Atomic decompositions of weak Orlicz-Lorentz martingale spaces

Chuanzhou Zhang,Mingming Zhong,Xueying Zhang

doi:10.1186/1029-242x-2014-66

Chuanzhou Zhang, Mingming Zhong + Show 1 more

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https://doi.org/10.1186/1029-242x-2014-66

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Abstract

In this paper we establish atomic decompositions of some weak Orlicz-Lorentz martingale spaces which are generalization of Orlicz martingale spaces and of Lorentz martingale spaces. With the help of atomic decompositions, the boundedness of sublinear operators is obtained. MSC:60G46, 47A30.

Highlights

Introduction and preliminariesThe idea of atomic decomposition in martingale theory is derived from harmonic analysis [ ]
Just as it does in harmonic analysis, the method is a key ingredient in dealing with many problems including martingale inequalities, duality, interpolation, and so on, especially for small-index martingale and multi-parameter martingales
Weisz [ ] gave some atomic decompositions on martingale Hardy spaces and proved many important theorems by atomic decompositions; Weisz [ ] made a further study of atomic decompositions for weak Hardy spaces consisting of Vilenkin martingales, and he proved a weak version of the Hardy-Littlewood inequality; Liu and Hou [ ] investigated the atomic decompositions for vector-valued martingales and some geometry properties of Banach spaces were characterized; Hou and Ren [ ] considered the vector-valued weak atomic decompositions and weak martingale inequalities

Summary

Introduction and preliminaries

The idea of atomic decomposition in martingale theory is derived from harmonic analysis [ ]. We define the Orlicz-Lorentz space LF,G as the set of all measurable f ’s on for which the Orlicz-Lorentz functional f F,G = f ∗ ◦ F ◦ G – G = inf λ : G f ∗(F (G – (t))) λ dt ≤. Denote by the collection of all sequences (λn)n≥ of non-decreasing, non-negative and adapted functions and set λ∞ = limn→∞ λn. We can define some weak Orlicz-Lorentz martingale spaces as follows: HFσ,∞ = f = (fn) : σ (f ) F,∞ < ∞ , QF,∞ = f = (fn) : ∃(λn)n≥ ∈ , s.t. Sn(f ) ≤ λn– , λ∞ ∈ LF,∞ , f. Let ak be defined as in the proof of Theorem.

By the definition of

It follows from the boundedness of T and that