Radial Fourier Multipliers and Associated Maximal Functions

Recent Progress in Fourier Analysis I. Peral and J.-L.Rubio de Francia (Editors)

0 Elsevier Science Publishers B.V.(North-Holland), 1985

RADIAL FOURIER hlU1,TIPLIERS AND ASSOCIATED h1AXIMAL FUNCTl ONS

Anthony C a r b e r y * C a l i f o r n i a I n s t i t u t of Technology

I n t h i s t a l k we i n t e n d t o show how a c e r t a i n s q u a r e f u n c t i o n ill

1.

t r o d u c e d by E . M . S t e i n c 3 n b e u s e d t o o b t a i n " g e n e r a l " m u l t i p l i e r and maximal m u l t i p l i e r t h e o r e m s Cor r a d i a l F o u r i e r m u l t i p l i e r s . T h e m u l t i p l i e r t h e o r e m e x t e n d s t h e t h e o r e m o f C a r l e s o n a n d S j a l i n , [5] t o r a d i a l m u l t i p l i e r s o f R 2 w h i c h a r e n o t s m o o t h away f r o m a o n e dimensional " s i n g u l a r i t y " (as a r e t h e Bochner-Riesz m u l t i p l i e r s (1 - 151 '):, u > 0 ) a n d t h e maximal t h e o r e m g e n e r a l i z e s t h e r e s u l t o f [l] c o n c e r n i n g a l m o s t - e v e r y w h e r e c o n v e r g e n c e o f B o c h n e r - R i e s z means on R 2 t o a w i d e r c l a s s o f f u n c t i o n s , a s well a s p r o v i d i n g a u n i f i e d a p p r o a c h t o c e r t a i n o t h e r o p e r t o r s a s s o c i a t e d t o maximal a n d p o i n t w i s e c o n v e r g e n c e p r o b l e m s , i n c l u d i n g S t e i n ' s s p h e r i c a l maximal f u n c t i o n , "41 and t h e s o l u t i o n o p e r a t o r t o t h e l i n e a r i s e d SchrGd i n g e r e q u a t i o n Au = i a u / a t , u ( x , 0 ) = f . Let us begin with t h e square f u n c t i o n .

P(5) =

=(lo

15 ( 1 5 1 - 1 1 y - l

m

*

10;

duced

G"

t h e maxima

f ( x ) I 2 dt/t)"2 i n [12]

and l e t

G"(f)(x)

For

let

a > 1/2,

=

( w h e r e $J,(x) = t - n $ ( x / t ) ) .

where he used i t t o s t u d y t h e

L2

Stein intrg

behaviour of

Bochner-Riesz o p e r a t o r . In f a c t , t h e easy r e s u l t about

a > 1/2, t h e n I G a ( f ) 1 2 5 Ca I f 1 2 . Sunouchi n + 1 t h e n c;" b e h a v e s a s a v e c t o r - v a l u e d a > -, 2 < Calder6n-Zygmund s i n g u l a r i n t e g r a l o p e r a t o r , a n d s o I G " ( f ) l P < C i f l p f o r 1 < p < a. T h i s seems t o b e a l l t h a t was known P,",U a b o u t G" u n t i l f a i r l y r e c e n t l y . However w e now h a v e f u r t h e r linow-

on

[18]

L'

is that i f

observed t h a t i f

ledge of G u t s Lp(Rn)

*

r a n g e s o f b o u n d e d n e s s when

n = 1

or

R e s e a r c h s u p p o r t e d i n p a r t b y NSF G r a n t MCS 8 2 0 - 3 3 1 9 . 49

2.

50 A. Carbery

Theorem 1. Let IGa(f)lp

n

=

1

5 Cp,n,alflp

-'

2 , a > 1/2, and 1 < p < 2n 2 n > P > n+2u-1'

m.

Then

The case a > n+ 1 of the theorem is Sunouchi's result, and the case a > 1/2, p = 2 is Stein's. The case a > n ,p 2 seems to be due to CBrdoba [7] when n = 1 , and is possibly new in general*. The case a > 1/2, 2 5 p 5 4 , n = 2 is a consequence of (This is the only "critical" case which does not at Carbery, [ l ] . present generalise to all dimensions). The remaining positive cases follow by complex interpolation using Stein's theorem (see [17]) for analytically varying families of operators. (Unfortunately we have to complicate G" a little by letting a take on complex values; but using the formula (1) below and arguing as in [17] we see

z-

that IGa(f)lp Re B > 0 and theorem).

