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

Abstract

We prove in this paper the following estimate for the maximal operator $T^{\ast }$ associated to the singular integral operator $T$: $\Vert T^{\ast }f{\Vert }_{L^{1,\infty }(w)}\lesssim \frac{1}{\epsilon }{\int }_{{\mathbb{R}}^n}|f(x)|M_{L(\log L{)}^{\epsilon }}(w)(x)dx$, for $w\ge 0,0<\epsilon \le 1.$ This follows from the sharp $L^p$ estimate $\Vert T^{\ast }f{\Vert }_{L^p(w)}\lesssim p^{\prime }(\frac{1}{\delta }{)}^{1/p^{\prime }}\Vert f{\Vert }_{L^p(M_{L(\log L{)}^{p-1+\delta }}(w))}$, for $1<p<\infty ,w\ge 0,0<\delta \le 1.$ As as a consequence we deduce that $\Vert T^{\ast }f{\Vert }_{L^{1,\infty }(w)}\lesssim [w{]}_{A_1}\log (e+[w{]}_{A_{\infty }}){\int }_{{\mathbb{R}}^n}|f|wdx,$ extending the endpoint results obtained in [LOP] and [HP] to maximal singular integrals. Another consequence is a quantitative two weight bump estimate.

21 pages, final version, accepted for publication in J. Math. Anal. Appl