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

In this paper we obtain for $T^+$, a one-sided singular integral given by a Calder\'on-Zygmund kernel with support in $(-\infty,0)$, a $L^p(w)$ bound when $w\in A_1^+$. A. K. Lerner, S. Ombrosi, and C. P\'erez proved in [ "$A_{1}$ Bounds for Calder\'on-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.