The inhomogeneous Calder\'on--Zygmund singular integral operators on weak local Hardy spaces associated with ball quasi-Banach function spaces

Abstract

Let $X$ be a ball quasi-Banach function space on $\mathbb{R}^n, W X$ be the weak ball quasi-Banach function space on $\mathbb{R}^n, h_X$ be the local Hardy space associated with $X$. In this paper, we introduce the weak local Hardy-type space $W h_X$, associated with $X$, via using maximal function characterization. Moreover we obtain the boundedness of inhomogeneous Calder\'on--Zygmund operators from $h_X$ to $W X$  or $W h_X$. All these results have a wide range of generality and, particularly, to our best knowledge, even when they are applied to the Morrey spaces, Orlicz-slice spaces and Mixed-norm Lebesgue spaces, the results in this paper are also new.