5 Clflp * IGa+@(f)lp 5 C(r(Re B1/Ir(@)[)lflp when R, which is good enough for us to apply Stein's

Than G" cannot be bounded unless p > 2n/n+2a-1 is a corollary of theorem 4 (see application 3 below), and the necessity of the condition p < 2n/n-2a follows from applying theorem 2 below to the Bochner-Riesz means. (These statements remain valid in all dimensions). Before proceeding to state and prove our main theorems, we need a variant of the formula

(valid when Re a > 11 + 1/2) which appears in the works of Stein on the maximal Bochner-Riesz and spherical maximal operators, [12] and [14]. This variant is contained in the following well-known RiemannLiouville formula:

*

Details will appear elsewhere.

Radial Fourier multipliers 51

Proof. For € , a > 0, h in the hypothesis of the lemma and f f: L2(R), introduce {(d/dt)z h}-(v) = (E-iv)a ;(v) and

1

m

I:

f(x)

=

X

c

(E-iv)-'

h

=

a

(t-x)a-' e-E(t-x) f(t)dt.

Notice that since is the Fourier transform of the L 1 function (-x):-'

ca I a€ (d/dt)z h

and moreover

(d/dt)z h

eEX,

=

= Ca(d/dt)Lalt1 I1-at[a]h. E Thus supp (d/dt): h C (--,a[, and an application of the dominated convergence theorem shows that (d/dt)" h + (d/dt)" h, and that for each -ESt-X) (t)(d/dt)z .+ x[~,-) (t)(=) d a h in L 2 as E 0. x, e X[x,m) Therefore, since the function (t-x)a-' X[,,~] (t) belongs to L2 when a > 1/2, we have for almost every x that -+

=

ca

[ (t-x)'-l

e-E(t-x)(d/dt):

h(t)dt

j

ca

(t-x)a-'(d/dt)

h(t)dt,

EO '

concluding the proof of the lemma. 11. We are now in a position to state the main inequality from which

-

multiplier theorems may be deduced. Let I$ be a fixed nonnegative smooth bump function supported in [1,2]. For 1 < q < and a > 1/q let R(q,a) = Im e Lm(O,-)l 1 . 1 R(q,a)

Theorem 2. Let = m(IcI) 1 ( 6 1 .

a >

1/2 and suppose t h a t

m

E

R(2,a).

Let Tf(6)

=

hen

Corollary 3. ( [ 2 ] ) . Let n = 2 , a > l/q and suppose that m E R(q,a). Then m(lS1) is an Lp(R 2) multiplier when

Remarks 1 . The most interesting special case of the corollary is the case q = 2,a > 1/2. It then states that a radial function, which, when regarded as a function on ( 0 , ~ ) belongs to R(2,a), a > 1/2, 2 is an Lp(R ) multiplier for 4 / 3 5 p 5 4. This extends the

52 A. Carbery

C a r l e s o n - S j 6 l i n t h e o r e m b e c a u s e ( 1 - t 2 ) A+ E R ( q , a ) when T h i s s p e c i a l c a s e f o l l o w s from t h e o r e m 2 , t h e c a s e Ig(Tf o f t h e o r e m 1 and t h e i n e q u a l i t y [ T f l < C P - P," which i s a s t a n d a r d r e s u l t from s i n g u l a r i n t e g r a l s

a <

+ l/q.

2. The c a s e q = 2 , a > 1 o f t h e c o r o l l a r y i s a v e r s i o n o f 11Grm a n d e r ' s m u l t i p l i e r theorem f o r r a d i a l f u n c t i o n s , a n d f o l l o w s from t h e c a s e n = 2 , a > 1 , p L 2 of theorem 1 .

