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

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.