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volume 3, issue 2, article 19, 2002.

Received 10 April, 2001;

accepted 16 November, 2001.

Communicated by:L. Losonczi

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Journal of Inequalities in Pure and Applied Mathematics

LITTLEWOOD’S INEQUALITY FOR p−BIMEASURES

NASSER M. TOWGHI

Raytheon System Company

528 Boston Post Road Mail Stop 2-142, Sudbury, MA 01776.

EMail:[email protected]

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2000Victoria University ISSN (electronic): 1443-5756 031-01

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Littlewood’s Inequality for p−Bimeasures

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J. Ineq. Pure and Appl. Math. 3(2) Art. 19, 2002

http://jipam.vu.edu.au

Abstract

In this paper we extend an inequality of Littlewood concerning the higher variations of functions of bounded Fréchet variations of two variables (bimeasures) to a class of functions that arep-bimeasures, by using the machinery of vector measures. Using random estimates of Kahane-Salem-Zygmund, we show that the inequality is sharp.

2000 Mathematics Subject Classification: Primary 26B15, 26A42, Secondary 28A35, 28A25.

Key words: Inequalities, Bimeasures, Fréchet variation,p-variations, Bounded variations.

1. Introduction

Let µ be a set function defined on the product σ(B₁)× σ(B₂) of 2 σ-fields, such that it is a finite complex measure in each coordinate. More precisely, for each fixedA ∈σ(B₁)the set functionµ(A,·)is a complex measure defined on σ(B₂). Similarly for eachB ∈σ(B₂), the set functionµgives rise to a measure in the first coordinate. Such set functions dubbed bimeasures by Morse and Transue were studied extensively by these and other authors (see [1, 2,3,5, 6, 7,10,11,12]). It is well known that such set functions need not be extendible to a measure on theσ−Algebra generated byσ(B₁)×σ(B₂). Now suppose thatµ is a set function defined onσ(B₁)×σ(B₂), such that it has finite semi-variation;

that is,

(1.1) kµk_F = sup







X

j,k

µ(A_j×B_k)r_j⊗r_k ∞







<∞,

where sup is taken over all measurable partitions {A_j}, {B_k} of Ω₁ and Ω₂, respectively. Here{r_j}is the usual system of Rademachers, realized as functions on the interval [0,1]. By a partition of Ω, we mean a finite collection of mutually disjoint measurable sets whose union isΩ. F in|| · ||_F is for Fréchet.

It is clear that a set function µ with finite semi-variation is also a bimeasure.

It is interesting that the converse also holds. That is, a bimeasure has finite semi-variation. This follows easily from the machinery of vector measure theory. On the other hand, it is well known that a set function which has finite semi-variation need not have finite total variation (in the sense of Vitali), hence it may not be extendible to a measure [2,9]. However, all is not lost, in his 1930

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paper, Littlewood showed that a bimeasure has finite 4/3-variation. To make this precise we first introduce the notion of mixed variation ofµ. Letp, q >0, and define the mixed(p, q)-variation ofµto be

(1.2) kµk_p,q= sup









 X

k

X

j

|µ(A_j×B_k)|^p

!^q_p



1 q





 ,

where the sup is taken over all finite measurable partitions {A_j} and{B_k}of Ω1 andΩ2 respectively. In the case thatp = q, we simply write kµk_p, that is kµk_p =kµk_p,p. We now state Littlewood’s4/3inequalities.

1.1. Littlewood’s Inequalities

(1.3) kµk_2,1+kµk_1,2 +kµk_4/3 ≤ckµk_F ,

wherecis a fixed universal constant. The result is sharp in the sense that, there exists µ ∈ such that kµk_p and kµk_q,1/q are infinite for all p < 4/3and for all q <2. Extension of Littlewood’s inequality to a larger class of functions of two variables is the main result of this paper.

Definition 1.1. A set function µdefined on product of two algebrasB₁×B₂ is called a pre-p-bimeasure, if it is finitely additive in each coordinate, and for each fixedA∈B1, the quantity

BV_p(µ(A,·)) := sup (

X

k

|µ(A×B_k|^p )

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is finite, and for each fixed B ∈B₂, BV_p(µ(·, B)) is finite. Here sup is taken over all finite measurable partitions ofΩ2.

