Stability of trace theorems on the sphere

TL;DR

This paper establishes stable versions of trace theorems on the sphere with optimal constants, providing detailed insights into near-extremisers and extending stability results to various contexts including $L^q$ spaces and Strichartz estimates.

Contribution

It introduces new stability results for trace theorems on the sphere, combining refined inequalities and duality techniques to enhance understanding of extremisers and stability in related inequalities.

Findings

Stable trace theorems with optimal constants on the sphere.

Stability results for trace into $L^q$ spaces for $q > 2$.

Extension of stability to Strichartz estimates for the kinetic transport equation.

Abstract

We prove stable versions of trace theorems on the sphere in $L^2$ with optimal constants, thus obtaining rather precise information regarding near-extremisers. We also obtain stability for the trace theorem into $L^q$ for $q > 2$, by combining a refined Hardy-Littlewood-Sobolev inequality on the sphere with a duality-stability result proved very recently by Carlen. Finally, we extend a local version of Carlen's duality theorem to establish local stability of certain Strichartz estimates for the kinetic transport equation.