Rough maximal singular integral and maximal operators supported by subvarieties on Triebel–Lizorkin spaces

Abstract

In this paper, our object of investigation is the maximal singular integrals with rough kernels associated to polynomial mapping P as well as the corresponding compound submanifolds, which is defined by Th,Ω,P∗f(x):=supϵ>0|∫|y|>ϵf(x−P(y))h(|y|)Ω(y)|y|ndy|.We show that the operator Th,Ω,P∗ is bounded on Triebel–Lizorkin spaces and Besov spaces when the rough kernel Ω∈L(log+L)β(Sn−1) for some β∈(0,1]. Similar results can be extended to Hardy–Littlewood maximal operators with rough kernels associated to the mapping P.