t e s i n R n a n d i n R m (considered as different spaces e v e n if n = m ) we m a y a s s u m e t h a t a is g i v e n b y

(a (~))j = ~j § aj0 , ] = 1 . . . m.

L e t T

ELpP(m)

a n d f o r m T 1 = e - ~ l T |

where 1 ( x ) = x 1 alo + ... + xm am o a n d ~ is t h e Dirac m e a s u r e i n t h e v a r i a b l e s xm+l . . . x~.

T h e F o u r i e r t r a n s f o r m of T1 as a d i s t r i b u t i o n i n R n is

TI

= ~ (~1-4- a l o , . . . , ~m A- a m o ) ,

so t h a t w h a t we h a v e to p r o v e is t h a t T 1ELvv(~) a n d has t h e same n o r m t h e r e as T has i n L~V(~). Now, if u E S(~),

T~ ~ u = e -2"~ ( T-)~(e ~ z u) ),

where t h e c o n v o l u t i o n i n t h e r i g h t h a n d side is t a k e n w i t h respect to x~, --. , xm, t h e o t h e r v a r i a b l e s b e i n g fixed. If C = L v ~ (T) we h a v e

f , T ~ u , v d X l ... d x . , ~ c ~ f ' u ' V d x ~ "" dx.~

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TI~ANSLATIOI~ II~IVARIANT OPERATORS 1 0 9

for f i x e d Xm+l, " " , xn. I n t e g r a t i n g w i t h r e s p e c t to these v a r i a b l e s , we g e t

which p r o v e s t h a t

T 1ELpp(n)

a n d has a n o r m w h i c h is a t m o s t C. T h a t t h e n o r m c a n n o t be s m a l l e r t h a n C is i m m e d i a t e l y seen be c o n s i d e r i n g f u n c t i o n s u w h i c h a r e p r o d u c t s of f u n c t i o n s of x 1, ...~ Xm a n d a f i x e d f u n c t i o n ~ 0 of xm+l, " " , x~. Since t h e m a p p i n g a has a n affine i n v e r s e if n = m , t h e p r o o f is c o m p l e t e .

W e o m i t t h e s i m i l a r b u t less s i m p l e a n d useful r e s u l t c o n c e r n i n g M , q w h e n p :~ q.

I n t h a t case one c a n o n l y t a k e m = n .

U n d e r c e r t a i n r e g u l a r i t y a s s u m p t i o n s it, will n o w b e p r o v e d t h a t t h e a s s u m p t i o n in T h e o r e m 1.13 t h a t t h e m a p p i n g a is affine is n e c e s s a r y . T h e e s s e n t i a l s t e p in t h e proof is t h e following l e m m a .

L E ~ M A 1.3. e ~AI~I* is not in M~" /or any p 4 2 i/ A is a real constant :~0.

Proo/. S u p p o s e t h a t e~AI~'~EM," a n d t h a t A ~:0, p 4 2 . Since M~P=M~2" we m a y a s s u m e t h a t p > 2, a n d since M p ~ is i n v a r i a n t for c o n j u g a t i o n we m a y also a s s u m e t h a t A < 0. I n v i r t u e of T h e o r e m 1.13 a p p l i e d t o t h e m a p p i n g ~ - - > ( - t / A ) 89 ~, t h e f u n c t i o n e -ftl~'l~ is in M p p for e v e r y t > 0 a n d M~ p (e -~tt~l~) is i n d e p e n d e n t of t. H e n c e we g e t w h e n u E S

if ut is d e f i n e d as in L e m m a 1.2 b y t h e e q u a t i o n e-itlr t ( ~ ) = a ( ~ ) . W h e n t - ~ (1.3.1) c o n t r a d i c t s L e m m a 1.2 w h i c h c o m p l e t e s t h e proof.

LEMMA 1.4. Let A (~) /ollows that A = O.

