Compactness for Commutator of Fractional Integral on Non-homogeneous Morrey Spaces

Abstract:

      The aim of this paper is to establish the necessary and sufficient conditions for the compactness of fractional integral commutator $[b, I_{\gamma}]$ which is generated by fractional integral $I_{\gamma}$ and function $b\in\mathrm{Lip}_{\beta}(\mu)$ on Morrey space over non-homogeneous metric measure space, which satisfies the geometrically doubling and upper doubling conditions in the sense of Hyt\"{o}nen. Under assumption that the dominating function $\lambda$ satisfies weak reverse doubling condition, the author proves that the commutator $[b,I_{\gamma}]$ is compact from Morrey space $M^{p}_{q}(\mu)$ into Morrey space $M^{s}_{t}(\mu)$ if and only if $b\in\mathrm{Lip}_{\beta}(\mu)$.