Generalized Fractional Integrals and Their Commutators over Non-homogeneous Metric Measure Spaces

Abstract

Let $(\mathcal{X},d,\hat{I}¼)$ be a metric measure space satisfying both the upper doubling and the geometrically doubling conditions. In this paper, the authors establish some equivalent characterizations for the boundedness of fractional integrals over $(\mathcal{X},d,\hat{I}¼)$. The authors also prove that multilinear commutators of fractional integrals with $\mathrm{RBMO}(\hat{\mathrm{I}}¼)$ functions are bounded on Orlicz spaces over $(\mathcal{X},d,\hat{I}¼)$, which include Lebesgue spaces as special cases. The weak type endpoint estimates for multilinear commutators of fractional integrals with functions in the Orlicz-type space ${\mathrm{Osc}}_{\exp L^r}(\hat{I}¼)$, where $r\in [1,\infty )$, are also presented. Finally, all these results are applied to a specific example of fractional integrals over non-homogeneous metric measure spaces.

Taiwanese J. Math. (to appear)