no. G1
Article de recherche - Analyse fonctionnelle, Équations aux dérivées partielles
On Gaussian interpolation inequalities
[Sur des inégalités d’interpolation Gaussiennes]
Brigati, Giovanni1 orcid ; Dolbeault, Jean1 orcid ; Simonov, Nikita2 orcid
1CEREMADE (CNRS UMR n ∘ 7534), PSL University, Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France
2LJLL (CNRS UMR n ∘ 7598) Sorbonne Université, 4 place Jussieu, 75005 Paris, France
Comptes Rendus. Mathématique, Tome 362 (2024) no. G1, 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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
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

Brigati, Giovanni  1   ; Dolbeault, Jean  1   ; Simonov, Nikita  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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {On {Gaussian} interpolation inequalities},
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Brigati, Giovanni; Dolbeault, Jean; Simonov, Nikita. On Gaussian interpolation inequalities. Comptes Rendus. Mathématique, Tome 362 (2024) no. G1, pp. 21-44. doi: 10.5802/crmath.488

[1] Arnold, Anton; Bartier, Jean-Philippe; Dolbeault, Jean Interpolation between logarithmic Sobolev and Poincaré inequalities, Commun. Math. Sci., Volume 5 (2007) no. 4, pp. 971-979 | MR | DOI | Zbl

[2] Arnold, Anton; Dolbeault, Jean Refined convex Sobolev inequalities, J. Funct. Anal., Volume 225 (2005) no. 2, pp. 337-351 | DOI | MR | Zbl

[3] Arnold, Anton; Markowich, Peter; Toscani, Giuseppe; Unterreiter, Andreas On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Commun. Partial Differ. Equations, Volume 26 (2001) no. 1-2, pp. 43-100 | DOI | MR | Zbl

[4] Bakry, Dominique; Émery, Michel Hypercontractivité de semi-groupes de diffusion, C. R. Math. Acad. Sci. Paris, Volume 299 (1984) no. 15, pp. 775-778 | MR | Zbl

[5] Bakry, Dominique; Émery, Michel Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84 (Lecture Notes in Mathematics), Volume 1123, Springer, 1985, pp. 177-206 | DOI | MR | Numdam | Zbl

[6] Bakry, Dominique; Émery, Michel Inégalités de Sobolev pour un semi-groupe symétrique, C. R. Math. Acad. Sci. Paris, Volume 301 (1985) no. 8, pp. 411-413 | MR | Zbl

[7] Bakry, Dominique; Gentil, Ivan; Ledoux, Michel Analysis and geometry of Markov diffusion operators, Grundlehren der Mathematischen Wissenschaften, 348, Springer, 2014, xx+552 pages | DOI | MR

[8] Beckner, William A generalized Poincaré inequality for Gaussian measures, Proc. Am. Math. Soc., Volume 105 (1989) no. 2, pp. 397-400 | DOI | MR | Zbl

[9] Beckner, William Sobolev inequalities, the Poisson semigroup, and analysis on the sphere 𝕊 n , Proc. Natl. Acad. Sci. USA, Volume 89 (1992) no. 11, pp. 4816-4819 | DOI | MR | Zbl

[10] Beckner, William Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. Math., Volume 138 (1993) no. 1, pp. 213-242 | DOI | MR | Zbl

[11] Beckner, William Geometric asymptotics and the logarithmic Sobolev inequality, Forum Math., Volume 11 (1999) no. 1, pp. 105-137 | DOI | MR | Zbl

[12] Beckner, William; Pearson, Michael On sharp Sobolev embedding and the logarithmic Sobolev inequality, Bull. Lond. Math. Soc., Volume 30 (1998) no. 1, pp. 80-84 | DOI | MR | Zbl

[13] Berger, Marcel; Gauduchon, Paul; Mazet, Edmond Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, 194, Springer, 1971, vii+251 pages | DOI | MR

[14] Bianchi, Gabriele; Egnell, Henrik A note on the Sobolev inequality, J. Funct. Anal., Volume 100 (1991) no. 1, pp. 18-24 | DOI | MR | Zbl

[15] Bidaut-Véron, Marie-Françoise; Véron, Laurent Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., Volume 106 (1991) no. 3, pp. 489-539 | DOI | MR | Zbl

[16] Bidaut-Véron, Marie-Françoise; Véron, Laurent Erratum: “Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations” [Invent. Math. 106 (1991), no. 3, 489–539], Invent. Math., Volume 112 (1993) no. 2, p. 445 | DOI | MR | Zbl

