Martingale inequalities on Musielak–Orlicz Hardy spaces

Lechen He, Lihua Peng and Guangheng Xie

Mathematische Nachrichten, 2023, vol. 296, issue 11, 5171-5189

Abstract: Given a probability space (Ω,F,P)$(\Omega ,\mathcal {F},\mathbb {P})$ and a Musielak–Orlicz function φ:Ω×[0,∞)→[0,∞]$\varphi :\ \Omega \times [0,\infty )\rightarrow [0,\infty ]$, we investigate martingale inequalities in the framework of Musielak–Orlicz spaces by constructing atomic decompositions. Especially, the obtained results for Musielak–Orlicz functions φ(x,t)$\varphi (x,t)$ with particular structure, including the variable Orlicz functions Φ(tp(x))$\Phi (t^{p(x)})$, [Φ(t)]p(x)$[\Phi (t)]^{p(x)}$, the variable double phase‐growth tp(x)+a(x)tq(x)$t^{p(x)}+a(x)t^{q(x)}$, and the perturbed variable exponent function tp(x)log(e+t)$t^{p(x)}\log (e+t)$ are also new. Hence, we handle the martingale inequality for the above functional in a universal way.

Date: 2023
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