Acta mathematica scientia,Series B ›› 2011, Vol. 31 ›› Issue (3): 1041-1050.doi: 10.1016/S0252-9602(11)60296-8

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SEVERAL WEAK-TYPE WEIGHTED INEQUALITIES IN ORLICZ MARTINGALE CLASSES

 CHEN Wei1,2,, LIU Pei-De2   

  1. 1. School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China|2. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
  • Received:2009-02-16 Revised:2010-03-02 Online:2011-05-20 Published:2011-05-20
  • Supported by:

    Supported by the National Natural Science Foundation of China (10671147; 11071190).

Abstract:

The aim of this paper is to establish several necessary and sufficient conditions in order that the weighted inequality

ρ(MfΦ(λ)≤C∫Ωψ(C|f|)σdu, ∨λ>0

or

ρ(MfΦ(λ)≤C∫ΩΦ(-1|f|)σdu, ∨λ>0

holds for every uniformly integral martingale f=(fn), where M is the Doob's maximal operator, ψψ are both Φ-functions, and ρ, σ are weights.

Key words: martingale space, maximal operator, weighted inequality, Orlicz norm

CLC Number: 

  • 60G46
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