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

We prove in this paper the following estimate for the maximal operator $T^*$ associated to the singular integral operator $T$: $\|T^*f\|_{L^{1,\infty}(w)} \lesssim \frac{1}ε \int_{\mathbb{R}^n} |f(x)| M_{L(\log L)^ε} (w)(x)dx$, for $w\geq 0, 0<ε\leq 1.$ This follows from the sharp $L^p$ estimate $\|T^*f \|_{ L^{p}(w) } \lesssim p' (\frac{1}δ)^{1/p'} \|f \|_{L^{p}(M_{ L(\log L)^{p-1+δ}} (w))}$, for $1<p<\infty, w\geq 0, 0<δ\leq 1.$ 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.

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