Proo/. A s s u m e t h a t A we m a y w r i t e

where a 1 4 O.

be a real quadratic /orm. I / e~AEM~ p vhere p~=2, it

does n o t v a n i s h i d e n t i c a l l y . A (~) = 31 ,~ + " " + a~ ~

I n v i e w of T h e o r e m 1.13 we m a y e v e n a s s u m e t h a t l a l l > l a 2 1 §

W i t h s u i t a b l e c o o r d i n a t e s

(1.3.2) I f k is a p e r m u t a t i o n (k 1, . . . , k~) of t h e i n t e g e r s 1, . . . , n we w r i t e

Ak

(~) = 31 ~t:2 ~_ ... _[_

an ~ k,2.

Since e~AEMp p we h a v e also e~A~EMp p in v i r t u e of T h e o r e m 1.13. H e n c e C o r o l l a r y 1.4 shows t h a t

I-[ e ~A~ E M p ".

k

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110 LAXS H()RMANDER

NOW ~ A k = ( n - - 1 ) ! ( a l + . . . + a n ) l~12=al~l 2 where a * 0 in view of (1.3.2). This contradicts L e m m a 1.3 a n d hence proves the ]emma.

W e are now able to prove t h e m a i n theorems in this section.

THEOREM 1.14. Let / be a real valued /unction EC ~. Suppose that there exists a sequence tk o/ real numbers such that tk--~+ oo and eit~fEM, p,

Mp v (e ~tkf) < C, k = 1, .-. (1.3.3)

where C is a constant and p # 2. Then / is a linear /unction.

On t h e other hand, if / ( ~ ) = a + 2 r ~ < h , ~> is a real linear function t h e n e " I is the Fourier t r a n s f o r m of the mass e ~t~ a t - t h , hence M~ "(eits) = 1 for every p.

(This follows for p = 1, 2 a n d oo f r o m Theorems 1.4 a n d 1.5, a n d t h e n in general f r o m the c o n v e x i t y in T h e o r e m 1.3.)

Proo/ o/ Theorem 1.14. We shall prove t h a t the second derivatives of / vanish.

I t is sufficient to do so for ~ = 0 , for every translation of / also satisfies (1.3.3) in view of T h e o r e m 1.13. Since / E C 2 we have

/ ( ~ ) = a + < h , ~ > + A ( ~ ) + o ( l ~ l ~ ) , ~-->0,

where a is a real n u m b e r , h a real vector, A a real q u a d r a t i c form. W r i t e g (~) = / (~) -- a -- <h, ~>.

I t follows f r o m Corollary 1.4 a n d the r e m a r k a b o v e after T h e o r e m 1.14 t h a t g satisfies the same a s s u m p t i o n s in the t h e o r e m as / does. B u t n o w we h a v e g (~) = A (~) +

+ o (I $ Is), a n d writing g~ (~) = tag (~/tk 89 it thus follows t h a t gk (~)-+A (~),

u n i f o r m l y on every c o m p a c t set. I t follows f r o m (1.3.3) a n d T h e o r e m 1.13 t h a t Mp" (e%) < C.

F r o m t h e following l e m m a it follows t h a t e ~A E M~ p. Hence A = 0 in view of L e m m a 1.4, which completes the proof.

LEMMA 1.5. The unit spheres in M~ ~ and in L~ q are closed in $'.

Proo/. O n l y t h e s t a t e m e n t concerning Lv q needs to be proved, for the Fourier t r a n s f o r m a t i o n is an isomorphism of $' m a p p i n g the u n i t sphere in Lp q o n t o t h a t in M , q. N o w the u n i t sphere in L , q is b y definition the set {T; T e $' a n d ] T ~ e u ~ v (0)[

<~ [luH, [[vi{q,; u, v e S}, a n d since t h e left h a n d side of t h e i n e q u a l i t y is the absolute value of a continous linear f o r m on S', the assertion is obvious.

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T R A N S L A T I O N I N V A R I A N T O P E R A T O R S 111 THEOREM 1.15. Let a be a C 2 mapping o/ R ~ into R m. Assume that a* maps MpP(m) into M~(~) and that p=~ 2. Then a i8 a/line and onto.

