Hypersingular Marcinkiewicz Integrals along Surface with Variable Kernels on Sobolev Space and Hardy-Sobolev Space

Abstract

Let , the authors introduce in this paper a class of the hypersingular Marcinkiewicz integrals along surface with variable kernels defined by , where with . The authors prove that the operator is bounded from Sobolev space to space for , and from Hardy-Sobolev space to space for . As corollaries of the result, they also prove the boundedness of the Littlewood-Paley type operators and which relate to the Lusin area integral and the Littlewood-Paley function.