Calderón-Zygmund operators with variable kernels acting on weak Musielak-Orlicz Hardy spaces

Abstract

Abstract. Let $\varphi : \mathbb R^n \times [0,\infty) \rightarrow [0,\infty)$ satisfy that $\varphi(x,\cdot)$ is an Orlicz function for any given $x \in \mathbb R^n$, and $\varphi(\cdot,t)$ is a Muckenhoupt $A_\infty$ weight uniformly in $t \in (0,\infty)$. The weak Musielak-Orlicz Hardy space $WH^\varphi(\mathbb R^n)$ is defined to be the set of all tempered distributions such that their grand maximal functions belong to the weak Musielak-Orlicz space $WL^\varphi(\mathbb R^n)$. In this paper, we discuss the boundedness of the Calderón-Zygmund operator with variable kernel from $WH^\varphi(\mathbb R^n)$ to $WL^\varphi(\mathbb R^n)$. These results are new even for the classical weighted weak Hardy space and probably new for the classical weak Hardy space.