Sharp interpolation inequalities on the sphere : new methods and consequences

TL;DR

This paper explores sharp interpolation inequalities on the sphere, connecting spectral properties of the Laplace-Beltrami operator with optimal constants, and introduces new methods involving symmetrization and ultraspherical analysis.

Contribution

It presents novel methods and insights into sharp interpolation inequalities on the sphere, linking spectral theory with inequality optimization.

Findings

Established connections between optimal constants and spectral properties.

Developed proofs using symmetrization and ultraspherical techniques.

Unified various inequalities in a comprehensive framework.

Abstract

These notes are devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension two and higher interpolate between Poincar\'e, logarithmic Sobolev and critical Sobolev (Onofri in dimension two) inequalities. We emphasize the connexion between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere. We shall address a series of related observations and give proofs based on symmetrization and the ultraspherical setting.