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

Abstract

Let $(\mathcal{X},d,\hat{I}¼)$ be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let $T$ be a Calderón-Zygmund operator with kernel satisfying only the size condition and some Hörmander-type condition, and $b\in \widetilde{\mathrm{RBMO}}(\hat{\mathrm{I}}¼)$ (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 ${\tilde{H}}^1(\hat{I}¼)$ with the discrete coefficient into the weak Lebesgue space $L^{1,\infty }(\hat{I}¼)$. The boundedness of the commutator generated by the generalized fractional integral $T_{\hat{I}}\pm (\hat{I}\pm \in (0,1))$ and the $\widetilde{\mathrm{RBMO}}(\hat{\mathrm{I}}¼)$ function from ${\tilde{H}}^1(\hat{I}¼)$ into $L^{1/{(1-α)},\,\fz}(μ)$ is also presented. Moreover, by an interpolation theorem for sublinear operators, the authors show that the commutator $T_b$ is bounded on $L^p(\hat{I}¼)$ for all $p\in (1,\infty )$.

34 pages, Submitted. arXiv admin note: text overlap with arXiv:1412.0190