Chord Sobolev inequalities

TL;DR

This paper introduces a new family of sharp analytic inequalities called chord Sobolev inequalities, connecting them with isoperimetric inequalities and extending existing fractional Sobolev results.

Contribution

It establishes the complete class of chord Sobolev inequalities, including their limiting cases and connections to integral geometry, expanding the theoretical framework of Sobolev inequalities.

Findings

Derived a new family of sharp analytic inequalities.

Connected chord Sobolev inequalities with isoperimetric inequalities in integral geometry.

Included endpoint cases and a logarithmic Sobolev-type inequality.

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.