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Variable Hardy–Lorentz spaces associated with operators satisfying Davies–Gaffney estimates on metric measure spaces of homogeneous type

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Abstract

Suppose that \((\mathcal{X}, d, \mu)\) is a metric measure space of homogeneous type and L is a one-to-one operator of type \(\omega\) on \(L^2(\mathcal{X})\), with \(\omega \in[0, \pi / 2)\), which has a bounded holomorphic functional calculus, and whose heat kernel satisfies the Davies–Gaffney estimates. Suppose that \(p(\cdot) \colon \mathcal{X} \rightarrow(0,1]\) is a variable exponent function with the globally log-Hölder continuous condition. In this paper, we introduce the variable Hardy–Lorentz space \({H}_L^{p(\cdot),q}(\mathcal{X})\) associated with L with \(0<p_{-} \leq p_{+} \leq 1\) and \(0<q<\infty\), and establish its molecular characterization. Moreover, we study the dual spaces of \(H_L^{p(\cdot), q}(\mathcal{X})\) with \(0<p_{-} \leq p_{+} \leq 1\) and \(0<q<\infty\).

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  1. School of Mathematics and Statistics, Central South University, Changsha, 410075, China

    Y. He

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He, Y. Variable Hardy–Lorentz spaces associated with operators satisfying Davies–Gaffney estimates on metric measure spaces of homogeneous type. Acta Math. Hungar. 170, 209–243 (2023). https://doi.org/10.1007/s10474-023-01334-6

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