Boundedness of Calderón--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 $\tilde{T}$ of the Calderón--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 $\tilde{T}$ is bounded on ${\mathcal{L}}_{X,q,s,d}({\mathbb{R}}^n)$ if and only if, for any $\gamma \in {\mathbb{Z}}_{+}^n$ with $|\gamma |\le s$, $T^{\ast }(x^{\gamma })=0$, which is hence sharp. Moreover, $\tilde{T}$ is proved to be the adjoint operator of $T$, which further strengthens the rationality of the definition of $\tilde{T}$. 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 ${\mathcal{L}}_{X,q,s,d}({\mathbb{R}}^n)$.

32 pages, Submitted. arXiv admin note: substantial text overlap with arXiv:2206.06551, arXiv:2206.06080