Molecular characterization of anisotropic weak Musielak-Orlicz Hardy spaces and their applications

Abstract

Let \begin{document}$ A $\end{document} be a real \begin{document}$ n\times n $\end{document} matrix with all its eigenvalues \begin{document}$ \lambda $\end{document} satisfy \begin{document}$ |\lambda|>1 $\end{document} . Let \begin{document}$ \varphi: \mathbb{R}^n\times[0, \, \infty)\to[0, \, \infty) $\end{document} be an anisotropic Musielak-Orlicz function, i.e., \begin{document}$ \varphi(x, \cdot) $\end{document} is an Orlicz function uniformly in \begin{document}$ x\in{\mathbb{R}^n} $\end{document} and \begin{document}$ \varphi(\cdot, \, t) $\end{document} is an anisotropic Muckenhoupt \begin{document}$ \mathcal {A}_\infty({\mathbb{R}^n}) $\end{document} weight uniformly in \begin{document}$ t\in(0, \, \infty) $\end{document} . In this article, the authors introduce the anisotropic weak Musielak-Orlicz Hardy space \begin{document}$ WH^{\varphi}_A(\mathbb{R}^n) $\end{document} via the grand maximal function and establish its molecular characterization which are anisotropic extensions of Liang, Yang and Jiang (Math. Nachr. 289: 634-677, 2016). As an application, the boundedness of anisotropic Calderon-Zygmund operators from \begin{document}$ H_A^\varphi(\mathbb{R}^n) $\end{document} to \begin{document}$ WH_A^\varphi(\mathbb{R}^n) $\end{document} in the critical case is presented.