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., $L^p$ for 1 < $p$ < ∞), a setting which excludes $L^1$. Our aim is to go beyond boundedness and to address the case $X$ = $L^1$. For this, we need to consider $T$ as an unbounded operator on $L^1$. 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 $L^1$, fails to be an ideal and properly contains the Zygmund space $L$log$L$ (which is the largest ideal of functions that $T$ maps boundedly into $L^1$).