Generalized fractional integrals and their commutators over non-homogeneous metric measure spaces

Abstract

Let (X, d, μ) 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 (X, d, μ). The authors also prove that multilinear commutators of fractional integrals with RBMO(μ) functions are bounded on Orlicz spaces over (X, d, μ), 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 OscexpLr (μ), where r ∈ [1,∞), are also presented. Finally, all these results are applied to a specific example of fractional integrals over non-homogeneous metric measure spaces.