An extension of Marcinkiewicz integrals with rough kernels

This paper considers the following Marcinkiewicz type integrals mu (Omega,beta),f(x) = ((0)integral(infinity) |integral 2 dt!1/2 Omega(x - y) |x - y|n-1- f(y)dy , 0 < 3 < n, t3 0|x-y|<= t which can be regarded as an extension of the classical Marcinkiewicz integral mu c introduced by Stein in [Trans. Amer. Math. Soc. 88(1958), 159-172], where Omega is a homogeneous function of degree zero on Rn with mean value zero in the unit sphere Sn-1. Under the assumption that Omega is an element of L infinity(Sn-1), the authors establish the L-q-estimate and weak (1,1) type estimate as well as the corresponding weighted estimates for mu (c), with 1 < q < infinity and 0 < 3 < (q- 1)n/q. Moreover, the bounds do not depend on beta and the strong (q, q) type and weak (1, 1) type estimates for the classical Marcinkiewicz integral mu (Omega) can be recovered from the above estimates of mu (Omega,beta), when beta -> 0.