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Real-variable Theory of Anisotropic Musielak-Orlicz-Lorentz Hardy Spaces with Applications to Calderón-Zygmund Operators (2410.06611v2)

Published 9 Oct 2024 in math.CA and math.FA

Abstract: Let $\varphi: \mathbb{R}{n}\times[0,\infty)\rightarrow[0,\infty)$ be a Musielak-Orlicz function satisfying the uniformly anisotropic Muckenhoupt condition and be of uniformly lower type $p-_{\varphi}$ and of uniformly upper type $p+_{\varphi}$ with $0<p-_{\varphi}\leq p+_{\varphi}<\infty$, $q\in(0,\infty]$, and $A$ be a general expansive matrix on $\mathbb{R}{n}$. In this article, the authors first introduce the anisotropic Musielak-Orlicz-Lorentz Hardy space $H{\varphi,q}_A(\mathbb{R}{n})$ which, when $q=\infty$, coincides with the known anisotropic weak Musielak-Orlicz Hardy space $H{\varphi,\infty}_A(\mathbb{R}{n})$, and then establish atomic and molecular characterizations of $H{\varphi,q}_A(\mathbb{R}{n})$. As applications, the authors prove the boundedness of anisotropic Calder\'{o}n-Zygmund operators on $H{\varphi,q}_A(\mathbb{R}{n})$ when $q\in(0,\infty)$ or from the anisotropic Musielak-Orlicz Hardy space $H{\varphi}_A(\mathbb{R}{n})$ to $H{\varphi,\infty}_A(\mathbb{R}{n})$ in the critical case. The ranges of all the exponents under consideration are the best possible admissible ones which particularly improve all the known corresponding results for $H{\varphi,\infty}_A(\mathbb{R}{n})$ via widening the original assumption $0<p-_{\varphi}\leq p+_{\varphi}\leq1$ into the full range $0<p-_{\varphi}\leq p+_{\varphi}<\infty$, and all the results when $q\in(0,\infty)$ are new and generalized from isotropic setting to anisotropic setting.

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