Improved interpolation inequalities and stability

TL;DR

This paper revisits optimal interpolation inequalities on the sphere, employing carré du champ methods to improve inequalities, estimate constants, and analyze stability and symmetry breaking, with some results applicable in Euclidean space.

Contribution

It introduces improved interpolation inequalities with stability estimates using carré du champ methods, providing lower bounds on constants and insights into symmetry breaking.

Findings

Improved bounds for optimal constants in interpolation inequalities.

Stability estimates for optimal functions.

Results applicable to Euclidean space via stereographic projection.

Abstract

For exponents in the subcritical range, we revisit some optimal interpolation inequalities on the sphere with carr\'e du champ methods and use the remainder terms to produce improved inequalities. The method provides us with lower estimates of the optimal constants in the symmetry breaking range and stability estimates for the optimal functions. Some of these results can be reformulated in the Euclidean space using the stereographic projection.