# 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.