$L^{p}$ bounds for a class of Marcinkiewicz integral operators

Abstract

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 Mintegral 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.