The $L(\log L)^{\epsilon}$ endpoint estimate for maximal singular integral operators

We prove in this paper the following estimate for the maximal operator $T^*$ associated to the singular integral operator $T$: $\|T^*f\|_{L^{1,\infty}(w)} \lesssim \frac{1}{\epsilon} \int_{\mathbb{R}^n} |f(x)| M_{L(\log L)^{\epsilon}} (w)(x)dx$, for $w\geq 0, 0<\epsilon \leq 1.$ This follows from the sharp $L^p$ estimate $\|T^*f \|_{ L^{p}(w) } \lesssim p' (\frac{1}{\delta})^{1/p'} \|f \|_{L^{p}(M_{ L(\log L)^{p-1+\delta}} (w))}$, for $1<p<\infty, w\geq 0, 0<\delta \leq 1.$ As as a consequence we deduce that $\|T^*f\|_{L^{1,\infty}(w)} \lesssim [w]_{A_{1}} \log(e+ [w]_{A_{\infty}}) \int_{\mathbb{R}^n} |f| w dx,$ extending the endpoint results obtained in [LOP] and [HP] to maximal singular integrals. Another consequence is a quantitative two weight bump estimate.