A characterization of Calder\'on-Zygmund operators on RBMO

TL;DR

This paper characterizes bounded Calderón-Zygmund operators on the RBMO space associated with a finite positive measure, using the behavior of the operator on the constant function 1, extending classical results to a more general setting.

Contribution

It provides a new characterization of Calderón-Zygmund operators on RBMO spaces for measures beyond Lebesgue, based on the analysis of T1.

Findings

Characterization of bounded Calderón-Zygmund operators on RBMO.

Extension of classical operator theory to measures with finite positive support.

Conditions on T1 that ensure boundedness on RBMO.

Abstract

Let $\mathrm{RBMO}(\mu) = \mathrm{RBMO}(\mathbb{R}^m, \mu)$ denote the regular BMO space introduced by X. Tolsa for an $n$-dimensional finite positive measure on $\mathbb{R}^m$, $0<n \le m$. We characterize the bounded Calder\'on-Zygmund operators $T:\mathrm{RBMO}(\mu) \to \mathrm{RBMO}(\mu)$ in terms of the function $T1$.