Sharp Stability of Log-Sobolev and Moser-Onofri inequalities on the Sphere

TL;DR

This paper establishes the stability of endpoint conformally invariant Sobolev inequalities on the sphere, specifically proving global stability for the log-Sobolev inequality and non-existence of global stability for the Moser-Onofri inequality.

Contribution

It provides the first proof of stability for Beckner's log-Sobolev and Moser-Onofri inequalities on the sphere, highlighting differences between global and local stability constants.

Findings

Global stability constant for log-Sobolev inequality is strictly smaller than the local stability constant.

No global stability exists for Moser-Onofri inequality on the sphere.

Sharp stability results are established for endpoint conformally invariant inequalities.

Abstract

In this paper, we are concerned with the stability problem for endpoint conformally invariant cases of the Sobolev inequality on the sphere $\mathbb{S}^n$. Namely, we will establish the stability for Beckner's log-Sobolev inequality and Beckner's Moser-Onofri inequality on the sphere. We also prove that the sharp constant of global stability for the log-Sobolev inequality on the sphere $\mathbb{S}^n$ must be strictly smaller than the sharp constant of local stability for the same inequality. Furthermore, we also derive the non-existence of the global stability for Moser-Onofri inequality on the sphere $\mathbb{S}^n$.