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

TL;DR

This paper establishes the boundedness criteria for Calderón--Zygmund operators on ball Campanato-type function spaces, extending the results to various classical and modern function spaces with sharp conditions.

Contribution

It introduces a reasonable version of Calderón--Zygmund operators on these spaces and proves their boundedness under sharp conditions, with broad applications to many function space settings.

Findings

Boundedness of the operator characterized by vanishing moments condition.

The operator's adjoint is identified as the modified Calderón--Zygmund operator.

Results apply to a wide range of classical and modern 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 $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$.