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

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…”

“…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…”