Chord Sobolev inequalities
Abstract
The paper establishes a new family of sharp analytic inequalities. Together with the fractional Sobolev inequalities of Almgren and Lieb, they form a complete class of analytic inequalities, referred to as the chord Sobolev inequalities. A close connection between these inequalities and chord isoperimetric inequalities in integral geometry is established through a functional extension of chord power integrals. The limiting cases of the chord Sobolev inequalities are derived, one of which yields a logarithmic Sobolev-type inequality. Combined with the work of Bourgain, Brezis, and Mironescu, these results complete the picture of the chord Sobolev inequalities, including their endpoint cases.
Cite
@article{arxiv.2602.15160,
title = {Chord Sobolev inequalities},
author = {Fernanda M. Baêta and Xiaxing Cai},
journal= {arXiv preprint arXiv:2602.15160},
year = {2026}
}
Comments
This revised version improves the exposition and provides more detailed proofs where needed