Rd~{O}. multiplier theorem

A n H 1 multiplier theorem

W. T. Sledd and D. A. Stegenga*

Introduction. Let H I ( R a) and H I ( T a) denote the usual H a r d y spaces on Euclidean space and the torus [18], [19, p. 283]. Given a function in H I ( R d) its Fourier transform is a continuous function on R d which vanishes at the origin.

Thus the transform may be integrable with respect to a measure which is singular at the origin.

We have two main results. One is a characterization of all such measures and the other is an application to r a n d o m Fourier series.

1. Main results. Denote by A the integer lattice in R d and Q~ the cube {xERd: eo~j--e/2<:xj<eo~j+e/2} where ~---(cq, ...,c~d)EA and e>O.

Theorem 1. Let # be a positive Borel measure ^{o n}

Rd~{O}.

Then

(1) s u p f Ifl @ <

where the supremum is taken over all f i n H I ( R d) o f norm 1 if and only if

(2) sup ( 2 / ~ (Q~)2)1/2 < ~ .

s

Moreover, the corresponding suprema are equivalent.

Corollary 1. Let {m~},ea be nonnegative numbers and define a measure on R e \ { 0 } by # = ~ # o m~6~ where 6~ is a point mass at x = e . Then

(3) sup ~ ' ~ , o lf(~)l m~ < ~

~>-0

where the supremum is taken over all f i n H * ( T d) of unit norm i f and only i f l~ satisfies condition (2).

Remarks. (a) F o r d = 1, Corollary 1 is an unpublished result o f C. Feffer- man, see [1]. It contains in particular the classical inequalities o f H a r d y [9] and Paley [13]. T h e o r e m 1 is a generalization of this result.

* The second author is supported by a grant from the National Science Foundation.

(b) Theorem 1 can easily be generalized by replacing the 1-norm condition in condition (1) by a p-norm for p ~ l . Corresponding to condition (2) is

(2") s u p { ~ o l t ( Q ~ ) 2 / 2 - p } 2-p/2p < ~ , 1 <= p < 2

6 > 0

sup #(5 <-

Ixl

~ 2~) 1/p < ~ , p => 2.

The case p = 2 is a result o f Stein and Zygmund [20].

The space o f functions o f bounded mean oscillation (BMO) introduced in [11]

are naturally involved in this problem by means of Fefferman's duality theorem [6], [7]. We restrict our attention to functions defined on the circle ( d = 1). The following corollary is a consequence of this duality and a result in [5, p. 105].

Corollary 2. Let {m,}~=_~ be a square summable sequence o f nonnegative num- bers. The function f with Fourier series ~ mne ~"~ is in BMO i f and only i f condition (2) is satisfied (for the corresponding measure IZ).

Theorem 2. There exists an IS-sequence {m,} with the property that ~ 2 , m , e i"~

is not in B M O f o r any sequence {2,} with

T2.1--~ for

all n,

Proof. Let E c Z + be a Sidon set which is not a lacunary set, see [10]. It fol- lows that there exists a sequence o f nonnegative numbers {m,}~l ~ supported on the set E which violates condition (2). Let f ~ m , e ~"~ be the corresponding func- tion in L 2.

By Corollary 2 f i s not in BMO. If {2,} is sequence with 12,[--1 then there is a measure p whose Fourier coefficients/~(n) agree with 2, on the set E [10]. Since BMO is closed under convolution with a measure it follows that ~ 2,m,e~"~

Remarks. (a) The motivation for the last theorem is the well known fact that the random L2-function is in L p for p < ~ o [21]. Since BMO is contained in all of the LP-spaces it is natural to extend this result. T h e o r e m 2 provides a strong counterexample to this conjecture. See also [16].

(b) The existence of the sequence

{ran}

could have been deduced from the converse to Paley's Theorem which was proved by Rudin [14], see also Stein and Zygmund [20].

(c) Another curiosity along the same lines is that a lacunary function (Hada- mard gaps) in BMO has some continuity properties, namely, a lacunary function in BMO is in the space of functions with vanishing mean oscillation (VMO) intro- duced by Sarason [15].

2. P r o o f of Theorem 1. An atom a(x) corresponding to a cube Q is a measur- able function supported on Q which has zero mean and is bounded by [01-1 (1" [=

An H ~ multiplier theorem 267 Lebesgue measure). By a result o f C o i f m a n [3] the sufficiency o f condition (2) will follow if there is a c < ~, with

(4)

for all atoms a.

Part of (4) is straightforward.

flald~<_- c

I f a is an a t o m corresponding to a cube of side length 6 then by a well-known estimate [d(y)l<=clyl6 for y in Q~ where e = 6 -1.

Here e is a dimensional constant independent o f a. N o w it is not hard to show that (2) implies

=-'f

Qg

^Ixl

dp(x) <= c (e > O) and hence (4) will follow f r o m

(4)'

where e is related to a as above.

S=<,\Q~ :i

ctu _-< c

This result is now easily seen to be a consequence of condition (2) and the following theorem.

Theorem 3. There is a constant c < co such that i f a(x) is an atom corresponding to a cube with side length 6 and e=fi -1 then

~ ' , sup [d[ 2 -<_ c.

QI

Proof. We only prove the result for d = 1. The general case involves an itera- tion technique which is somewhat more complicated to describe. In addition, it suffices to assume that a is s m o o t h and supported in the interval [ - 6 / 2 , 6/2].

Fix an interval I of length e a n d assume that f is continuously differentiable on I. It is elementary that sups [ f - b l < = f r ]f'l where b is the average [I]-~fIf.

