Weighted boundedness for a class of rough Marcinkiewicz integrals

Suppose that Omega(x') is an element of L-1(Sn-1) (n greater than or equal to 2) and is homogeneous of degree zero satisfying the cancellation conditi on. Then the Marcinkiewicz integral operator mu(Omega) related to the Littl ewood-Paley g-function is defined by mu(Omega)f(x) = (integral(0)(infinity)\F-t(x)\(2)dt/t3)(1/2), where [GRAPHICS] b(r) is an element of L-infinity(R+). In this paper we prove that the operator mu(Omega) is bounded on L-p(omega) for 1 < p < infinity, where omega belongs to the Muckenhoupt weight class. Moreover, we give also the weighted L-p-boundedness for a class of Marcink iewicz integral operators with rough kernels mu(Omega,lambda)* and mu(Omega ),S related to the Littlewood-Paley g(lambda)*-functionand the area integra l S, respectively. Our results are substantial improvements and extensions of known weighted r esults on the Marcinkiewicz integral operator introduced by E. M. Stein.