If the set function is defined on the product of two σ-algebras with above properties, then it is called ap-bimeasure.

Definition 1.2. A pre-p-bimeasureµdefined on product of two algebrasB₁×B₂, is said to be bounded, if there exists a positive constantM such thatBVp(µ(A,·))+

BV_p(µ(·, B))≤M, for allA∈B₁ and for allB ∈B₂. We prove the following result.

Theorem 1.1. Suppose that eitherµis ap-bimeasure defined onσ(B₁)×σ(B₂), or that it is a bounded pre-p-bimeasure defined onB₁×B₂. If1≤p≤2then

(1.4) kµk_2,p+kµk_p,2+kµk ^4p

2+p <∞.

In the case thatp≥2, then

(1.5) kµk_p <∞.

Furthermore, the result is sharp, in the sense that, there exists a p-bimeasure such thatkµk_q =∞, for allq < _2+p^4p .

To prove Theorem1.1 we collect some definitions and results about vector measures. Much of the following can be found in Chapter 1 of [4].

Definition 1.3. A functionµfrom a fieldBof a setΩto a Banach space is called a finitely additive vector measure, or simply a vector measure, if wheneverA1

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and A₂ are disjoint members ofB then µ(A₁S

A₂) = µ(A₁) +µ(A₂). The variation of a vector measureµis the extended nonnegative function|µ|whose value on the setEis given by

|µ|(A) = sup

π

X

A∈π

||µ(A)||,

where the sup is taken over all partitionsπofAinto a finite number of disjoint members of B. If |µ|(Ω) is finite, thenµwill be called a measure of bounded variation.

A different type of variation related to a vector measureµ is the so called semi-variation of µ. More precisely, the semi-variation ofµ is the extended nonnegative functionkµk_F whose value on a measurable setAis given by

kµk_F (A) = sup{|x^∗(µ)|(A) :x^∗ ∈X^∗,kx^∗k ≤1},

where|x^∗(µ)|is the variation of the real-valued measure (finitely additive measure) x^∗(µ). Ifkµk_F(Ω) is finite, then µwill be called a measure of bounded semi-variation.

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2. Proof of Theorem 1.1

We now prove Theorem 1.1. Suppose that1 ≤ p < 2. LetX₁ be the space of finitely additive set functions defined onσ(B1), which have finitep-variations.

Similarly let X₂ be the set finitely additive functions defined on σ(B₂) which have finitep-variations. It can be shown that equipped with p-variation norm, X₁ and X₂ are Banach spaces. Let L be the X₁-valued function defined on σ(B₂)as follows: L(A) = µ(·, A), whereA ∈ σ(B₂). LetR be theX₂-valued function defined onσ(B₁)as follows: R(A) =µ(A,·), whereA ∈σ(B₁). Ifµ is ap-bimeasure then by the Nikodym Boundedness Theorem (see [4, Theorem 1, page 14]), L and R have finite semi-variations. If µ is a bounded pre-p- bimeasure then by general properties of vector measures (see e.g., [4, Propo- sition 11, page 4]), L and R have finite semi-variations. Let {A_n}be a finite measurable partition ofΩ₂and{B_k}be a finite measurable partition ofΩ₁, then

∞>||L||_F(Ω₂) (2.1)

≥

BV_p X

n

r_nµ(A_n,·)

! _∞

≥

X

k

X

n

r_nµ(A_n, B_k)

p!¹_p ∞

≥ Z 1

0

X

k

X

n

r_n(x)µ(A_n, B_k)

p

dx

!¹_p

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(Khinchin’s inequality) ⇒≥c



 X

k

X

n

|µ(An, Bk)|²

!^p₂



1 p

.

Similarly,

(2.2) ∞>||R||F(Ω1)≥c



 X

n

X

k

|µ(An, Bk)|²

!^p₂



1 p

.

