Boundedness of Commutators of Marcinkiewicz Integrals on Nonhomogeneous Metric Measure Spaces

Abstract

Let (X,d,μ) be a metric measure space satisfying the upper doubling condition and geometrically doubling condition in the sense of Hytönen. The aim of this paper is to establish the boundedness of commutator Mb generated by the Marcinkiewicz integral M and Lipschitz function b. The authors prove that Mb is bounded from the Lebesgue spaces Lp(μ) to weak Lebesgue spaces Lq(μ) for 1≤p<n/β, from the Lebesgue spaces Lp(μ) to the spaces RBMO(μ) for p=n/β, and from the Lebesgue spaces Lp(μ) to the Lipschitz spaces Lip(β-n/p)(μ) for n/β<p≤∞. Moreover, some results in Morrey spaces and Hardy spaces are also discussed.