ZHANG Chuan-zhou, WANG Jiu-feng, ZHANG Xue-ying

(College of Science, Wuhan University of Science and Technology, Wuhan 430065, China)

Abstract: In this paper, we study the atomic decompositions of weak and strong dyadic martingale spaces with variable exponents.By atomic decompositions, we prove that sublinear operator T is bounded fromto wLp(·); Cesro operator is bounded from Hp(·) to Lp(·) and from Lp(·) to Lp(·), which generalize the boundedness of operators in constant exponent case.

Keywords: atomic decompositions; variable exponents; Cesro operator

1 Introduction

It’s well known that variable exponent Lebesgue spaces have been got more and more attention in modern analysis and functional space theory.In particular, Fan and Zhao [1]investigated various properties of variable exponent Lebesgue spaces and Sobolev spaces.Diening [2,3]and Cruz-Uribe [4]proved the boundedness of Hardy-Littlewood maximal operator on variable exponent Lebesgue function spaces Lp(·)(Rn) under the conditions that the exponent p(·) satisfies so called log-Hlder continuity and decay restriction.Many other authors studied its applications to harmonic analysis and some other subjects.

The situation of martingale spaces is different from function spaces.For example, the good-λ inequality method used in classical martingale theory can not be used in variable exponent case.However, recently, variable exponent martingale spaces have been paid more attention too.Aoyama [5]proved that, if p(·) is F0-measurable, then there exists a positive constant c such thatNakai and Sadasue[6]pointed out that the inverse is not true,namely,there exists a variable exponent p(·)such that p(·)is not F0-measurable, and the above inequality holds, under the assumption that every σ-algebra Fnis generated by countable atoms.Zhiwei Hao [7]established an atomic decomposition of a predictable martingale Hardy space with variable exponents defined on probability spaces.Motivated by them, we research dyadic martingale Hardy space with variable exponents.

2 Preliminaries and Notations

In this paper the unit interval [0,1) and Lebesgue measure P are to be considered.Throughout this paper, Z, N denote the integer set and nonnegative integer set.By a dyadic interval we mean one of the form [k2?n,(k+1)2?n) for some k ∈ N,0 ≤ k < 2n.Given n ∈ N and x ∈ [0,1),let In(x)denote the dyadic interval of length 2?nwhich contains x.The σ-algebra generated by the dyadic intervals {In(x) : x ∈ [0,1)} will be denoted by Fn, more precisely,

Obviously, (Fn) is regular.Define F = σ(∪nFn) and denote the set of dyadic intervals by A(Fn) and write A = ∪nA(Fn).The conditional expectation operators relative to Fnare denoted by En. For a complex valued martingale f =(fn)n≥0, denote dfi=fi? fi?1(with convention df?1=0) and

Remark 2.1(see [8]) If (Fn) is regular, then for all nonnegative adapted processesthere exist a constant c>0 and a stopping time τλsuch that

Let p(·) : [0,1) → (0,∞) be an F-measurable function, we define= ess inf{p(x) :x ∈ B},= ess sup{p(x) : x ∈ B}, B ? [0,1). We use the abbreviationsand

when d(x,y)≤1/2.

The Lebesgue space with variable exponent p(·) denoted by Lp(·)is defined as the set of all F-measurable functions f satisfying

If f =(fn) is a martingale, we define

Remark 2.2(see [7]) If p is log-Hlder continuous, then we have

The weak Lebesgue space with variable exponent p(·) denoted by wLp(·)is defined as the set of all F-measurable functions f satisfying

Then we define the strong and weak variable exponent dyadic martingale Hardy spaces as follows

We always denote by c some positive constant, but its value may be different in each appearance.

3 Weak Atomic Decompositions

Definition 3.1A measurable a is called a weak- atom if there exists a stopping time ν such that

(1) En(a)=0,n ≥ ν,

Theorem 3.2Suppose that p is log-Hlder continuous and 0

where the infimum is taken over all preceding decompositions of f.

ProofAssume thatLet us define the stopping times νk:=inf{n ∈N:sn+1(f)>2k}. Consequently, fncan be written as

Thus

and p(·)/p?≥ 1 implies that s(f)≤ s(g)+s(h) and

Since s(ak)≤3·2k, thus we get

and so {s(g)>3y}? {s(g)>3·2j}= ?.

which implies

Thus we complete the proof.

Theorem 3.3Suppose that p is log-Hlder continuous, 1/2 < p?≤ p+≤ 1 and suppose that sublinear T is bounded from L2to L2. If

for all weak atom a supported on the interval I, then

ProofWe may supposeTaking the atomic decomposition and the martingales g and h given in the proof of Theorem 3.2 we get that for any given y >0,

(I) If 0

Define B1={Tg >y}, thus we have

(II) If y >1. Thus

We also have

Combining (3.15) and (3.17) we get

On the other hand, let B2={Tak>0}, Ikis support of ak, we have

which implies

By (3.18) and (3.19), we have

Thus we complete the proof of Theorem 3.3.

4 Boundedness of Cesro Operator

First we introduce the Walsh system.Every point x ∈[0,1) can be written in the following way

In case there are two different forms, we choose the one for which

For x,y ∈ [0,1) we definex ⊕ y =which is also called dyadic distance.

The product system generated by these functions is the Walsh system: ωn(x) :=

If f ∈L1[0,1), then the numberis said to be the n-th Walsh-Fourier coefficient of f.

Denote by snf the n-th partial sum of the Walsh-Fourier series of a martingale f,namely,

Recall that the Walsh-Dirichlet kernelssatisfy

Moreover,for any measurable function f,the sequence{f ?D2n=s2nf =fn}is a martingale sequence.

where x ∈ [0,1),n,N ∈ N,2N?1≤ n<2N(see [9]).

Moreover,

For n ∈N and a martingale f, the Cesro mean of order n of the Walsh-Fourier series of f is given by

It is simple to show that in case f ∈L1[0,1) we have

Definition 4.1A pair (a,B) of measurable function a and B ∈ A(Fn) is called a p(·)-atom if (1) En(a)=0, (2)

Lemma 4.2(see [10]) Suppose that p is log-Hlder continuous and 0 < p?< p+≤ 1.For any f =(fn) ∈ Hp(·), there exist (aB,B)B∈Aof p(·)-atoms and (uB)B∈Aof nonnegative real numbers such that

Lemma 4.3(see[10]) Suppose that the operator T is sublinear and for each p00 such that

for every p(·)-atom (a,B).If T is bounded from L∞to L∞, then

Theorem 4.4Suppose that p is log–Hlder continuous and 1/2

ProofBy Lemma 4.3, Theorem 4.4 will be complete if we show that the operator σ?satisfies (4.3) and is bounded from L∞to L∞.

Moreover for 2N?1≤ n<2Nand n>2τ(which implies N ? 1 ≥ τ),

Since x ∈[0,1)B, we have

By the definition of an atom, for n ≥ 2τ,2N?1≤ n<2N, we have

To verify(4.3)we have to investigate the integral of(supn|a|?|Kn|(x))p(·)over[0,1)B.Integrating over [0,1)B, we obtain

By Lemma 4.3 , the proof is completed.

Theorem 4.5Suppose that p is log-Hlder continuous and 1

ProofWe assumeIf else, we let f replaced bySince

where x ∈ [0,1),n,N ∈ N,2N?1≤ n<2N.Thus we have

Thus by Doob’s inequality of variable exponents martingale spaces , we have

Thus we complete the proof.