Rellich identities for the Hilbert transform

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 weighted estimates for the Hilbert transform, including a sharp bound for its norm as a bounded operator in weighted in terms of a weight constant associated to the Helson-Szegö theorem.