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

We prove in this paper the following estimate for the maximal operator $T^{\ast }$ 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.$Thisfollowsfromthesharp$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.