Hence

Normalizing the Fourier transform so that I f l z = l f l 2 we obtain

Z. sup t< I 2 =2 [+ f - l<it=+= :'i =]

1 612 2 612

from which the theorem follows.

Remark. The results in [2], [12] concerning the behavior o f d follow f r o m T h e o r e m 3.

In order to prove that condition (2) is necessary we require s o m e examples of functions in HI(Rd). The following lemma is sufficient for our purposes.

L e m n m 4 . Let gEL~(R a) and assume that ~ = 0 on l y l ~ l . I f f=gf[B(o,x) then f E H 1 and }lflIa~=el]g]Jz. (Here Xs(o,1) is the characteristic function for the unit ball centered at the origin.)

Proof. Assume that ~ is a C=-function with compact support in IY] >1. Then f is the convolution ~ . XB(0,1) and hence is a rapidly decreasing function which vanishes in a neighborhood of the origin. Thus, f i s in H 1. If uEBMO (R a) and b is its average over B(0, 1) then by the Schwarz inequality

I flu I =

f I" l u - b ~ dx )1/2

<= c I g 2 ij.l ~-~Va-~--i

-<- r Jl glJ~ II

ull,,~,o.

The first inequality is a well-known estimate for ~B(0,,) and the second a slight extension of inequality (1.2) in [7]. By duality the proof is complete.

The proof of Theorem 1 will be complete once we establish the necessity of condition (2). However, if (1) holds with supremum A then from Lemma 4 we deduce that

(5) f [p (y + B (0, 1))3 ~ dy ~ cA 2.

It follows easily that there is an M<o% 6 > 0 for which ~l~l_~M p(Qa~)2<=cAZ where c is a dimensional constant. But then a dilation argument gives this inequality for all 6 > 0 and (2) now follows in an elementary way. Thus the proof of Theorem 1 is complete.

3. Proof of the Corollary. The space H01(T a) is th'. subspace of H I ( T ~) con- sisting of functions with zero mean. Given f~HX(R a) we define

P f ( x ) = ~'~e a f ( x + a ) .

Since f ~ U ( R d) we have Pf~LI(T d) and by the Poisson summation formula it follows that f ( e ) = ( P f ) ^ (c 0 for e~A. Here the Fourier coefficients for functions on T d are given for e~ A by

P(~) = f T~ F(x) e-~"'~'~ clx

where T d is identified with the d-fold product of the unit interval.

The proof of the corollary is an immediate consequence of the following theorem.

Theorem S.

P(H~(R~))

=H](T~).

An H ~ multiplier theorem 269

Proof.

Let q~ be a nonnegative rapidly decreasing function for which q3 has support contained in the open unit ball centered at the origin and ~3(0)= 1. Put

q~(x) =ed9 (eX)

for 0 < 5 < 1. F o r a polynomial

F(x) = ~ a~e 2€

let f~ = F . q),.

Then f ~ 6 H I ( R d) a n d f , ( ~ ) = a ~ for ccEA.

Claim.

lim,_ o

[I frtlHI(Ra)<=IfFIIHI(Ta).

We start with the easily derived fact that

(6) l i m f ~ 0 ,) R d

glq~I=fTdgdx

for all continuous functions g on

T a.

Observe that j]q),[ll=l.

Let

S], Rj

denote the j t h Riesz transforms on T a, R d. Then by (6) we obtain lira sup

[llLlla+ Z~

IIRjAII1] ~ ]l FIIH~(T,)

~ 0

+ lira sup Zax

li Rjf~- (Sj F) ~o~ll~

~ 0

so that we must show that the second term on the right is zero. Since F is a poly- nomial it suffices to fix aEA with ~ 0 , put

h~(y-~) = (yj/lyl-~jl~l)O,(y-=)

for some

l<=j<=d

and show that lim,.oll/~l]x=0.

Now ~, is supported in the ball of radius e centered at the origin. Thus, we may assume that

h,(y)=m(y)~b~(y)

where m is smooth, all derivatives up to order d + l are bounded by a dimensional constant, and

[m(y)]<=c]y].

The conditions on m imply that

IIDh,llx<=c

where

D =bd+I/~y~+~+... +3d+~/Oy~+~.

Hence I~,(x)l-~

clxl -(d§

Clearly, lira,_+0 []h~[Ix=0 so that lim~_~0

I1~11=--0

and thus the above estimate implies that lim~_~0 lIh~lIa=o. This proves the claim.

To complete the p r o o f we fix F in Ho~(T d) and note that there are polynomials

Fo~H2OJ)

with Z

I1F~llz~<~

and F = ~ F,. Using the above we

findfo~H~(R d)

with

Z IIf~ll~,X(R~)<oo

and

Pf,=F,.

Thus,

Pf=F

where

f = ~ f ,

is a function in H 1 (Rd).

Remarks.

(a) Theorem 5 is an extension o f a result o f deLeeuw [4], see Gold- berg [8] for a similar result.

(b) A sharpening of the lemma is that given

FCH2(T d)

and ~>0, there exist

f~H~(R d)

with

Pf=F

and

IIFIIn~(T~) =< I/fllr*~(R~) => (1 +~)IIFIIH~(T~).

This is best possible since ItPf[I/-/I(T,)<I{ f I I , t ~ , ) in general.

(c) A similar argument to the above shows that

P(L~(Rd))=L~o(Td).

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Received October 11, 1979 Michigan State University

E. Lansing, Michigan 48 823 Indiana University

Bloomington, Indiana 47405