On certain estimates for Marcinkiewicz integrals and extrapolation

We obtain $L^p$ estimates for parametric Marcinkiewicz integrals associated to polynomial mappings and with rough kernels on the unit sphere as well as on the radial direction. These estimates will allow us to use an extrapolation argument to obtain some new and improved results on Marcinkiewicz integrals. Also, such estimates provide us with a unifying approach in dealing with Marcinkiewicz integrals when the kernel function $\Omega$ belongs to the class of block spaces $B_q^{(0,\alpha)}(S^n-1)$ as well as when $\Omega$ belongs to the class $L(log L) (S^n-1)$. Our conditions on the kernels are known to be the best possible in their respective classes.