L^p Estimates of certain rough parametric Marcinkiewicz functions

We prove L^P estimates of a class of parametric Marcinkiewicz integral operators when their kernels satisfy only the L^1(S^n-1)integrability condition.The obtained L^P estimates resolve a problem left open in previous work.Our argument is based on duality technique and direct estimation of operators.As a consequence of our result,we deduce the L^P boundedness of a class of fractional Marcinkiewicz integral operators.