A note on parameterized Marcinkiewicz integrals with variable kernels

In this paper, the parameterized Marcinkiewicz integrals with variable kernels defined by [GRAPHICS] are investigated. It is proved that if Omega is an element of L(infinity)(R(n)) x L(r) (S(n-1)) (r > (n-1)p'/n) is an odd function in the second variable y', then the operator mu(rho)(Omega) is bounded from L(p)(R(n)) to L(p)(R(n)) for 1 < p <= max{(n+1)/2, 2}. It is also proved that, if Omega satisfies the L(1)-Dini condition, then mu(rho)(Omega) is of type (p, p) for 1 < p <= 2, of the weak type (1, 1) and bounded from H(1) to L(1).