Boundedness of the Hilbert transform on weighted Lorentz spaces

TL;DR

This paper characterizes the boundedness of the Hilbert transform and its maximal operator on weighted Lorentz spaces, providing necessary and sufficient conditions for weak and strong types, with applications to Lorentz spaces $L^{p,q}(u)$.

Contribution

It offers a complete characterization of the boundedness of Hilbert transform operators on weighted Lorentz spaces, extending previous results to new weighted settings.

Findings

Complete characterization of weak-type boundedness for $H$ and $H^*$ when $u ext{ in }A_1$

Characterization of strong-type boundedness for $p>1$

Applications to Lorentz spaces $L^{p,q}(u)$

Abstract

We study the boundedness of the Hilbert transform $H$ and the Hilbert maximal operator $H^*$ on weighted Lorentz spaces $\Lambda^p_u(w)$. 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 $u\in A_1$. 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.