Boundedness of Calder\'on--Zygmund operators on ball Campanato-type function spaces

Abstract

Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ satisfying some mild assumptions. In this article, the authors first find a reasonable version $\widetilde{T}$ of the Calder\'on--Zygmund operator $T$ on the ball Campanato-type function space $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ with $q\in[1,\infty)$, $s\in\mathbb{Z}_+^n$, and $d\in(0,\infty)$. Then the authors prove that $\widetilde{T}$ is bounded on $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ if and only if, for any $\gamma\in\mathbb{Z}^n_+$ with $|\gamma|\leq s$, $T^*(x^{\gamma})=0$, which is hence sharp. Moreover, $\widetilde{T}$ is proved to be the adjoint operator of $T$, which further strengthens the rationality of the definition of $\widetilde{T}$. All these results have a wide range of applications. In particular, even when they are applied, respectively, to weighted Lebesgue spaces, variable Lebesgue spaces, Orlicz spaces, Orlicz-slice spaces, Morrey spaces, mixed-norm Lebesgue spaces, local generalized Herz spaces, and mixed-norm Herz spaces, all the obtained results are new. The proofs of these results strongly depend on the properties of the kernel of $T$ under consideration and also on the dual theorem on $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$.