Parabolic methods for ultraspherical interpolation inequalities

TL;DR

This paper develops a parabolic flow approach to prove interpolation inequalities with explicit constants on the sphere, addressing symmetry breaking and regularity issues for weighted inequalities.

Contribution

It introduces the first complete parabolic proof for weighted interpolation inequalities using a nonlinear flow with regularization, within the ultraspherical framework.

Findings

Established explicit constants for inequalities on the sphere.

Provided a regularized nonlinear flow approach for weighted inequalities.

Addressed symmetry breaking issues in Caffarelli-Kohn-Nirenberg inequalities.

Abstract

The carr\'e du champ method is a powerful technique for proving interpolation inequalities with explicit constants in presence of a non-trivial metric on a manifold. The method applies to some classical Gagliardo-Nirenberg-Sobolev inequalities on the sphere, with optimal constants. Very nonlinear regimes close to the critical Sobolev exponent can be covered using nonlinear parabolic flows of porous medium or fast diffusion type. Considering power law weights is a natural question in relation with symmetry breaking issues for Caffarelli-Kohn-Nirenberg inequalities, but regularity estimates for a complete justification of the computation are missing. We provide the first example of a complete parabolic proof based on a nonlinear flow by regularizing the singularity induced by the weight. Our result is established in the simplified framework of a diffusion built on the ultraspherical operator, which amounts to reduce the problem to functions on the sphere with simple symmetry properties.