Boundedness of CalderónZygmund operators on ball Campanatotype function spaces
Abstract
Let $X$ be a ball quasiBanach function space on ${\mathbb R}^n$ satisfying some mild assumptions. In this article, the authors first find a reasonable version $\widetilde{T}$ of the CalderónZygmund operator $T$ on the ball Campanatotype 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, Orliczslice spaces, Morrey spaces, mixednorm Lebesgue spaces, local generalized Herz spaces, and mixednorm 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 $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$.
 Publication:

arXiv eprints
 Pub Date:
 August 2022
 arXiv:
 arXiv:2208.06266
 Bibcode:
 2022arXiv220806266C
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Analysis of PDEs;
 Mathematics  Classical Analysis and ODEs;
 Primary 42B20;
 Secondary 42B25;
 42B30;
 42B35;
 46E35;
 47A30
 EPrint:
 32 pages, Submitted. arXiv admin note: substantial text overlap with arXiv:2206.06551, arXiv:2206.06080