An Endpoint Estimate for Rough Maximal Singular Integrals

We study the rough maximal singular integral $$T^{\#}_\Omega\big(\,f\big)\big(x\big)=\sup_{\varepsilon>0} \left| \int_{\mathbb{R}^{n}\setminus B(0,\varepsilon)}|y|^{-n} \Omega(y/|y|)\,f(x-y) \mathrm{d}y\right|,$$where $\Omega$ is a function in $L^\infty (\mathbb{S}^{n-1})$ with vanishing integral. It is well known that the operator is bounded on $L^p$ for $1<p<\infty ,$ but it is an open question whether it is of the weak type 1-1. We show that $T^{\#}_\Omega$ is bounded from $L(\log \log L)^{2+\varepsilon }$ to $L^{1,\infty }$ locally.