(2.2) and (2.3) imply that, kµk_2,p is finite. Applying Minkowski’s inequality we obtain kµk_p,2 ≤ kµk_2,p < ∞. We now show that kµk ^4p

2+p is finite. Let an,k = µ(An, Bk). Applying Hölder’s inequality with exponents ^2+p_p and ^2+p₂ , we obtain

X

n,k

|a_n,k|^2+p^4p =X

n,k

|a_n,k|^2+p^2p |a_n,k|^2+p^2p (2.3)

≤X

n

"

X

k

|a_n,k|²

#_2+p^p "

X

k

|a_n,k|^p

#_p+2²

≤

"

X

n

(X

k

|a_n,k|²)^p²

#_2+p² 

 X

n

X

k

|a_n,k|^p

!²_p



p 2+p

≤

kµk_2,pkµk_p,2_p+2^2p

<∞.

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This proves inequality (1.5). Ifp≥2thenp/2≥1, consequently

||R||_F(Ω₁)≥c



 X

n

X

k

|µ(A_n, B_k)|²

!^p₂



1 p

(2.4)

≥c X

k

X

n

|µ(An, Bk)|^p

!!¹_p .

Similarly

||L||_F(Ω₂)≥c X

k

X

n

|µ(A_n, B_k)|^p

!!¹_p . This proves inequality (2.1).

We now show that the exponent _p+2^4p is sharp. We only consider the case 1 < p < 2. Sharpness of Theorem 1.1 for the case p = 1 is known [9].

Sharpness of Theorem1.1forp≥2is trivial.

We need the following result, which is a consequence of Kahane-Salem- Zygmund estimates (see [8, Theorem 3, p. 70]).

Lemma 2.1. Let X_n₁_,n₂_,...n_s be a subnormal collection of independent random variables. Given complex numbersc_n₁_,n₂_,...,n_s, where the multi-index(n₁, n₂, ..., n_s) satisfies|n₁|+|n₂|+· · ·+|n_s| ≤N, then

(2.5) Pr

sup

t1,...,ts

XX_n₁_,n₂_,...n_sc_n₁_,n₂_,...,n_seⁱ⁽ⁿ¹^t¹^+···n^s^t^s⁾

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≥C h

sX

|c_n₁_,n₂_,...,n_s|²logN i¹₂

≤N⁻²e^−s,

whereCis an independent constant.

To apply Lemma2.1, we will need to construct an appropriate sequence of independent subnormal random variables. We will construct a Radamacher type of system, which we will call the 4-level Radamacher system.

2.1. 4-level Radamacher System

4-level Radamacher system is the sequence of independent random variables, {w_j(x)}^∞_j=1, defined on the unit interval [0,1], such that each w_j takes on 4 discrete values{2,−2,1,−1}, each with probability ¹₄. Such a system can be constructed similar to the usual Radamacher system. Observe that, M 4-level Radamacher system generate 4^M distinct vectors of length M. On the other hand the set{1,2, ..., M}has2^M distinct subsets.

By Lemma 2.1, for j, k = 1, ..., M, there exists a vector~t = (t₁, t₂) and choice of scalers{b_jk}^M_j,k=1 (approximately as many as 1− _M¹2

4^M² −2^M²), such thatb_jk ∈ {2,−2,1,−1}, and for any subsetAof{1,2, ..., M},

(2.6)

X

j∈A

b_jke^i(kt¹^+jt²⁾

≤C[4Mlog(2M)]¹²,

and (2.7)

X

k∈A

b_jke^i(kt¹^+jt²⁾

≤C[4Mlog(2M)]¹².

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Let

(2.8) (a) ={a_jk}_j,k =

b_jke^i(jt¹^+kt²⁾ ^M

j,k=1. LetA, B ⊂ {1,2, ..., M}and define

(2.9) a(A, B) =X

j∈A

X

k∈B

a_jk,

then by virtue of inequalities (2.7) and (2.8), (2.10) ||a||F ≤CpM¹²⁺¹^pp

log(2M).

On the other hand for anyr >0,

(2.11) ||a||_r =

" _M X

j=1 M

X

k=1

|a_jk|^r

#¹_r

≥M²^r.

We see that ifr < _p+2^4p ,

(2.12) lim

M→∞

||a||_r

||a||_F =∞.

This shows that _p+2^4p is sharp. The proof for the case that µ is a bounded-pre- bimeasure is similar.

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3. Functions of Bounded p-Variations and Related Function Spaces

Letp≥1andf be a function defined on[0,1]². Let V_p⁽²⁾(f,[0,1]²) = sup

π1,π2

X

i,j

|∆^π_i,j¹^,π²f|^p

!1/p

.

