Rellich identities for the Hilbert transform

TL;DR

This paper establishes new identities for the Hilbert transform using Rellich identities and conformal maps, leading to improved weighted $L^2$ estimates and a sharp norm bound in weighted spaces.

Contribution

It introduces novel Hilbert transform identities involving conformal maps and Rellich identities, providing sharp weighted $L^2$ bounds and advancing understanding of the transform's operator norm.

Findings

New $L^2$-weighted estimates for the Hilbert transform

A sharp bound for the Hilbert transform norm in weighted $L^2$ spaces

Connections to the Helson-Szeg"o theorem

Abstract

We prove Hilbert transform identities involving conformal maps via the use of Rellich identity and the solution of the Neumann problem in a graph Lipschitz domain in the plane. We obtain as consequences new $L^2$-weighted estimates for the Hilbert transform, including a sharp bound for its norm as a bounded operator in weighted $L^2$ in terms of a weight constant associated to the Helson-Szeg\"o theorem.