The finite Hilbert transform acting in the Zygmund space LlogL

TL;DR

This paper investigates the finite Hilbert transform's behavior on the Zygmund space LlogL, establishing its optimal domain and providing inversion formulas crucial for solving related integral equations.

Contribution

It extends classical inversion results of the finite Hilbert transform to the Zygmund space LlogL and proves the transform's optimal domain within LlogL for mapping into L^1.

Findings

Finite Hilbert transform maps LlogL continuously into L^1.

Inversion formulas are extended to the LlogL setting.

The transform cannot be extended beyond LlogL while maintaining its properties.

Abstract

The finite Hilbert transform T is a singular integral operator which maps the Zygmund space $LlogL:=LlogL(-1,1)$ continuously into $L^1:=L^1(-1,1)$. By extending the Parseval and Poincar\'e-Bertrand formulae to this setting, it is possible to establish an inversion result needed for solving the airfoil equation $T(f)=g$ whenever the data function $g$ lies in the range of $T$ within $L^1$ (shown to contain $LlogL$). Until now this was only known for $g$ belonging to the union of all $L^p$ spaces with $p>1$. It is established (due to a result of Stein) that $T$ cannot be extended to any domain space beyond $LlogL$ whilst still taking its values in $L^1$, i.e., $T:LlogL\to L^1$ is optimally defined.