# On some inequalities for Doob decompositions in Banach function spaces

Research paper by **Masato Kikuchi**

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

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### 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.