Non-extendability of the finite Hilbert transform

Abstract

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

arXiv comment8 pages. Minor changes from previous version. This is the final version, to be published in Monatshefte für Mathematik