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 Λ u p ( 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 ∈ A 1 . For the strong-type case, we also get the characterization of both operators when p > 1 . Applications to the case of Lorentz spaces L p , q ( u ) are presented.