The r e m a i n i n g c a s e s f o l l o w from embedding a n d i n t e r p o l a t i o n p r o p e r t i e s o f R(q,a) s p a c e s . (For t h e s e p r o p e r t i e s and a comparis o n w i t h WBV s p a c e s , see [ 3 ] ) . 3 . The c o r o l l a r y i s b e s t p o s s i b l e i n t h e s e n s e t h a t i n d e x 1 1 a p p e a r i n g i n i t s s t a t e m e n t cannot be i n c r e a s e d . 1 / 2 ( a - (- 9 T h i s may b e s e e n i n t h e c a s e q 5 2 by e x a m i n i n g t h e B o c h n e r - R i e s z m u l t i p l i e r s , and i n t h e c a s e q 2 2 smooth m u l t i p l i e r s o f t h e form eilEla/lclaa, (large l c l ) , 0 c a < 1 which b e l o n g t o R ( q , a ) f o r 1 < q < m . The p r e c i s e r a n g e o f p ' s f o r w h i c h t h e s e i a t t e r m u l t i p l i e r s g i v e bounded o p e r a t o r s on Lp was d e t e r m i n e d i n [lo].

z)+)

1

P r o o f o f Theorem 2 . Apply t h e lemma w i t h @(SUI m

(5) =

ca

a

S

(t-s)a-l(-&)a

by t h e a s s u m p t i o n on =

m.

h(s)

=

[@(tu) m ( t ) ] d t ,

Therefore,

$(Su) m

supp ( d / d t ) a [ @ ( u t ) m ( f ) ] C_ ( - - , 2 / u l .

5 'a as r e q u i r e d .

lmlR(2,a) Ga(f)*(x)

which i s v a l i d

(lCl)?(EI

I E / u @ ( I c I u m) ( 6 )

since

@(su) m (s) :

Thus

=

Radial Fourier m u l t i p l i e r s

53

W e t u r n now t o t h e maximal o p e r a t o r a s s o c i a t e d t o a r a d i a l F o u r i e r m u l t i p l i e r m( 15 I ) . If 111.

T* f ( x ) =

sup

ITtf(x)

O
1.

We s e e k " g e n e r a l " c o n d i t i o n s o n a m u l t i p l i e r a s s e r t an i n e q u a l i t y of t h e form

ITx fl

p r i a t e d i l a t i o n invariant spaces

X

m

t h a t a l l o w us t o

L p ( R n ) 5 C p , n l m l X l f l LP(R") f o r a n a p p r o p r i a t e Banach s p a c e X o f b o u n d e d f u n c t i o n s d e f i n e d on (0,~). S i n c e I T f ( x ) I 5 T b f ( x ) , o n e i s l e d t o b e l i e v e t h e a p p r o -

R(q,a)

t o c o n s i d e r a r e embedded i n

- i n f a c t we n e e d p o t e n t i a l s p a c e s o f w h i c h t h e 2 l e t La b e t h e coma r e l o c a l i z e d v e r s i o n s . For a > 1 / 2 ,

the spaces

R(q,a)

pletion of the

Cm

t h e norm

=

[ml:2

{*}2I$.

f u n c t i o n s o f compact s u p p o r t i n

I, I a+' s

(d/ds)a

(0,m)

under

A simple computation

at1 ( d / d s ) "

shows t h a t t h e a M e l l i n t r a n s f o r m o f s t i a l l y ( i t + l ) a times t h e h l e l l i n t r a n s f o r m o f

{q}

is essenand so t h e 2 L,(R), (as

m, 1; is j u s t the usual Bessel p o t e n t i a l space f o u n d f o r e x a m p l e i n S t e i n [ I 3 1 Ch. 5 ) u n d e r t h e c h a n g e o f v a r i a b l e s s + exps. space

Theorem 4 . S u p p o s e

m

E

L2

a

f o r some

a

>

1/2.

With

T*

as above

L

a 2 o r 2 , a > 1 / 2 a n d s u p p o s e t h a t m E La. Then t h e maximal o p e r a t o r a s s o c i a t e d w i t h m i s b o u n d e d on Lp(Rn) when 2n 2n n-2cr > p > -n + 2 a - 1 C o r o l l a r y 5 . Let

n = 1

Remarks 1 . The c o r o l l a r y f o l l o w s d i r e c t l y f r o m Theorems 1 a n d 4 . By c o n s i d e r i n g t h e Bochner-Riesz means, w e see t h a t t h e i n d e x 2n/(n-2a) cannot b e r a i s e d ; t o see t h a t 2n/(n+2a-1) cannot be lowered, s e e a p p l i c a t i o n 3 below.

2.

If

D s f ( 5 ) = 161'

?(S)

ITtf(x)l

5

for Ca t S

s E R

w e see i m m e d i a t e l y t h a t

I q l 1 . 1 La

Ga ( D s f ) ( x ) .