Proo/. Since t h e m a p p i n g a* is obviously closed, it follows from the t h e o r e m on the closed g r a p h t h a t a* m a p s M~'(m) c o n t i n u o u s l y into MpP(~). I f l is ~ linear function in R m, the n o r m of e ~t~ in Mp~(m) is 1 for every t. I n view of t h e c o n t i n u i t y of a*, it follows t h a t the n o r m of e ~t~*~ in Mp'(n) is b o u n d e d for all t. Hence Theo- rem 1.14 shows t h a t a * l is a linear function. A p p l y i n g this with 1 equal to the ]th coordinate in R m, it follows t h a t (a(~))j is a linear function of ~ E R n for ] = 1 , . . . , m.

This proves t h e theorem. F o r if a were n o t onto, its range were a null set a n d every f u n c t i o n would be in M~P(n).

F o r p = 1, t h a t is, for the algebra of Fourier-Stieltjes transforms, a m u c h more precise result has been given b y Beurling a n d Helson [1] (see also Helson [4]). I n particular, it is n o t necessary in t h a t case to assume t h a t a EC 2. (These a u t h o r s also treat more general h o m o m o r p h i s m s . However, the proof of T h e o r e m 1.14 i m m e d i a t e l y extends to t h a t case if a smoothness a s s u m p t i o n replacing t h e a s s u m p t i o n a E C 2 is made.) F o r p:t: 1 a n d ~o, however, some smoothness a s s u m p t i o n is needed in Theo- rem 1.14. I t m a y be sufficient to assume t h a t a E C ~ b u t n o t merely t h a t a is Lip- schitz continuous. I n fact, using Riesz' t h e o r e m on conjugate functions (see Chapter II), Corollary 1.4 a n d T h e o r e m 1.13 it is easily seen t h a t if a is pieceweise linear (and has only a finite n u m b e r of pieces) t h e n a* m a p s M~ ~ into itself, if 1 < p < oo. F o r f u r t h e r details see Schwartz [13].

1.4. Analytic operations in M~ ~

Our purpose here is to prove an analogue of t h e W i e n e r - L ~ v y t h e o r e m concerning M11, or r a t h e r the subspace of M11 consisting of t h e Fourier t r a n s f o r m s of functions in L 1. This subset of M11 can also be regarded as the closure of S in M11 a n d we are thus led to introduce the following definition.

D E ~ ' I N I T I O N 1.4. The closure o/ S in M~ p will be denoted by mp ~.

I t is clear t h a t m; p is also a n o r m e d ring. Since MpP([)>~[[[[[~ according to (1.2.6), it follows t h a t m ~ P c C (1L~. On t h e other hand, we can prove an opposite result which is o n l y slightly weaker.

THEOREM 1.16. I]

I 1 / q - 1 / 2 1 < 1 1 / p - 1 / 2 1 we

have M / n c n L r c m j .

Remark. If p = 1 the result is n o t valid with q = 1 since there are singular measures with F o u r i e r t r a n s f o r m s converging to 0 a t infinity. ~u do n o t k n o w if it is possible to t a k e q = p for some other value of p :t: 2.

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112 L ~ S HORMANDER

Proo/ o/ the theorem. L e t

/EMp~N

C a n d a s s u m e first t h a t / has c o m p a c t support. T a k e a n o n n e g a t i v e f u n c t i o n ~ E C ~ such t h a t S ~ d ~ = 1 a n d f o r m

1~ (~) = f / ( ~ - s ~ ) ~ (~) d ~ .

