Comptes Rendus
Research article - Functional analysis, Partial differential equations
On Gaussian interpolation inequalities
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 21-44.

This paper is devoted to Gaussian interpolation inequalities with endpoint cases corresponding to the Gaussian Poincaré and the logarithmic Sobolev inequalities, seen as limits in large dimensions of Gagliardo–Nirenberg–Sobolev inequalities on spheres. Entropy methods are investigated using not only heat flow techniques but also nonlinear diffusion equations as on spheres. A new stability result is established for the Gaussian measure, which is directly inspired by recent results for spheres.

Cet article est consacré à des inégalités d’interpolation Gaussiennes, avec comme cas extrêmes l’inégalité de Poincaré Gaussienne et l’inégalité de Sobolev logarithmique, vues comme limites en grandes dimensions des inégalités de Gagliardo–Nirenberg–Sobolev sur les sphères. Les méthodes d’entropie sont abordées en utilisant non seulement des techniques basée sur l’équation de la chaleur mais aussi sur des équations de diffusion non-linéaires, comme pour les sphères. Un nouveau résultat de stabilité est établi pour les mesures Gaussiennes, qui s’inspire directement de résultats récents sur les sphères.

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DOI: 10.5802/crmath.488
Classification: 26D10, 46T12, 46E35, 39B62, 43A90, 49J40, 58E35
Keywords: logarithmic Sobolev inequality, Gagliardo–Nirenberg–Sobolev inequalities, Gaussian Poincaré inequality, sphere, spectral decomposition, entropy methods, Ornstein–Uhlenbeck operator, nonlinear diffusions, improved inequalities, stability
Giovanni Brigati 1; Jean Dolbeault 1; Nikita Simonov 2

1 CEREMADE (CNRS UMR n ∘  7534), PSL University, Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France
2 LJLL (CNRS UMR n ∘  7598) Sorbonne Université, 4 place Jussieu, 75005 Paris, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Giovanni Brigati; Jean Dolbeault; Nikita Simonov. On Gaussian interpolation inequalities. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 21-44. doi : 10.5802/crmath.488. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.488/

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