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

Abstract

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 2 2 − C 2 N ϕ 2 2N N −2 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 + × 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. * Work partially supported by U.S. National Science Foundation grants DMS 1201354 and DMS 1501007 2 , where β < 2.Recently, Bakry, Gentil, and Ledoux proved a sharp extension of the Sobolev inequality to "fractional dimensions," and showed how it relates to certain optimal Gagliardo-Nirenberg inequalities. V. H. Nguyen has determined all of the extremals in the version of the inequality for real-valued functions. The goal of the present paper is to prove an analogue of Theorem 1.2 for this extended Sobolev inequality. Actually, the case we treat is more general, because we generalize Nguyen's classification of extremals from real-valued functions to complex-valued functions. We then prove the analogue of Theorem 1.2 for this generalization of Bakry, Gentil, and Ledoux's Theorem with classification of extremals for complex-valued functions. This is notable, because Bianchi and Egnell prove their stability estimate for real-valued functions only, while our stability estimate is for complex-valued functions. This is one of the aspects that make our proof more intricate than Bianchi and Egnell's, but it is hardly the most notable or the most difficult aspect to deal with. Some of the steps in the proof of our extension of the Bianchi-Egnell Stability Estimate are a fairly direct adaptation of steps in Bianchi and Egnell's proof. Others are not. To help highlight these differences, we outline a proof of Theorem 1.2 based upon the steps of the proof to our extension of the Bianchi-Egnell Stability Estimate. This outline is provided in subsection 1.5. In the outline, we point out where our approach differs from Bianchi and Egnell's, and in particular, which parts require new arguments.

Duality and Stability for Functional Inequalities

“…The method used by Bianchi and Egnell to prove (3.8) can be elaborated to give a quantitative stability bound for the Sobolev inequality, but only locally, near to X 0 : As discussed above, their local estimate comes from an eigenvalue calculation and control of remainder terms in a Taylor expansion, and no essential use of qualitative compactness arguments is made up to this point. Though they did not obtain quantitative control over the remainder terms in the Taylor expansion, this can be done using uniform convexity inequalities; see Section 2 of [18]. The optimal logarithmic HLS inequality on R 2 can be obtained from the HLS inequality by taking the limit α → 1; see [9].…”

The sharp Gagliardo–Nirenberg–Sobolev inequality in quantitative form

“…One of the main ingredients in our proof is the stability version of the weighted Sobolev inequality on half space R n+1 due to Seuffert [50]. The sharp weighted Sobolev inequality on half space was proved by the author in [46] by means of the mass transportation technique which generalizes one result of Bakry, Gentil and Ledoux in [2].…”

“…They also mentioned in their paper that their method can be used to obtain the stability results for whole family of GNS inequality (1.2). This was completely done in recent work of Seuffert [51] by using the technique of Carlen and Figalli and his stability version for the weighted Sobolev inequality on half space [50]. For 1 < t < n/(n − 2), denote 2(t) = 2(4t + n − nt)/(n + 2 + 2t − nt) and…”

“…Our method used in this paper is the modification of the one given by Carlen and Figalli [10] and Seuffert [51]. We combining the stability version of the weighted Sobolev inequality established in [50] by Seuffert and the dimension reduction argument of Bakry [2] to obtain Theorem 1.1. The main different between our proof and the one of Carlen and Figalli, and of Seuffert is that after applying the stability version of the weighted Sobolev inequality on the half space, we do not apply the weighted Sobolev inequality to the remainder term.…”

“…for some c ∈ R, a > 0 and x 0 ∈ R n . In [50], by adapting the proof of Bianchi and Egnell, Seuffert established a stability version of (1.15) as follows…”