As is well known, /~ E C~ a n d converges to / u n i f o r m l y when s-->0, hence M2 ~ ( / - [~)-->0 as s-->0 (Theorem 1.5). T h e c o n v e x i t y of the n o r m in M r p a n d T h e o r e m 1.13 also give t h a t M r r ([~) ~< M r r ([), hence M r r ( / - / ~ ) ~< 2 M r r ([). R e p l a c i n g if necessary q b y q' we m a y a s s u m e t h a t l / q = ~ / p + ( 1 - ~ ) / 2 where 0 ~ < ~ < 1. H e n c e t h e logarithmic c o n v e x i t y of t h e i q q n o r m as function of 1/q ( T h e o r e m 1.3) shows t h a t

Mq q (f - f~) <, (My ~ (f ^- f~))~ (M~ ~ ( / - ]~))1-~-+0 as e-->O,

which p r o v e s t h a t / ~ m q q. N e x t let / be a n a r b i t r a r y function in M r r N C N L ~ . L e t

~pEC~ be equal to 1 w h e n I$1 ~<1 a n d set /, ($) = f ($) yJ (e $). Since M r p is a n algebra containing $ we get / , E M r r, a n d f r o m w h a t we h a v e a l r e a d y p r o v e d it t h u s follows t h a t f, E mq q. Since Mr r (f, - f) <, Mr r (f) (1 + M r r (W)) a n d

/ 2 ~ ( [ - / , ) ~ < ( 1 + sup I~1) sup I/I- 0 as

EI~I>I

it follows again f r o m the logarithmic c o n v e x i t y of the Mq q n o r m as a f u n c t i o n of 1/q t h a t Mqq(/-/~)--~O as ~-->0. H e n c e / E m q q.

THEOI~EM 1.17. The maximal ideal space o/ the algebra mr r can be identi]ied with R~; the characters are the mappings /--->[ (~), ~ E R ~.

Proo[. T h e restriction of a continuous c h a r a c t e r in mr r to ml 1 is a continuous c h a r a c t e r in m l 1, hence of t h e f o r m [->[ (~) since mx 1 is t h e Fourier t r a n s f o r m of L 1.

I n view of t h e definition of mr r, t h e set S a n d a f o r t i o r i m l I is dense in mr ~. H e n c e all continuous characters on mp r are of the f o r m [--->[ (~). Since S ~ m r r ~ C it is obvious t h a t t h e t o p o l o g y of t h e space of m a x i m a l ideals is t h e usual t o p o l o g y in R ".

Remark. I t is n o t k n o w n to the a u t h o r w h e t h e r R n is t h e m a x i m a l ideal space of M , r N C for some p 4 : 2 . T h a t this is n o t t r u e for p = l is well k n o w n . (Cf.

~reider [14].)

F r o m T h e o r e m 1.17 a n d t h e basic results on c o m m u t a t i v e B a n a c h algebras (see L o o m i s [6], pp. 78 a n d 79), we o b t a i n t h e following t h e o r e m .

T H E O R E M 1.18. I[ [ E m r r and q) is analytic in a neighbourhood o/ the closed range o/ [ and @ (0)= O, then (I)([)~mr r.

C o m b i n a t i o n of T h e o r e m s 1.16 a n d 1.18 also gives

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T1KANSLATION I N V A R I A : N T OPEI~ATOtCS 113 THEOREM 1.19. I / / E M p ~ N C N L ~ and 9 is analytic in a neighbourhood o/ the closed range of ] and r (0) = O, then r (/) E Mq q i ] ] l / q - 1/2] < ] 1 / p - 1 / 2 ].

F o r a n o t h e r subalgebra of M~ ~ we shall in Chapter I I discuss similar results, which are closely related to some theorems of Calder6n a n d Z y g m u n d [3].

C H A P T E R I I

E s t i m a t e s for s o m e special operators 2.1. Main theorem

Corollary 1.2 shows t h a t T E L ~ (or T E M if a = 1)implies TEL~q if 1 ~ p < ~ q ~ ~ a n d

1 / p - l / q = 1 - 1/a. (2.1.1)

T h e o r e m 1.4 shows t h a t these conditions on T are also necessary in order t h a t T E L p q for all p, q satisfying (2.1.1) a n d l ~ p < ~ q ~ ~ . The purpose here is to show t h a t if the condition T E L ~ (or M) is slightly weakened we still have T E L l " if (2.1.1) is fulfilled a n d 1 < p ~< q < c~.

