Endpoint estimates for commutators of singular integral operators associated with admissible functions

Abstract

Let $(\mathcal X, d, \mu)$ be a space of homogeneous type and let $\rho$ be an admissible function on $\mathcal X$. In this paper, we introduce a new class of singular integral operators associated with $\rho$, including a wide range of operators arising in harmonic analysis. For such an operator $T$ and a function $b$ belonging to suitable localized $\mathrm{BMO}$ spaces associated with $\rho$, which are strictly larger than the classical space $\mathrm{BMO}(\mathcal X)$, we establish the boundedness of the commutator $[b, T]$ from the Hardy space $H^1_\rho(\mathcal X)$ into $L^{1,\infty}(\mathcal X)$, $L^1(\mathcal X)$, and $H^1_\rho(\mathcal X)$. Moreover, the boundedness of $[b,T]$ on $H^1_\rho(\mathcal X)$ is characterized by necessary and sufficient conditions. We then apply this to investigate the boundedness of commutators of singular integrals in various settings, including Schr\"odinger operators on stratified Lie groups and Laguerre operators of convolution type. Our results are new even for Schr\"odinger operators on $\mathbb R^n$.