Marcinkiewicz functions along flat surfaces with hardy space kernels

In this paper, we study Marcinkiewicz integral operators along subvarieties determined by flat surfaces with kernels in the Hardy space H¹(S^n-1). We establish the L^p boundedness of our operators under weak convexity assumptions on the surfaces. Moreover, we establish the L^p boundedness of the corresponding Marcinkiewicz integral operators that are related to area integral and Littlewood-Paley g^z.ast;_λ functions. Our results offer substantial improvements of previously known results.