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 \(\varLambda^{p}_{u}(w)\), 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 Λp(w) and Arino-Muckenhoupt weights Bp.