Compactness for Commutator of Fractional Integral on Non-homogeneous Morrey Spaces
Received:January 28, 2021  Revised:May 20, 2021
Key Word: non-homogeneous metric measure space   compactness   commutator of fractional integral   $\mathrm{Lip}_{\beta}(\mu)$   Morrey space
Fund ProjectL:Supported by the Scientific Startup Foundation for Doctors of Northwest Normal University (Grant No.0002020203) and the Innovation Fund Project for Higher Education of Gansu Province (Grant No.2020A-010).
 Author Name Affiliation Guanghui LU College of Mathematics and Statistics, Northwest Normal University, Gansu 730070, P. R. China
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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 $b\in\mathrm{Lip}_{\beta}(\mu)$.