Non-extendability of the finite Hilbert transform

Guillermo P. Curbera; Susumu Okada; Werner J. Ricker

Non-extendability of the finite Hilbert transform

Functional Analysis 2021-05-14 v2

Authors: Guillermo P. Curbera , Susumu Okada , Werner J. Ricker

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.

Cite

@article{arxiv.1911.08375,
  title  = {Non-extendability of the finite Hilbert transform},
  author = {Guillermo P. Curbera and Susumu Okada and Werner J. Ricker},
  journal= {arXiv preprint arXiv:1911.08375},
  year   = {2021}
}

Comments

8 pages. Minor changes from previous version. This is the final version, to be published in Monatshefte f\"ur Mathematik

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