On the Commutators of Marcinkiewicz Integral with a Function in Generalized Campanato Spaces on Generalized Morrey Spaces

Abstract

This paper is devoted to exploring the mapping properties for the commutator μΩ,b generated by Marcinkiewicz integral μΩ with a locally integrable function b in the generalized Campanato spaces on the generalized Morrey spaces. Under the assumption that the integral kernel Ω satisfies certain log-type regularity, it is shown that μΩ,b is bounded on the generalized Morrey spaces with variable growth condition, provided that b is a function in generalized Campanato spaces, which contain the BMO(Rn) and the Lipschitz spaces Lipα(Rn) (0<α≤1) as special examples. Some previous results are essentially improved and generalized.