Weighted estimates for Marcinkiewicz integrals with non-doubling measures

Let µ be a nonnegative Radon measure on R d which satisfies the growth condition µ(B(x,r)) � C0r n for all x 2 R d and r > 0, where C0 is a fixed constant and 0 < nd. The purpose of this paper is to establish the boundedness of the Marcinkiewicz integrals from L p (u) to L p,1 (u), where u is a weight function of Muckenhoupt type associated with µ.