Inversion of the Unbounded Finite Hilbert Transform on $L^1$

Abstract

The finite Hilbert transform T is a classical singular integral operator with its roots in aerodynamics, elasticity theory and image reconstruction. The setting has always been to consider T as acting in those rearrangement invariant spaces X over (−1, 1) which T maps boundedly into itself (e.g., Lp for 1 < p < ∞), a setting which excludes L1. Our aim is to go beyond boundedness and to address the case X = L1. For this, we need to consider T as an unbounded operator on L1. Is there a “suitable” domain for T? Yes. Remarkably, for T acting on this domain, we prove a full inversion theorem, together with refined versions of both the Parseval and Poincaré-Bertrand formulae, which are crucial results needed for the proof. This domain, a somewhat unusual space, turns out to be a rather extensive subspace of L1, fails to be an ideal and properly contains the Zygmund space LlogL (which is the largest ideal of functions that T maps boundedly into L1).