Acta mathematica scientia, Series B >
SEVERAL WEAK-TYPE WEIGHTED INEQUALITIES IN ORLICZ MARTINGALE CLASSES
Received date: 2009-02-16
Revised date: 2010-03-02
Online published: 2011-05-20
Supported by
Supported by the National Natural Science Foundation of China (10671147; 11071190).
The aim of this paper is to establish several necessary and sufficient conditions in order that the weighted inequality
ρ(Mf>λΦ(λ)≤C∫Ωψ(C|f|)σdu, ∨λ>0
or
ρ(Mf>λΦ(λ)≤C∫ΩΦ(Cλ-1|f|)σdu, ∨λ>0
holds for every uniformly integral martingale f=(fn), where M is the Doob's maximal operator, ψ, ψ are both Φ-functions, and ρ, σ are weights.
Key words： martingale space; maximal operator; weighted inequality; Orlicz norm
CHEN Wei , LIU Pei-De . SEVERAL WEAK-TYPE WEIGHTED INEQUALITIES IN ORLICZ MARTINGALE CLASSES[J]. Acta mathematica scientia, Series B, 2011 , 31(3) : 1041 -1050 . DOI: 10.1016/S0252-9602(11)60296-8
[1] Muckenhoupt B. Weighted norm inequalities for the Hardy maximal function. Trans Amer Math Soc, 1972, 165: 207--226
[2] Gallardo D. Weighted weak type integral inequalities for the Hardy-Littlewood maximal operator. Israel J Math, 1989, 67: 95--108
[3] Bagby R J. Weak bounds for the maximal function in the weighted Orlicz spaces. Studia Math, 1990, 95: 195--204
[4] Pick L. Two-weight weak type maximal inequalities in Orlicz class. Studia Math, 1991, 100: 207--218
[5] Kokilashvili V, Krbec M. Weighted Inequalities in Lorentz and Orlicz Spaces. Singapore: World Scientific, 1991
[6] Lai Q S. Two weight mixed Φ-inequalities for the Hardy and the Hardy-Littlewood maximal operator. J London Math Soc, 1992, 46(2): 301--318
[7] Lai Q S. Two weight Φ-inequalities for the Hardy operator, Hardy-Littlewood maximal operator, and fractional integrals. Proc Amer Math Soc, 1993, 118(1): 129--142
[8] Sun Q Y. Weighted norm inequalities on spaces of homogeneous type. Studia Math, 1992, 101(3): 241--251
[9] Kerman R A, Torchinsky A. Integral inequalities with weight for Hardy maximal function. Studia Math, 1982, 71: 277--284
[10] Long R L. Martingale Spaces and Inequalities. Beijing: Peking University Press, 1993
[11] Mei T, Liu P D. On the maximal inequalities for martingales involving two functions. Proc Amer Math Soc, 2002, 130(3): 883--892
[12] Zuo H L, Liu P D. The minimal operator and weighted inequalities for martingales. Acta Math Sci, 2006, 26B(1): 31--40
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