L^p bounds for a class of Marcinkiewicz integral operators

In this paper, we introduce a class of mappings that is more general than the class of polynomials as well as the class of convex functions. We prove L^p estimates of Marcinkiewicz integral operators along surfaces generated by mappings belong to this class. Moreover, we establish the L^p boundedness of the corresponding maximal functions. Our results extend as well as improve previously known results on Marcinkiewicz integral operators and maximal functions.