The L(log L)e endpoint estimate for maximal singular integral operators

Abstract

We prove in this paper the following estimate for the maximal operator T ∗ associated to the singular integral operator T: kT ∗ fkL 1,∞ (w) . 1 ǫ Z Rn | f(x)| ML(log L) ǫ (w)(x) dx, w ≥ 0, 0 < ǫ ≤ 1. This follows from the sharp L p estimate kT ∗ fkLp (w) . p ′ ( 1 δ ) 1/p ′ kfk L p (ML(log L) p−1+δ (w)), 1 < p < ∞, w ≥ 0, 0 < δ ≤ 1. As as a consequence we deduce that kT ∗ fkL 1,∞ (w) . [w]A1 log(e + [w]A∞ ) Z Rn | f | w dx, extending the endpoint results obtained in [LOP] A. Lerner, S. Ombrosi and C. Pérez, A1 bounds for Calderón-Zygmund operators related to a problem of Muckenhoupt and Wheeden, Mathematical Research Letters (2009), 16, 149–156 and [HP] T. Hytönen and C. Pérez, Sharp weighted bounds involving A∞, Analysis and P.D.E. 6 (2013), 777–818. DOI 10.2140/apde.2013.6.777 to maximal singular integrals. Another consequence is a quantitative two weight bump estimate.