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

Abstract

Let $\phi :{\mathbb{R}}^n\times [0,\infty )\to [0,\infty )$ be a Musielak-Orlicz function satisfying the uniformly anisotropic Muckenhoupt condition and be of uniformly lower type $p_{\phi }^{-}$ and of uniformly upper type $p_{\phi }^{+}$ with $0<p_{\phi }^{-}\le p_{\phi }^{+}<\infty$, $q\in (0,\infty ]$, 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 $H_A^{\phi ,q}({\mathbb{R}}^n)$ which, when $q=\infty$, coincides with the known anisotropic weak Musielak-Orlicz Hardy space $H_A^{\phi ,\infty }({\mathbb{R}}^n)$, and then establish atomic and molecular characterizations of $H_A^{\phi ,q}({\mathbb{R}}^n)$. As applications, the authors prove the boundedness of anisotropic Calderón-Zygmund operators on $H_A^{\phi ,q}({\mathbb{R}}^n)$ when $q\in (0,\infty )$ or from the anisotropic Musielak-Orlicz Hardy space $H_A^{\phi }({\mathbb{R}}^n)$ to $H_A^{\phi ,\infty }({\mathbb{R}}^n)$ 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 $H_A^{\phi ,\infty }({\mathbb{R}}^n)$ via widening the original assumption $0<p_{\phi }^{-}\le p_{\phi }^{+}\le 1$ into the full range $0<p_{\phi }^{-}\le p_{\phi }^{+}<\infty$, and all the results when $q\in (0,\infty )$ are new and generalized from isotropic setting to anisotropic setting.

52 pages