New real-variable characterizations of anisotropic weak Hardy spaces of Musielak-Orlicz type

Abstract

A real $n\times n$ matrix $A$ is called an expansive dilation if all of its eigenvalues $\lambda $ satisfy $|\lambda |\!\!>\!\!1$. Let $\varphi : \mathbb {R}^n\times [0,\infty )\to [0,\infty )$ be a Musielak-Orlicz function. The aim of this article is to find an appropriate general space which includes the weak Hardy space of Fefferman and Soria, the weighted weak Hardy space of Quek and Yang}, the anisotropic weak Hardy space of Ding and Lan, the Musielak-Orlicz Hardy space of Ky and the anisotropic Hardy space of Musielak-Orlicz type of Li, Yang and Yuan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ with $m\in \mathbb {N}$ and obtain some new real-variable characterizations of $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ in terms of the radial, the non-tangential and the tangential maximal functions via a new monotone convergence theorem adapted to the weak anisotropic Musielak-Orlicz space $L^{\varphi , \infty }({\mathbb {R}}^n)$. These maximal function characterizations generalize the known results on the anisotropic weak Hardy space $H^{p, \infty }_A({\mathbb {R}}^n)$ with $p\in (0, 1]$ and are new even for their weighted variants or weak Orlicz-Hardy variants. As an application, the authors show the boundedness of a class of multilinear operators formed by the anisotropic Calderon-Zygmund operators from product weighted Lebesgue space to $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ with $\varphi (x,t):=t^p\omega (x)$ and $\omega \in \mathbb {A}_1(A)$, which is a weighted and non-isotropic extension of Grafakos.