Degenerate Stability of the Caffarelli-Kohn-Nirenberg Inequality along the Felli-Schneider Curve

TL;DR

This paper proves the optimal quartic stability of the Caffarelli-Kohn-Nirenberg inequality along the Felli-Schneider curve, revealing degenerate stability due to zero modes of the Hessian, and completes the leading order stability analysis.

Contribution

It establishes the best possible quartic remainder term for the CKN inequality along the FS curve, including degenerate cases with non-constant optimizers.

Findings

The remainder term is quartic in the distance to optimizers.

The quartic stability is optimal and due to zero modes of the Hessian.

First instance of degenerate stability for non-constant optimizers on a non-compact domain.

Abstract

We show that the Caffarelli-Kohn-Nirenberg (CKN) inequality holds with a remainder term that is quartic in the distance to the set of optimizers for the full parameter range of the Felli-Schneider (FS) curve. The fourth power is best possible. This is due to the presence of non-trivial zero modes of the Hessian of the deficit functional along the FS-curve. Following an iterated Bianchi-Egnell strategy, the heart of our proof is verifying a `secondary non-degeneracy condition'. Our result completes the stability analysis for the CKN-inequality to leading order started by Wei and Wu. Moreover, it is the first instance of degenerate stability for non-constant optimizers and for a non-compact domain.