Boundedness of the Hilbert transform on weighted Lorentz spaces

We study the boundedness of the Hilbert transform H and the Hilbert maximal operator H∗ on weighted Lorentz spaces Λup(w). We start by giving several necessary conditions that, in particular, lead us to the complete characterization of the weak-type boundedness of both H and H∗, whenever u∈A1. For the strong-type case, we also get the characterization of both operators when p>1. Applications to the case of Lorentz spaces Lp,q(u) are presented.