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

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.