Boundedness of the Hilbert transform on weighted Lorentz spaces

We study the boundedness of the Hilbert transform $H$ and the Hilbert maximal operator $H^{\ast }$ on weighted Lorentz spaces ${\mathrm{\Lambda }}_u^p(w)$. 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 $H^{\ast }$, whenever $u\in A_1$. 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.