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Variable Hardy-Lorentz spaces associated to operators satisfying Davies-Gaffney estimates. (English) Zbl 1423.42043

Summary: Let \(L\) be a one-to-one operator of type \(w\) with \(w\in[0,\pi/2)\), which satisfies the Davies-Gaffney estimates and has a bounded holomorphic calculus, and let \(p(\cdot)\) be a measurable function on \(\mathbb{R}^n\) with \(0<p_-:=\operatorname{ess\,inf}_{x\in\mathbb{R}^n}p(x)\leq\operatorname{ess\,sup}_{x\in\mathbb{R}^n}p(x)=:p_+<\infty\). Under the assumption that \(p(\cdot)\) satisfies the global log-Hölder condition, we introduce the variable Hardy-Lorentz space \(H_L^{p(\cdot),q}(\mathbb{R}^n)\) for \(0<q<\infty\) and construct its molecular decomposition. Furthermore, we investigate the dual spaces of the variable Hardy-Lorentz space \(H_L^{p(\cdot),q}(\mathbb{R}^n)\) with \(0<p_-\leq p_+\leq1\) and \(0<q<\infty\). These results are new even when \(p(\cdot)\) is a constant.

MSC:

42B30 \(H^p\)-spaces
42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis

References:

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