Rough Singular Integral Operators Along Submanifolds

Abstract

This paper is a summary of the research on the mapping properties of singular integrals with rough kernels and the corresponding maximal operators. More precisely, the author presents some recent progress, interesting problems and typical methods used in the theory concerning the boundedness and continuity for the rough singular integral operators and maximal singular integral operators along certain submanifolds such as polynomial mappings, polynomial curves, homogeneous mappings, surfaces of revolution and real-analytic submanifolds on the Lebesgue spaces, Triebel–Lizorkin spaces, Besov spaces and mixed radial-angular spaces.