Characterization of the weak-type boundedness of the Hilbert transform on weighted Lorentz spaces

Abstract

We characterize the weak-type boundedness of the Hilbert transform H on weighted Lorentz spaces Lambda_p(u,w) , with p>0, in terms of some geometric conditions on the weights u and w and the weak-type boundedness of the Hardy-Littlewood maximal operator on the same spaces. Our results recover simultaneously the theory of the boundedness of H on weighted Lebesgue spaces Lp(u) and Muckenhoupt weights Ap , and the theory on classical Lorentz spaces Lambda_p(w) and Ariño-Muckenhoupt weights Bp .