Generalized Calder\'on-Zygmund operators on the Hardy space $H^1_\rho(\mathcal X)$

TL;DR

This paper characterizes the boundedness of a new class of generalized Calderón-Zygmund operators of log-Dini type on Hardy spaces over RD-spaces, extending existing theories and unifying recent results.

Contribution

It establishes necessary and sufficient conditions for these operators' boundedness on $H^1_ ho( ext X)$, advancing the understanding of singular integrals in this context.

Findings

Provides a complete characterization of boundedness conditions.

Extends previous results to a broader class of operators.

Unifies various recent findings in the theory of singular integrals.

Abstract

Let $(\mathcal X, d,\mu)$ be an RD-space, and let $\rho$ be an admissible function on $\mathcal X$. We establish necessary and sufficient conditions for the boundedness of a new class of generalized Calder\'on-Zygmund operators of log-Dini type on the Hardy space $H^1_\rho(\mathcal X)$, introduced by Yang and Zhou. Our results extend and unify some recent results, providing further insights into the study of singular integral operators in this setting.