An Extension of the Bianchi-Egnell Stability Estimate to Bakry, Gentil, and Ledoux's Generalization of the Sobolev Inequality to Continuous Dimensions

TL;DR

This paper generalizes the Bianchi-Egnell stability estimate for the Sobolev Inequality to a new form that applies to continuous dimensions, building on recent extensions by Bakry, Gentil, and Ledoux.

Contribution

It extends the classical stability estimate to a Sobolev Inequality in continuous dimensions, providing a quantitative stability result for this generalized setting.

Findings

Extended stability estimate to continuous dimensional Sobolev Inequality.

Confirmed stability bounds hold in the generalized setting.

Connected stability analysis with recent sharp inequality extensions.

Abstract

This paper extends a stability estimate of the Sobolev Inequality established by G. Bianchi and H. Egnell in their paper "A note on the Sobolev Inequality." Bianchi and Egnell's Stability Estimate answers the question raised by H. Brezis and E. H. Lieb: "Is there a natural way to bound $\| \nabla \varphi \|_2^2 - C_N^2 \| \varphi \|_\frac{2N}{N-2}^2$ from below in terms of the 'distance' of $\varphi$ from the manifold of optimizers in the Sobolev Inequality?" Establishing stability estimates - also known as quantitative versions of sharp inequalities - of other forms of the Sobolev Inequality, as well as other sharp inequalities, is an active topic. In this paper, we extend Bianchi and Egnell's Stability Estimate to a Sobolev Inequality for "continuous dimensions." Bakry, Gentil, and Ledoux have recently proved a sharp extension of the Sobolev Inequality for functions on $\mathbb{R}_+ \times \mathbb{R}^n$, which can be considered as an extension to "continuous dimensions." V. H. Nguyen determined all cases of equality. The present paper extends the Bianchi-Egnell stability analysis for the Sobolev Inequality to this "continuous dimensional" generalization.