Hardy-type space estimates for multilinear commutators of Calderón–Zygmund operators on nonhomogeneous metric measure 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 Calderón–Zygmund operator and let $\overrightarrow{b}:=(b_1,\dots ,b_m)$ be a finite family of $\backslash widetilde\{RBMO\}\left(\mu \right)$ functions. In this article, the authors establish the boundedness of the multilinear commutator $T_{\overrightarrow{b}}$, generated by $T$ and $\overrightarrow{b}$ from the atomic Hardy-type space ${\tilde{H}}_{\mathrm{fin},\overrightarrow{b},m,\rho }^{1,q,m+1}\left(\mu \right)$ into the Lebesgue space $L^1\left(\mu \right)$. The authors also prove that $T_{\overrightarrow{b}}$ is bounded from the atomic Hardy-type space ${\tilde{H}}_{\mathrm{fin},\overrightarrow{b},m,\rho }^{1,q,m+2}\left(\mu \right)$ into the atomic Hardy space ${\tilde{H}}^1\left(\mu \right)$ via the molecular characterization of ${\tilde{H}}^1\left(\mu \right)$.