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Duality and Stability for Functional Inequalities
Eric A. Carlen
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 2, p. 319-350

We develop a general framework for using duality to “transfer” stability results for a functional inequality to its dual inequality. As an application, we prove a stability bound for the Hardy–Littlewood–Sobolev inequality, which is related by duality, and the results proved here, to a stability inequality for the Sobolev inequality proved by Bianchi and Egnell, and extended by Chen, Frank and Weth. We also discuss how the results proved here can be combined with the proof of functional inequalities by means of flows to prove stability bounds with computable constants.

Nous développons un cadre général pour l’utilisation d’une dualité permettant de “transférer” des résultats de stabilité pour une inégalité fonctionnelle à son inégalité duale. Comme application, nous donnons un résultat de stabilité pour l’inégalité de Hardy–Littlewood–Sobolev qui est lié, par la dualité et les résultats prouvés ici, à une inégalité de stabilité pour l’inégalité de Sobolev prouvée par Bianchi et Egnell, et prolongée par Chen, Frank et Weth. Nous discutons également comment les résultats donnés ici peuvent étre combinés à la preuve d’inégalités fonctionnelles utilisant des flots, afin de démontrer les limites de stabilité avec des constantes calculables.

Published online : 2017-04-13
DOI : https://doi.org/10.5802/afst.1535
Classification:  81V99,  82B10,  94A17
Keywords: uniform convexity, entropy
@article{AFST_2017_6_26_2_319_0,
     author = {Eric A. Carlen},
     title = {Duality and Stability for Functional Inequalities},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {2},
     year = {2017},
     pages = {319-350},
     doi = {10.5802/afst.1535},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2017_6_26_2_319_0}
}
Carlen, Eric A. Duality and Stability for Functional Inequalities. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 2, pp. 319-350. doi : 10.5802/afst.1535. afst.centre-mersenne.org/item/AFST_2017_6_26_2_319_0/

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