Marcinkiewicz integrals with variable kernels on Hardy and weak Hardy spaces

In this article, we consider the Marcinkiewicz integrals with variable kernels defined by mu Omega(f)(x) = (integral(infinity)(0) vertical bar integral(vertical bar x-y vertical bar <= t) Omega(x,x-y)/vertical bar x-y vertical bar(n-1) f(y)dy vertical bar(2) dt/t(3))(1/2), where Omega(x,z) is an element of L(infinity) (R(n)) x L(q) (S(n-1)) for q > 1. We prove that the operator mu Omega is bounded from Hardy space, H(p)(R(n)) , to L(p)(R(n)) space; and is bounded from weak Hardy space, H(p,infinity) (R(n)), to weak L(p)(R(n)) space for max {2n/2n+1, n/n+alpha} < p < 1, if Omega satisfies L(1,alpha)-Dini condition with any 0 < alpha <= 1.