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

This paper is devoted to exploring the mapping properties for the commutator mu(Omega,b) generated by Marcinkiewicz integral mu(Omega) with a locally integrable function b in the generalized Campanato spaces on the generalized Morrey spaces. Under the assumption that the integral kernel Omega satisfies certain log-type regularity, it is shown that mu(Omega,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(R-n) and the Lipschitz spaces Lip(alpha)(R-n) (0 < alpha <= 1) as special examples. Some previous results are essentially improved and generalized.