Boundedness of Commutators on Hardy Spaces over Metric Measure Spaces of Non-homogeneous Type

TL;DR

This paper proves the boundedness of commutators involving Calderón-Zygmund operators and fractional integrals on Hardy spaces over non-homogeneous metric measure spaces, extending classical results to more general settings.

Contribution

It establishes boundedness results for commutators on Hardy spaces over non-homogeneous metric measure spaces, including fractional integrals, using regularized BMO spaces.

Findings

Boundedness of commutators from Hardy space to weak Lebesgue space.

Boundedness of fractional integral commutators into Lebesgue spaces.

Interpolation results showing boundedness on all L^p spaces.

Abstract

Let $(\mathcal{X},d,\mu)$ be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let $T$ be a Calder\'{o}n-Zygmund operator with kernel satisfying only the size condition and some H\"ormander-type condition, and $b\in\rm{\widetilde{RBMO}(\mu)}$ (the regularized BMO space with the discrete coefficient). In this paper, the authors establish the boundedness of the commutator $T_b:=bT-Tb$ generated by $T$ and $b$ from the atomic Hardy space $\widetilde{H}^1(\mu)$ with the discrete coefficient into the weak Lebesgue space $L^{1,\,\infty}(\mu)$. The boundedness of the commutator generated by the generalized fractional integral $T_\alpha\,(\alpha\in(0,1))$ and the $\rm{\widetilde{RBMO}(\mu)}$ function from $\widetilde{H}^1(\mu)$ into $L^{1/{(1-\alpha)},\,\fz}(\mu)$ is also presented. Moreover, by an interpolation theorem for sublinear operators, the authors show that the commutator $T_b$ is bounded on $L^p(\mu)$ for all $p\in(1,\infty)$.