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On some inequalities for Doob decompositions in Banach function spaces

Research paper by Masato Kikuchi

Indexed on: 13 May '09Published on: 13 May '09Published in: Mathematische Zeitschrift



Abstract

Let \({\Phi : \mathbb{R} \to [0, \infty)}\) be a Young function and let \({f = (f_n)_n\in\mathbb{Z}_{+}}\) be a martingale such that \({\Phi(f_n) \in L_1}\) for all \({n \in \mathbb{Z}_{+}}\) . Then the process \({\Phi(f) = (\Phi(f_n))_n\in\mathbb{Z}_{+}}\) can be uniquely decomposed as \({\Phi(f_n)=g_n+h_n}\) , where \({g=(g_n)_n\in\mathbb{Z}_{+}}\) is a martingale and \({h=(h_n)_n\in\mathbb{Z}_{+}}\) is a predictable nondecreasing process such that h0 = 0 almost surely. The main results characterize those Banach function spaces X such that the inequality \({\|{h_{\infty}}\|_{X} \leq C \|{\Phi(Mf)} \|_X}\) is valid, and those X such that the inequality \({\|{h_{\infty}}\|_{X} \leq C \|{\Phi(Sf)} \|_X}\) is valid, where Mf and Sf denote the maximal function and the square function of f, respectively.