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 h 0 = 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.
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This research was supported by Grant-in-aid for Scientific Research (C) 17540152.
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Kikuchi, M. On some inequalities for Doob decompositions in Banach function spaces. Math. Z. 265, 865–887 (2010). https://doi.org/10.1007/s00209-009-0546-3
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DOI: https://doi.org/10.1007/s00209-009-0546-3
Keywords
- Martingale
- Doob decomposition
- Banach function space
- Interpolation space
- Rearrangement-invariant function space