ZHANG Chuan-zhou， JIAO Fan， LI Tian-tian

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

(2.Hubei Province Key Laboratory of Systems Science in Metallurgical Process，Wuhan University of Science and Technology， Wuhan 430081， China)

Abstract: In this paper， we study the two-parameter B-valued strong martingale on weak Orlicz space，with emphasis on the strong atomic decomposition theorem of two-parameter B-valued strong martingale space w ， by using atomic decomposition theorem， the sufficient conditions for boundedness of sublinear operator ‖Tf‖wLΦ ≤C‖f‖ is given. The results as above generalize the conclusion of weak Lp martingale space.

Keywords: atomic decompositions; weak Orlicz spaces; strong martingale; two-parameter B-valued martingale

1 Introduction

In the paper，we will discuss the atomic decomposition for two-parameter B-valued weak Orlicz strong martingale space.

2 Preliminaries and Notations

Moreover， for the two-parameter B-valued strong martingale f = (fn，n ∈N2)， we introduce a conditional p-mean square function about F-n:

We define the weak Orlicz spaces as follows:

Lemma 2.2([6]) Let B be a Banach space， 1 ＜p ≤2， 0 ＜α ≤p， then the following statements are equivalent :

3 Atomic Decomposition

Theorem 3.1 Let B be a Banach space， 1 ＜p ≤2， Φ satisfies the condition of Δ2. If B is p smooth， 0 ＜α ≤p， then for any two-parameter B-valued strong martingale f = (fn，n ∈N2) ∈w， there is a strong (α，p) atom (g(k)，k ∈Z) and a column of non-negative real number μ=μk，k ∈Z ∈lα， so that for all n ∈N2:

where C is only related to p and α.

Because B has RN property， there exists g(k)∈Lpsuch that Eng(k)= g(k)n. From the definition of strong atom we know that g(k)is a strong (α，p) atom， which proved (3.1).

For any k ∈Z， we have

where C is a constant independent of f. Thus Theorem 3.1 is proved.

Theorem 3.2 Let B be a Banach space， 1 ＜p ≤min{qΦ，2}， Φ satisfies the condition of Δ2， the statements are equivalent

(1) B is p-uniform smooth;

Set m →∞，n →∞and k0→∞， therefore (fn，n ∈N2) is LαCauchy convergence.Thus fnconverges according in probability. From Lemma 2.1， B is p-smooth.

Theorem 3.2 is proved.

4 Boundedness of Sublinear Operators

Suppose T : X →Y is the mapping， where X is the weak Orlicz strong martingale space on(Ω，F(xiàn)n，P)， and Y is the space of measurable function on(Ω，F(xiàn)，P). T is sub-linear，if

Theorem 4.1 Let T :Lp→Lpbe a bounded sublinear operator， B be isomorphic to p-smooth Banach space which satisfies for any (α，p) atom a，

The basic thing is Φ(at)≤aΦ(t)， ?t ＞0， 0 ＜a ≤1， which derive from the convexity of Φ. For g =(gn，n ∈N2)， we know ‖g*‖p≤C‖(p)(g)‖p(1 ≤p ≤2) from Lemma 2.2. Since T :Lp→Lpis a bounded sublinear operator， then