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

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 $b\in \widetilde{\mathrm{R}\mathrm{B}\mathrm{M}\mathrm{O}}(\mu )$ (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 ${\tilde{H}}^1(\mu )$ with the discrete coefficient into the weak Lebesgue space $L^{1,\mathrm{\infty }}(\mu )$. The boundedness of the commutator generated by the generalized fractional integral $T_{\alpha }(\alpha \in (0,1))$ and the $\widetilde{\mathrm{R}\mathrm{B}\mathrm{M}\mathrm{O}}(\mu )$ function from ${\tilde{H}}^1(\mu )$ into $L^{1\mathrm{/}(1-\alpha ),\text{fz}}(\mu )$ 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 $p\in (1,\mathrm{\infty })$.