Characterization of theWeak-Type Boundedness of the Hilbert Transform onWeighted Lorentz Spaces

摘要

We characterize the weak-type boundedness of the Hilbert transform H on weighted Lorentz spaces Λ_u~p(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 L~p(u) and Muckenhoupt weights A_p, and the theory on classical Lorentz spaces Λ~p(w) and Ari?o- Muckenhoupt weights B_p.