Weighted inequalities related to a Muckenhoupt and Wheeden problem for one-side singular integrals

Abstract

In this paper we obtain for $T^{+}$, a one-sided singular integral given by a Calderón-Zygmund kernel with support in $(-\infty ,0)$, a $L^p(w)$ bound when $w\in A_1^{+}$. A. K. Lerner, S. Ombrosi, and C. Pérez proved in [ "$A_1$ Bounds for Calderón-Zygmund operators related to a problem of Muckenhoupt and Wheeden", Math. Res. Lett. \textbf{16} (2009), no. 1, 149-156] that this bound is sharp with respect to $||w|{|}_{A_1}$ and $p$ . We also give a $L^{1,\infty }(w)$ estimate, for a related problem of Muckenhoupt and Wheeden for $w\in A_1^{+}$ . We improve the classical results, for one-sided singular integrals, by putting in the inequalities a wider class of weights.