Generalized Calderón-Zygmund operators on the Hardy space $H^1_ρ(\mathcal X)$

TL;DR

This work addresses boundedness criteria for a broad class of log-Dini Calderón-Zygmund operators on the Hardy space $H^1_\rho(\mathcal X)$ over RD-spaces equipped with an admissible function $\rho$. The authors develop a robust framework based on atoms and log-type molecules, culminating in a molecular characterization of $H^1_\rho(\mathcal X)$ and a suite of equivalent conditions: $T$ bounded on $H^1_\rho(\mathcal X)$, a quantitative decay condition on $\langle T^*1,\mathfrak a\rangle$ for atoms, and $T^*1\in BMO_\rho^{\log}(\mathcal X)$ (with related $BMO_\rho$ criteria) under the log-Dini hypothesis on the kernel regularity. These results unify and extend prior Schrödinger- and inhomogeneous CZ-type results to the RD-space setting and provide a flexible toolkit for analyzing singular integrals adapted to $\rho$. The Appendix broadens the scope to generalized $(\omega_1,\omega_2,s)_\rho$-CZ operators, preserving the core boundedness and duality principles and linking to classical operators in special cases. Overall, the paper advances the theory of localized Hardy spaces and singular integrals in non-Euclidean contexts with variable smoothness scales.

Abstract

Let $(\mathcal X, d,μ)$ be an RD-space, and let $ρ$ be an admissible function on $\mathcal X$. We establish necessary and sufficient conditions for the boundedness of a new class of generalized Calderón-Zygmund operators of log-Dini type on the Hardy space $H^1_ρ(\mathcal X)$, 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.

Theorems & Definitions (47)

• Definition 1.1
• Remark 1.1
• Theorem 1.1
• Corollary 1.1
• Theorem 1.2
• Corollary 1.2
• Remark 1.2
• Definition 2.1
• Definition 2.2
• Remark 2.1