L e t k be a locally integrable function. I f ]c E Lp q a n d we set with t > 0

kt (a) (x) = t -n/a k (x/t), (2.1.2)

we h a v e also ]gt(a) ELp q and, if (2.1.1) holds,

Lp q (kt (~)) = L , q (/c). (2.1.3)

This follows from T h e o r e m 1.10 when p = q a n d in fact b y a trivial c o m p u t a t i o n for all p a n d q. I t is therefore n a t u r a l t h a t we n o w introduce a condition involving t h e f a m i l y of functions let (a).

D E F I N I T I O N 2.1. We shall say that the locally integrable /unction ]c is almost in L a and write k E K a i/ there is a compact set M , a neighbourhood N o/ 0 and a con.

stant C such that

( f []ct(a) (x--y)-]2t(a) (x)ia

y E N , O < t . (2.1.4)

CM

Remarlc. W h e n a = 1 it would have been e n o u g h to assume t h a t k is a measure a n d t h a t the analogue of (2.1.4) is valid. However, we do n o t consider this simple generalization in order n o t to complicate t h e notations.

9 8 -- 60173032.

Acta mathematica.

104. I i n p r i m 4 le 23 s e p t e m b r e 1960

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114 LARS H()RMANDER

E x a m p l e s of f u n c t i o n s in K a will b e g i v e n in t h e n e x t section. T h e m a i n re- s u l t we shall p r o v e is

T H E O R E M 2.1. Let k E K ~. Then k E L p q either /or all p and q satis/ying (2.1.1) with l < p <~ q < ~ or else /or no such p and q.

I n t h e a p p l i c a t i o n s we shall use T h e o r e m 1.5 o r 1.11 t o p r o v e t h a t k E L p q for s o m e p a n d q.

W e shall p r e p a r e t h e p r o o f of T h e o r e m 2.1 b y r e w r i t i n g t h e p r o p e r t y (2.1.4) in a m o r e useful form. L e t u E L 1 v a n i s h o u t s i d e N a n d f o r m t h e c o n v o l u t i o n

(kt(a) ~ u ) (x) = f kt (~) (x - y) u (y) d y (2.1.5) w h i c h e x i s t s a l m o s t e v e r y w h e r e ( a n d is t h e d e n s i t y of t h e c o n v o l u t i o n i n t h e d i s t r i - b u t i o n sense.) I f

f u d x = O (2.1.6)

we can also w r i t e

(kt(a)-)eu) (x) = f (kt (a) (x - y) - kt (a) (x) ) u (y) d y.

(2.1.5)'

U s i n g M i n k o w s k i ' s i n e q u a l i t y for i n t e g r a l s a n d (2.1.4) we t h u s o b t a i n

\ l l a

( f lkt(~ <-C f luldy. ^(2.1.7)

CM

T h a t (2.1.7) is p r a c t i c a l l y e q u i v a l e n t to (2.1.4) is seen b y l e t t i n g u in (2.1.7) con- v e r g e t o t h e difference b e t w e e n t h e D i r a c m e a s u r e s a t y E N a n d 0. (2.1.4) t h e n follows w i t h C r e p l a c e d b y 2 C.

L e t I 0 be a c u b e c N w i t h c e n t r e a t 0 a n d l e t I ~ be a n o t h e r cube w i t h c e n t r e a t 0 c o n t a i n i n g M . I f I is a n a r b i t r a r y c u b e we d e n o t e b y I* t h e cube w i t h t h e s a m e c e n t r e such t h a t m ( I * ) / m ( I ) = m ( I ~ ) / m ( I o ) = ~ , . ( B y a c u b e we a l w a y s m e a n a c u b e w i t h edges p a r a l l e l t o t h e c o o r d i n a t e axes.) W h e n I = I o i t t h e n follows f r o m (2.1.7) t h a t

( f lk:~176 f luldy if f udx=O and u=O outside I.

^(2.1.8)

r *

This i n e q u a l i t y is in f a c t v a l i d for e v e r y cube I . F o r since (2.1.8) is i n v a r i a n t for t r a n s l a t i o n we m a y a s s u m e t h a t I h a s i t s c e n t r e a t 0 so t h a t I = s - l l o for s o m e s.

