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

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 ‖∇φ‖22−CN2‖φ‖2NN−22 from below in terms of the 'distance' of φ 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+×Rn, 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.