A Littlewood Paley inequality for analytic measures
Louis Pigno and Brent Smith*
Let T be the circle group and Z the additive group of integers; designate by M ( T ) the customary space o f Borel measures on T, and, given p E M f T ) and nE Z, let
fl (n) = f T e- inO d# (0).
A measure p E M ( T ) is said to be o f analytic type if /~(n)=0 for all n < 0 ; as usual, H i ( T ) will denote the classical space o f all measures o f analytic type on T.
In 1933, Paley [4] published this remarkable inequality:
Theorem P. There is a C > 0 such that if (nk)oCZ + and nk+l/nk~2 for all k then
{ Z ~ 0 I~(nk)l'} ~/~ <- c I1~tl provided #E H i ( T ) .
F o r generalizations of Paley's Theorem see the work of J. Fournier [3] and the references cited therein.
In this paper we shall prove yet another generalization o f Paley's inequality;
before we describe our result we shall require some notation concerning quotient norms for M ( T ) :
For ~0EM(T) and E c Z put
IIo~11~ = inf{llv[l: 9 = cb on E};
here I1"
II
is the usual total variation norm on M(T). We say ( D , ) o is a sequence of positive dyadic intervals in Z if there exists a sequence (nk)o C Z +, nk+ Jnk---->2 for all k and Dk=[n~k, nZk+~). I f the sequence o f positive dyadic intervals (Dk~o* The work of the authors was partially supported by NSF grants MCS-8102061 and MCS-8102547.
272 Louis Pigno and Brent Smith
satisfies
nk+X/nk<--2
for some ;t and all k we say(Dk)o
is astandard sequence of pos- itive dyadic intervals.
Our generalization of Theorem P is the following Littlewood--Paley type inequality for the quotient norms of an analytic measure:
Theorem P'.
There is
adyadic intervals
provided
#CHI(T).C > 0
such that for any sequence (Dn) o of positive
The proof o f Theorem P' uses a variant on the construction of Cohen--Daven- port [1, 2] as well as some ideas in [3] and [5]. Before beginning the proof o f Theo- rem P' we shall need the following lemma.
Lennna.
Let
0_<-a=<l,t, zEC and
[t[=<l, [zl~_l.Then
a t( 1 - - ~ J z - - - ~ I
10 + -<_ 1.
~0 [ a ~ ' ~ a l
Proof.
Consider the function F ( z ) =~,+ 1 - - - ~ - - ~ z ;
since O_-_a<-I it is easy to check that [F(z)[~l for [zl~l and [tl_~l.Assume for the moment [z]= 1 ; as a consequence o f IF(z)[ ~- 1 we gather that
-fif + 1 - z - - f - f f z ~ ~= 1.
Our result now follows from the maximum modulus principle for analytic functions.
We turn to the proof of Theorem P': Let p ~ H I ( T ) and let (Dk)~" be any sequence of positive dyadic intervals. Suppose (a~)~" satisfies
(1) z~olan[ ~ _~ 1.
Let (t~)o be any sequence o f trigonometric polynomials on T such that supp fk c - Dk and [Itk[Ioo <- 1 for all k. We also arrange for f r
tk(O)dla(O)--#(t~) >---0
for all k. PutFo : 1 laol to and define inductively for n = 1, 2 . . .
As a consequence o f inequality (1), [a~[~l, so we may infer from the Lemma
(2) l[Fnl[~ <- 1 for all n.
A Littlewood--Paley inequality for analytic measures 273 Well, on the one hand,
I frF.(O)d~(O) I <-II~ll,
(3)
because of inequality (2), while on the other hand,
^ a 2
+ 1-- 1 5 " ta"-21/~("-2)+"" ,
+ { 1 - - ~ } (1 la"-1t2}5 '" ( 1 - J ~ } la~176 because the sequence (Dk)o is dyadic and pEHI(T).
As a consequence of inequalities (3) and (4), we obtain (5) 10[Ip[, _->-{/fo ( 1 - ~ } } { ~ ' o lakIft(tk)}
since I~(tk)>:O for all k. It now follows from inequalities (1) and (5) that
{~o II~Jl~,ff/~ --< CIIMII
for some universal constant C>0. Our proof is complete.
Corollary. (Meyer [7].) Let ~D;) o be a sequence of standard symmetric dyadic intervals, i.e., D~=[n2k, n2k+0U(--nzk+l,--n~k] where nk+llnk>--2 for all k and nk+llnk<=2 for some 2. Then there exists a C(2)>0 such that
{~o
I[~ll~*} 1/2 <--c(;.)II~ll
provided #E H I(T).
Comments. (a) Theorem P" tells us that the quotient norms of an analytic measure vanish very quickly at "+co"; cf. [5].
(b) Versions of Theorem P" hold for all compact abelian groups with ordered duals.
(c) For a different approach to Littlewood--Paley inequalities which generalize Theorem P see [6].
(d) The method of proof of Theorem P" can easily be adapted to give this generaliza- tion of Paley's inequality: Suppose q~ is a multiplicative linear functional with representing measure m; by Hl(dm) we mean the closure in L~(dm) of A and by H o the weak-*closure of A 0 in L ~ (dm).
274 Louis Pigno and Brcnt Smith: A Littlewood--Paley inequality for analytic measures
There is a universal constant C > 0 such that if {#k}o is any sequence of uni- modular elements of H~(dm) such that (~k_l)21~kEHo for all k then, for any hE Hl(dm),
{ Z o I f hflkdm 2}1/2<-- - Cllhlll.
Acknowledgement
T h e a u t h o r s wish to t h a n k J o h n F o u r n i e r for h e l p f u l c o r r e s p o n d e n c e .
References
1. COHEN, P., On a conjecture of Littlewood and idempotent measures, Amer. J. Math. 82 (1960), 191--212.
2. DAVENPORT, H., On a theorem of P. J. Cohen, Mathematika 7 (1960), 93--97.
3. FOURN~ER, J. J. F., On a theorem of Paley and the Littlewood conjecture, Arkiv frr Mate- matik 17 (1979), 199--216.
4. PALEr, R. E. A. C., On the lacunary coefficients of power series, Ann. of Math. 34 (1933), 615--616.
5. PIGNO, L. and SMITH, B., Quantitative behaviour of the norms of an analytic measure Proc.
Amer. Math. Soc. (1982), in press.
6. STEIN, E. M., Classes H p, multiplicateurs et fonctions de Littlewood--Paley, C. R. Acad.
Sci. Paris (Set. A) 263 (1966), 780--781.
7. M~YEg, Y., Compl~ment ~ un th~or~me de Paley. C. R. Acad. Sci. Paris (Ser. A) 262 (1966), 281---282.
Received August 17, 1981. Louis Pigno
Kansas State University Manhattan
KANSAS 66506 USA
Brent Smith
California Institute of Technology
Passadena CALIFORNIA USA