Hereπ₁ ={0 =x₀ < x₁, <· · · < x_m = 1}, andπ₂ ={0 =y₀ < y₁, <· · · <

y_n= 1}, are partitions of[0,1]and

∆^π_i,j¹^,π²(f) = f(x_i, y_j)−f(x_i, yj−1)−f(xi−1, y_j) +f(xi−1, yj−1).

LetWp⁽²⁾([0,1]²) =Wp⁽²⁾ denote the class of functionsf on[0,1]² such that,

||f||_W_p² =V_p⁽²⁾(f,[0,1]²) +V_p⁽²⁾(f(0,·),[0,1]) +V_p⁽¹⁾(f(·,0),[0,1]) +|f(0,0,0)|

<∞.

Let~x= (x₁, x₂), ~y= (y₁, y₂)∈[0,1]², andf be a function defined on[0,1]². Let

f_~_y(~x) = f(x₁, x₂)−f(x₁, y₂)−f(y₁, x₂) +f(y₁, y₂).

We say thatf is a Lipschitz function of orderαof first type, if there exists a constantCsuch that for all~xand~yin[0,1]²,

(3.1) |f(~x)−f(~y)| ≤C||~x−~y||^α₂.

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Here|| · ||₂refers to the usuall₂-norm. The class of Lipschitz functions of order αof first type is denoted byΛ¹_α(2). We say thatfis a Lipschitz function of order αof second type, if there exists a constantCsuch that for all~xand~yin[0,1]², (3.2) |f_~_y(~x)| ≤C||~x−~y||^α₂.

The class of Lipschitz functions of orderαof second type is denoted byΛ²_α(2).

Iff ∈Λ¹_α(2)then

|f_~_y(~x)| ≤4Cmin{|x_j −y_j|^α: 1≤j ≤2} ≤C₂||~x−~y||^α₂}.

Therefore,Λ¹_α(2) ⊂Λ²_α(2). Using Theorem1.1we obtain

Corollary 3.1. Let f be a function defined on [0,1]². Suppose that for any 1≤j ≤nand for any fixed partitionsπ1 andπ2of the interval[0,1], we have

(3.3) sup

π

"

X

i,j

|∆^π_i,j¹^,πf|^p

#1/p

+ sup

π

"

X

i,j

|∆^π,π_i,j²f|^p

#1/p

≤M <∞,

thenf ∈W⁽²⁾4p 2+p

.

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References

[1] R.C. BLEI, Fractional dimensions and bounded fractional forms, Mem.

Amer. Math. Soc., 57 (1985), No. 331.

[2] R.C. BLEI, Multilinear measure theory and the Grothendieck factorization theorem, Proc. London Math. Soc., 56(3) (1988), 529–546.

[3] R.C. BLEI, An Extension theorem concerning Fréchet measures, Can.

Math. Bull., 38 (1995), 278–285.

[4] J. DIESTELANDJ.J. UHL, Jr., Vector Measures, Math Surveys 15, Amer.

Math. Soc., Providence, (1977).

[5] M. FRÉCHET, Sur les fonctionnelles bilinéaires, Trans. Amer. Math. Soc., 16 (1915), 215–234.

[6] A. GROTHENDIECK, Résumé de la théorie métirque des produits ten- soriels topologiques, Bol. Soc. Matem. São Paulo, 8 (1956), 1–79.

[7] C.C. GRAHAM AND B.M. SCHREIBER, Bimeasure algebras on LCA groups, Pacific Journal of Math., 115 (1984), 91–127.

[8] J.P. KAHANE, Some Random Series of Functions, 2nd edition, Cambridge University Press, Heath (1986).

[9] J.E. LITTLEWOOD, On bounded bilinear forms in an infinite number of variables, Quart. J. Math. Oxford, 1 (1930), 164–174.

[10] M. MORSE AND W. TRANSUE, Functionals of bounded Fréchet varia- tion, Canad. J. Math., 1 (1949), 153–165.

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[11] M. MORSEANDW. TRANSUE, Integral representations of bilinear func- tionals, Proc. Nat. Acad. Sci., 35 (1949), 136–143.

[12] M. MORSE AND W. TRANSUE,C-bimeasuresΛ and their integral ex- tensions, Ann. Math., 64 (1956), 89–95.