The $L(\log L)^ε$ endpoint estimate for maximal singular integral operators (1503.04008v1)
Abstract: 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 $Lp$ 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.