Rough Singular Integrals with Kernels Supported by Submanifolds of Finite Type

Abstract

Our point of departure is the following $L^{p}$ boundedness result form [St], Theorem 1.1, ... Recently, the results in Theorem 1.1 were improved by Fan, Guo, and Pan in [FGP] who showed that the $L^{p}$ boundedness of $T_{\Phi}$ and $M_{\Phi}$ continues to hold if the condition $\Omega \in \mathbb{C}^{1}(Sn-1)$ is replaced by the weaker condition $\Omega \in L^{q}(Sn-1)$ for some $q > 1$. Also, the authors of [FGP] were able to establish the $L^{p}$ boundedness of the maximal operator $T^{*}_{\Phi}$ under the condition $\Omega \in L^{q}(Sn-1)$ for some $q > 1$. The main purpose of this paper is to present further improvements of the above results in which the condition $\Omega \in L^{q}(Sn-1)$ is replaced by a weaker condition $\Omega \in B^{0,0}_{q}(Sn-1)$ ...