Non-extendability of the finite Hilbert transform

TL;DR

This paper proves that the finite Hilbert transform cannot be extended beyond its current domain when acting on certain rearrangement invariant spaces with non-trivial Boyd indices, establishing its optimality.

Contribution

It demonstrates the non-extendability of the finite Hilbert transform on a broad class of rearrangement invariant spaces, clarifying its maximal domain.

Findings

Finite Hilbert transform is already optimally defined on these spaces.

No larger domain space exists where the transform can be extended.

The result applies to spaces with non-trivial Boyd indices.

Abstract

It is proved that the finite Hilbert transform $T\colon X\to X$, which acts continuously on every rearrangement invariant space $X$ on $(-1,1)$ having non-trivial Boyd indices, is already optimally defined. That is, $T\colon X\to X$ cannot be further extended, still taking its values in $X$, to any larger domain space.