A c o r r e s p o n d i n g s i m p l e maximal i n e q u a l i t y ( v a l i d i n a l l d i m e n s i o n s

54 A. Carbery

and not reflecting the full '!Bochner-Riesz" characteristics of

where

a >

112

and

2n-2 n-20. < P

<

Ga)

2n

n+2a-1*

Proof of Theorem 4. We proceed as in the proof of Theorem 2. Let and apply the lemma: m e C:(O,m)

Finally, applying the Cauchy-Schwarz inequality, we obtain

T, f(x) 5

ca

ImI

L Za 161 ):,

~"(f)(x).

2 a > 0, 6 e R . Applications 1 . Let M ( E ) = (1 4 2 corresponding maximal operator is bounded on L (R ) . [l]

Then the

2. Let I$ be a Cm bump function supported on the annulus 16 e R2 : I 151 - 1 I < 6). Then for small 6 , the corresponding maximal operator T, satisfies IT: fl 5 cE ( 1 / 6 I E lfI4 for all anklysis (see the lecture of C62 E > 0. In fact, a slightly finer doba in these proceedings) gives a power of log(1/6) in place of O((l/6)E)

VE > 0 .

mB (51,

-n/2 < B 5 0, be a smooth multiplier vanishing n+ 1 T+ 8 near zero and of the form C B ei 151 for 161 2 1, and let TB f(6) = mB(c) I t s ) . Modulo a term belonging to S , the kernels of the multiplier operators TB are the characteristic function of the unit ball when f3 = 0 and the uniform surface measure on the unit sphere in Rn (n F 2) when B = - 1 respectively and s o we are considering the Hardy-Littlewood maximal function and Stein's spherical maximal function [14] in these special c a s e s . Combining remark 2 with the obvious L" estimate when B = - 1 3.

Let

yields that

IT* flLp(Rn)

'

I cg,n

max{~, *I < - < ~n n+B for 1 2+B o < - < 7 for - 1 < B 5 0).

P

when LP -n/2 < B < 0 . (When n = 2 , read When - 1 5 B 5 0 , it is known 1161

Radial Fourier m u l t i p l i e r s 55

t h a t t h e range o f p a r a m e t e r s n/(n+B) < p c a n n o t b e improved - con2n s e q u e n t l y G" c a n n o t b e bounded o n Lp(Rn) u n l e s s p > n+2a-1. L

Let m(6) = e i l s I . Then t h e l i n e a r i s e d S c h r a d i n g e r e q u a t i o n A u ( x , t ) = i au / a t ( x , t ) , t > 0 , u ( x , O ) = f ( x ) , has a s o l u t i o n o p e r a t o r u ( x , t ) = TJF. f ( x ) . C o n d i t i o n s o n f s u f f i c i e n t t o i m p l y a s t -+ 0 h a v e b e e n s t u d i e d b y C a r l e s o n , that u(x.t) + f(x) a.e. [ 4 ] , D a h l b e r g a n d K e n i g , [9] a n d Kenig a n d R u i z [ l l ] The r e s u l t s 2 n) , a o f t h e s e p a p e r s show t h a t i f f e L,(R n/4, then s u p I T t f ( x ) I i s l o c a l l y i n L 2 ( R n ) , ( [ l l ] ) . Here we show t h a t O 1 . sup ITt f ( x ) I O
.

ISt f ( x ) I 5 Ccr t s l q l when

a > 1/2

when

s > 2a.

1.1

by r e m a r k 2 , ( w h e r e

s

Ga(Dsf)(x) La g(6) = n ( E ) ; ( S ) ) .

Consequently

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[l]

A.

[2]

A. Carbery, G. Gasper and W. T r e b e l s , R a d i a l F o u r i e r M u l t i 2 p l i e r s o f L P ( R 1, p r e p r i n t .

[3]

A.

[4]

L . C a r l e s o n , Some a n a l y t i c p r o b l e m s r e l a t e d t o S t a t i s t i c a l Me-

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c h a n i c s , i n L e c t u r e N o t e s i n M a t h . , 779 ( 1 9 7 9 ) , 5 - 4 5 . [5]

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[6]

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( 1 9 7 9 ) , 505-51 1 .

56 A. Carbery

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73,

(1978),

-

Spherical means, P.N.A.S.

2174-5.

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E . M.

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