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TRA~SLATIOI~ I~VA~IA~CT OPERATORS 115 If U vanishes outside I it follows t h a t v ( x ) = s n u (sx) vanishes outside I o a n d an easy c o m p u t a t i o n gives t h a t

(]~t(a)-~V) (X) = S n/a (~st(a)"~'U) (8X),

A p p l y i n g (2.1.8) w i t h u replaced b y v a n d I b y I 0, a n d s u b s t i t u t i n g x for s x we t h u s o b t a i n (2.1.8) with t replaced b y st. Since t is arbitrary, this proves (2.1.8) for all t. I n particular, for t = 1 we obtain the following lemma.

L E ~ M A 2.1. L e t k E K ~. T h e n with the same constant C as in De/inition 2.1, we have /or every cube I with I* de/ined as above

]k-~ul a d z ) l u l d x

i/

u d x = O and u = 0 outside I. (2.1.9)

GI* I

F o r the proof of T h e o r e m 2.1 we also need a f u n d a m e n t a l " c o v e r i n g l e m m a "

due to Calder6n a n d Z y g m u n d [2] (see also Z y g m u n d [18] a n d Stein [15]). We give it a slightly different form.

LEMMA 2.2. Let u E L 1 and let s be a number > 0 . Then we can write

oo

u = v + ~ wk, (2.1.10)

1 where v and all w~ E L 1,

NVIII ~- ~[IWklll ~< 3 IlUII 1, (2.1.11)

1

I v (x) I ~ 2n s almost everywhere, (2.1.12)

and /or certain disjoint cubes I s

f w k d x = O , and w k ( x ) = 0 i/ x q l k , (2.1,13)

f

~ m ( l k ) ~ 8 -1 luldx.

(2.1.14)

1

I / u has compact support, the supports o/ v and all wk are contained in a /ixed compact set.

Proo/. Divide t h e whole space R ~ into a mesh Of cubes of volume > s -1

f lul

d x.

The m e a n value of ]u] over every cube is t h u s < s. Divide each cube into 2 ~ equal cubes a n d let 111, Ix2, Ila . . . . be those (open) cubes so o b t a i n e d over which t h e m e a n value of l ul is ~>s. We have

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116 LARS ttORMANDER P

s m (Ilk) < I [ u [ d x < 2 n s m (Ilk). (2.1.15) Ilk

F o r if I1~ was obtained b y subdivision of the cube I ' , the c o n s t r u c t i o n gives

Ilk I"

We set v ( X ) - - m ( i l k ) u d y , XEIlk; w l k ( x ) = , x~iIl~"

Ilk

N e x t we m a k e a n e w subdivision of the cubes which are n o t a m o n g t h e cubes Ilk, select those new cubes /21, I~2 . . . . over which t h e m e a n value of ]u] is ~> s, a n d e x t e n d t h e definitions (2.1.16) to these cubes. Continuing in this w a y be o b t a i n disjoint cubes Ijk a n d functions wsk; for convenience in n o t a t i o n s we r e a r r a n g e t h e m as a sequence. If the definition of v is completed b y setting v ( x ) = u (x) w h e n x C O =

I.J Ik, it is clear t h a t (2.1.10) holds. To prove (2.1.11) we first note t h a t

Ik lk

Since t h e cubes are disjoint, wk vanishes outside Ik a n d

fcolvldx=fooluldx,

^we

i m m e d i a t e l y get (2.1.11). F u r t h e r (2.1.12) follows from (2.1.15) if x E O . On t h e other hand, if x ~ O, there are arbitrarily small cubes containing x over which to m e a n value of l u I is < s. H e n c e l u (x) l ~< s a t e v e r y Lebesgue point in C O, that is, a l m o s t everywhere. (2.1.13) follows f r o m the construction. To prove (2.1.14) we only n o t e t h a t since the cubes Ik are disjoint we get b y a d d i n g t h e inequalities (2.1.15)

lul dx.

