Variable Hardy–Lorentz martingale spaces Hp(·),q(·)$\mathcal {H}_{p(\cdot ),q(\cdot )}$

Jianzhong Lu and Tao Ma

Mathematische Nachrichten, 2024, vol. 297, issue 1, 8-37

Abstract: Let (Ω,F,P)$(\Omega ,\mathcal {F},\mathbb {P})$ be a complete probability space and let p(·),q(·):[0,1]→(0,∞)$p(\cdot ),q(\cdot ):[0,1]\rightarrow (0,\infty )$ be two variable exponents. In this paper, we investigate several new variable Hardy–Lorentz martingale spaces associated with the variable Lorentz spaces Lp(·),q(·)(Ω)$\mathcal {L}_{p(\cdot ),q(\cdot )}(\Omega )$, which are defined by nonincreasing rearrangement functions. To be precise, we first formulate the atomic decomposition theorems for these Hardy–Lorentz martingale spaces, and then establish martingale inequalities among them with the help of the boundedness of a σ‐sublinear operator. Furthermore, we prove the boundedness of fractional integrals on variable Hardy–Lorentz martingale spaces Hp(·),q(·)(Ω)$\mathcal {H}_{p(\cdot ),q(\cdot )}(\Omega )$.

Date: 2024
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