On maximal functions with rough kernels in $L(\log L)^{1/2}(\mathbb{S}^{n-1})$

In this paper, we study the $L^p$ mapping properties of maximal functions with rough kernels that are related to certain class of singular integral operators. We prove that our maximal functions are bounded on $L^p$ provided that their kernels are in $L(\log L)^{1/2}(\mathbb{S}^{n-1})$. Moreover, we present an example showing that our size condition on the kernel is optimal. As a consequence of our result, we substantially improve previously known results on maximal functions, singular integral operators, and Parametric Marcinkiewicz integral operators