8 ~ m ( I k ) ~ o T h e proof is complete.

We n o w prove a n estimate for t h e case p = 1, q = a , which is t h e n a substitute f o r T h e o r e m 2.1. Using this result it will be easy t o prove T h e o r e m 2.1.

THEOREM 2.2. Let k E K ~ and assume that k E L p q /or some p and q satis/ying (2.1.1) with l < p ~ < q < oo. T h e n we have, when u has compact support and u E L 1,

re{x; [ k ~ - u ( x ) [ > a } <C~ (l[u[[x/a)% (~>0, (2.1.17) where m denotes Lebesgue measure and C 1 a constant.

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T R A N S L A T I O N I N V A R I A I ~ I T O P E R A T O R S 117

Proo/.

We m a y assume in t h e proof t h a t I l U l [ l = l . To simplify t h e n o t a t i o n s we write ~2 ( x ) = k ~ u (x), which exists almost everywhere as an absolutely convergent integral. F o r m the decomposition of u given b y L e m m a 2.2. T h e n we h a v e

1~2 (x) l ~< I~ (x) l + ~ I ~ (x) l , (2.1.18)

1

for every x such t h a t

f l k ( z - - y ) l ( I v ( y ) l + 5 1 w k ( y ) ) d y < ~ ,

hence a l m o s t everywhere. I n virtue of L e m m a 2.1 we have

c~2

a n d if O = U Ik* it follows from (2.1.14) t h a t

m(o)<~-111~lll=~8 -1.

If we restrict the integration in t h e left h a n d side of (2.1.19) to C 0 a n d use Min- kowski's inequality, we get writing z~ = Z [ ~ k [

~~ <~[Iwkll~<nCIlulh=3U.

Co

H e n c e t h e measure of t h e set of points in C 0 where ~(x)~> 89 is at m o s t ( 6 C / a )

a.

Choosing s = aa we t h u s h a v e z~ (x) < 89 ~ except in a set of measure at m o s t (y + (6

c)a)/o ~

N o w the a s s u m p t i o n t h a t /c E Lp q for some p a n d q means t h a t

IIk~ullo<o'll~ll,,

~ e s .

(2.1.2o)

$ is dense in L p since p < oo. H e n c e (2.1.20) follows for every u EL p with c o m p a c t s u p p o r t (with t h e convolution defined in t h e distribution sense, which however is well k n o w n to be equivalent to the classical sense since /c is locally integrable). I n particular, ,(2.1.20) m a y be applied to v which gives

I1,~1to ~< c' I1,,11,, < (2,~)~

~ , [I v I[y~ < c " ~ , ~ , o - - ~ = c " ~ - o ~

(2.1.21)

in view of (2.I.11), (2.1.12) a n d (2.1.I). Hence t h e measure of t h e set where [ , ~ l > 8 9 is at m o s t ( 2 C " ) q a -~ Since (2.1.18) shows t h a t t h e set where ] ~ [ > a is contained in t h e u n i o n of t h e set where [ ~l > 89 a a n d t h a t where ~ > 89 a, t h e i n e q u a l i t y (2.1.17) follows.

Proo[ o/ Theorem

2.1. L e t

k E K ~

a n d /cEL~, ~ where P0 a n d

%

satisfy (2.1.1).

T h e n it follows from T h e o r e m 2.2 t h a t (2.1.17) holds. B u t this means t h a t Marcin-

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118 LARS HORMANDER

kiewicz' interpolation theorem (Zygmund [18], Theorem 1) can be applied, and we obtain if l < p < p 0 and q is defined b y (2.1.1)

IIk ull <Cllull

if

u E L v

and u has compact support. (2.1.22) I n particular, this holds when u E C~ and since C~ is dense in S for the L p norm we obtain (2.1.22) for u E S. Hence k E Lp q. To remove the restriction p < P0 we only have to use Theorem 1.3. The proof is complete.

Remark.

