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Let $(\mathcal{X},d,\mu)$ be a metric measure space satisfying the so-calledupper doubling condition and the geometrically doubling condition. Let $T$ be aCalder\'{o}n-Zygmund operator with kernel satisfying only the size conditionand some H\"ormander-type condition, and $b\in\rm{\widetilde{RBMO}(\mu)}$ (theregularized BMO space with the discrete coefficient). In this paper, theauthors 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 thediscrete coefficient into the weak Lebesgue space $L^{1,\,\infty}(\mu)$. Theboundedness 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 showthat the commutator $T_b$ is bounded on $L^p(\mu)$ for all $p\in(1,\infty)$.

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