Rellich identities for the Hilbert transform

MarÃ­a J. Carro; Virginia Naibo; MarÃ­a Soria-Carro

doi:10.1016/j.jfa.2023.110271

paper

Rellich identities for the Hilbert transform

arXiv:2405.02725 · doi:10.1016/j.jfa.2023.110271

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-SzegÃ¶ theorem.

15 pages, 2 figures