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

TL;DR

This paper develops a real-variable theory for anisotropic Musielak–Orlicz–Lorentz Hardy spaces $H^{\varphi,q}_A(\mathbb{R}^n)$ associated with an expansive matrix $A$ and a Musielak–Orlicz function $\varphi$. It defines $H^{\varphi,q}_A$ via a non-tangential grand maximal function and proves robust atomic and molecular decompositions, enabling precise norm controls and reconstruction. The authors establish boundedness results for anisotropic Calderón–Zygmund operators on $H^{\varphi,q}_A$ (for $q\in(0,\infty)$) and in the critical case from $H^{\varphi}_A$ to $H^{\varphi,\infty}_A$, with exponent ranges that are shown to be optimal and extend prior isotropic results to the anisotropic setting. These contributions broaden endpoint estimates and provide a comprehensive anisotropic framework for Musielak–Orlicz Hardy spaces, with potential impact on Calderón–Zygmund theory and related harmonic analysis in non-Euclidean dilations.

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 Calderón-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.

Theorems & Definitions (53)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 1.9
• Theorem 1.10