I t is i m p o r t a n t in the applications t h a t the proof gives an estimate of L~ q (k) which only depends on p, q, P0, q0, Lp0 ~~ (k), the constants C and ~ connected with (2.1.4) and the dimension n. This fact is often useful in estimating

Lvq(k)even

when k E L a. (See for example the proof of Theorem 2.5 below.)

2.2. Applications

Our first example is t h a t of Calder6n and Z y g m u n d [2], where the methods of section 2.1 were originally introduced. Thus k is a locally integrable function satisfying

k(x)~O

if

Ixl<l, k(tx)=t-nk(x)

if t~>l,

Ixl >l.

^(2.2.1)

Assume further t h a t k E K 1. (In virtue of the footnote on p. 95 in Calder6n a n d Z y g m u n d [2], this follows if k satisfies a Dini condition when Ix[ = 1.) We have to examine when the Fourier transform $ is in L ~ so t h a t k E L2 ~. First note t h a t if y E N we have

k ( x - y ) - k(x)E L 1

as a function of x, since k E K 1. Hence the Fourier transform (e 2.~<~.~> 1)/~ is continuos. Since this is true for all y in the neigbourhood N of 0 it follows t h a t ~ is a continuous function for ~ # 0 a n d bounded when ~ - - > ~ . I t remains to study the behaviour of $($) as ~-+0. Noting t h a t the Fourier transform of ~t r is ~ (t~), we obtain

Hence

1

Iml=l t

k(eo) dw dr/r.

~-~olim ( ~ ( t ~ ) - ~(~)) = - l o g

tf k(~o) din.

Letting t-->O we find t h a t if ~ is a bounded function we m u s t have

f k(w)dw=O.

I~1=1

(2.2.2)

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T R A N S L A T I O N I N V A R I A N T O I ' E R A T O I ~ S 119 Conversely, if this condition is fulfilled, one can write

1

]c (t ~ : ) - fc ( ~ ) =

f f ( e - 2 ~ i r < ~ ' ~ > = l ) k ( w ) d o ) d r / r .

Io~1=1 t

I f we t a k e ~ as a u n i t v e c t o r a n d 0 < t ~ < 1, the right h a n d side is b o u n d e d b y

2~flk(co)ldeo ,

which p r o v e s t h e boundedness of $ for ~ * 0 . T h u s Ic is the s u m of a b o u n d e d function a n d - - p o s s i b l y - - a distribution with s u p p o r t a t 0, t h a t is, a linear c o m b i n a t i o n of t h e Dirae m e a s u r e a t 0 a n d its derivatives. A c o m p o n e n t of t h a t f o r m is impossible, however. To see this it is sufficient to show t h a t if

qJ~(x)=q~(x/s),

where ~ E C~ r it follows t h a t f c ( ~ ) - + 0 as e->0. N o w we h a v e

( ~ ) = k ( ~ ) = f k (x) q~ (e x) e n d x.

t h e facts t h a t

I~(y)l<~C/(ly[§

~ e S a n d t h a t

f l k ( r o ) ) l d c o < C / r n ,

Using

we get b y introducing p o l a r coordinates a n d c o m p u t i n g t h e integral ] fc (~9,)I ~< C e'~ log (1 + l / e ) ,

a n d this tends to 0 as e--*0. H e n c e the following t h e o r e m follows f r o m T h e o r e m 2.1.

T H E O R E M 2.3.

I/ k is in K 1 and satis/ies

(2.2.1),

it /oUows that lcEL~ p /or 1 < p < ~ i]

(2.2.2)

is /ul/illed whereas k is not in L~ ~ /or any p i]

(2.2.2)

is not valid.

W e briefly recall the consequences of this result for the singular integrals corresponding to k. First n o t e t h a t it follows f r o m (2.2.2) t h a t k0 a) = l i m

kt a)

exists in t h e t o p o l o g y of S'. I n fact, if u E S we h a v e

kta)(u) ~ f f u(ro~)k(co)deodr/r,

a n d in view of (2.2.2)

f u (rog) k (w) do) = f (u (re)) - u

(0)) k (co) dco = 0 (r) as r - > 0 . H e n c e the integral

0

k (co) dco