An extension of the Bianchi-Egnell stability estimate to Bakry, Gentil, and Ledoux's generalization of the Sobolev inequality to continuous dimensions

作者：Seuffert Francis^*

来源：Journal of Functional Analysis, 2017, 273(10): 3094-3149.

DOI：10.1016/j.jfa.2017.07.001

• Stability estimate
• Weighted Sobolev inequality
• Concentration compactness

This paper extends a stability estimate of the Sobolev Inequality established by Bianchi and Egnell in [3]. Bianchi and Egnell's Stability Estimate answers the question raised by H. Brezis and E.H. Lieb in [5]: "Is there a natural way to bound vertical bar vertical bar del phi vertical bar vertical bar(2)(2) - C-N(2)vertical bar vertical bar phi vertical bar vertical bar(2)(2N/N-2) from below in terms of the 'distance' of phi 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 inequalities, is an active topic. See [9,11], and [12], for stability estimates involving Sobolev inequalities and [6,11], and [14] for stability estimates on other inequalities. 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 R+ x 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.

• 出版日期2017-11-15
• 单位rutgers