On some inequalities for Doob decompositions in Banach function spaces |
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Authors: | Masato Kikuchi |
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Institution: | 1. Department of Mathematics, University of Toyama, 3190 Gofuku, Toyama, 930-8555, Japan
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Abstract: | Let ${\Phi : \mathbb{R} \to 0, \infty)}Let
F: \mathbbR ? 0, ¥){\Phi : \mathbb{R} \to 0, \infty)} be a Young function and let
f = (fn)n ? \mathbbZ+{f = (f_n)_n\in\mathbb{Z}_{+}} be a martingale such that F(fn) ? L1{\Phi(f_n) \in L_1} for all
n ? \mathbbZ+{n \in \mathbb{Z}_{+}} . Then the process
F(f) = (F(fn))n ? \mathbbZ+{\Phi(f) = (\Phi(f_n))_n\in\mathbb{Z}_{+}} can be uniquely decomposed as F(fn)=gn+hn{\Phi(f_n)=g_n+h_n} , where
g=(gn)n ? \mathbbZ+{g=(g_n)_n\in\mathbb{Z}_{+}} is a martingale and
h=(hn)n ? \mathbbZ+{h=(h_n)_n\in\mathbb{Z}_{+}} is a predictable nondecreasing process such that h
0 = 0 almost surely. The main results characterize those Banach function spaces X such that the inequality ||h¥||X £ C ||F(Mf) ||X{\|{h_{\infty}}\|_{X} \leq C \|{\Phi(Mf)} \|_X} is valid, and those X such that the inequality ||h¥||X £ C ||F(Sf) ||X{\|{h_{\infty}}\|_{X} \leq C \|{\Phi(Sf)} \|_X} is valid, where Mf and Sf denote the maximal function and the square function of f, respectively. |
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Keywords: | |
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