On the Functions of Marcinkiewicz Integrals along Surfaces of Revolution on Product Domains via Extrapolation

Abstract

In this paper, we establish certain Lp bounds for several classes of rough Marcinkiewicz integrals over surfaces of revolution on product spaces. By using these bounds and using an extrapolation argument, we obtain the Lp boundedness of these Marcinkiewicz integrals under very weak conditions on the kernel functions. Several previous results on Marcinkiewicz operators are essentially extended or improved. Our results represent natural extensions and improvements of several known results on Marcinkiewicz integrals over symmetric spaces.