On Gaussian interpolation inequalities

TL;DR

This paper explores Gaussian interpolation inequalities, connecting them to classical inequalities on spheres, and introduces new stability results for the Gaussian measure using entropy and diffusion methods.

Contribution

It establishes a new stability result for Gaussian measures inspired by recent sphere inequalities, and extends entropy methods with nonlinear diffusion equations.

Findings

New stability result for Gaussian measure

Connection between Gaussian and sphere inequalities

Extension of entropy methods with nonlinear diffusion

Abstract

This paper is devoted to Gaussian interpolation inequalities with endpoint cases corresponding to the Gaussian Poincar\'e and the logarithmic Sobolev inequalities, seen as limits in large dimensions of Gagliardo-Nirenberg-Sobolev inequalities on spheres. Entropy methods are investigated using not only heat flow techniques but also nonlinear diffusion equations as on spheres. A new stability result is established for the Gaussian measure, which is directly inspired by recent results for spheres.