[17] Blanchet, Adrien; Bonforte, Matteo; Dolbeault, Jean; Grillo, Gabriele; Vázquez, Juan Luis Asymptotics of the fast diffusion equation via entropy estimates, Arch. Ration. Mech. Anal., Volume 191 (2009) no. 2, pp. 347-385 | DOI | MR | Zbl

[18] Brigati, Giovanni; Dolbeault, Jean; Simonov, Nikita Logarithmic Sobolev and interpolation inequalities on the sphere: Constructive stability results, Ann. Inst. Henri Poincaré, Anal. Non Linéaire (2023) (33 pages), online-first, https://doi.org/10.4171/AIHPC/106 | DOI

[19] Carrillo, José A.; Jüngel, Ansgar; Markowich, Peter A.; Toscani, Giuseppe; Unterreiter, Andreas Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., Volume 133 (2001) no. 1, pp. 1-82 | DOI | MR

[20] Chafaï, Djalil Entropies, convexity, and functional inequalities: on Φ-entropies and Φ-Sobolev inequalities, J. Math. Kyoto Univ., Volume 44 (2004) no. 2, pp. 325-363 | MR | Zbl

[21] Demange, Jérôme Des équations à diffusion rapide aux inégalités de Sobolev sur les modèles de la géométrie, Ph. D. Thesis, Université Paul Sabatier Toulouse 3 (2005)

[22] Demange, Jérôme Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature, J. Funct. Anal., Volume 254 (2008) no. 3, pp. 593-611 | DOI | MR | Zbl

[23] Dolbeault, Jean; Esteban, Maria J. Improved interpolation inequalities and stability, Adv. Nonlinear Stud., Volume 20 (2020) no. 2, pp. 277-291 | DOI | MR | Zbl

[24] Dolbeault, Jean; Esteban, Maria J.; Figalli, Alessio; Frank, Rupert L.; Loss, Michael Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence (2022) | arXiv

[25] Dolbeault, Jean; Esteban, Maria J.; Loss, Michael Nonlinear flows and rigidity results on compact manifolds, J. Funct. Anal., Volume 267 (2014) no. 5, pp. 1338-1363 | DOI | MR | Zbl

[26] Dolbeault, Jean; Esteban, Maria J.; Loss, Michael Interpolation inequalities on the sphere: linear vs. nonlinear flows, Ann. Fac. Sci. Toulouse, Math., Volume 26 (2017) no. 2, pp. 351-379 | DOI | MR | Numdam | Zbl

[27] Dolbeault, Jean; Li, Xingyu φ-entropies: convexity, coercivity and hypocoercivity for Fokker–Planck and kinetic Fokker-Planck equations, Math. Models Methods Appl. Sci., Volume 28 (2018) no. 13, pp. 2637-2666 | DOI | MR | Zbl

[28] Dolbeault, Jean; Nazaret, Bruno; Savaré, Giuseppe On the Bakry–Emery criterion for linear diffusions and weighted porous media equations, Commun. Math. Sci., Volume 6 (2008) no. 2, pp. 477-494 | MR | DOI | Zbl

[29] Gianazza, Ugo; Savaré, Giuseppe; Toscani, Giuseppe The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal., Volume 194 (2009) no. 1, pp. 133-220 | DOI | MR | Zbl

[30] Gigli, Nicola; Ketterer, Christian; Kuwada, Kazumasa; Ohta, Shin-ichi Rigidity for the spectral gap on RCD(K,∞)-spaces, Am. J. Math., Volume 142 (2020) no. 5, pp. 1559-1594 | DOI | MR | Zbl

[31] Grisvard, Pierre Elliptic problems in nonsmooth domains, Classics in Applied Mathematics, 69, Society for Industrial and Applied Mathematics, 2011, xx+410 pages (reprint of the 1985 original edition, with a foreword by Susanne C. Brenner) | DOI | MR

[32] Gross, Leonard Logarithmic Sobolev inequalities, Am. J. Math., Volume 97 (1975) no. 4, pp. 1061-1083 | DOI | MR

[33] Jüngel, Ansgar Entropy methods for diffusive partial differential equations, SpringerBriefs in Mathematics, Springer, 2016, viii+139 pages | DOI | MR

[34] Latała, R.; Oleszkiewicz, K. Between Sobolev and Poincaré, Geometric aspects of functional analysis (Lecture Notes in Mathematics), Volume 1745, Springer, 2000, pp. 147-168 | DOI | MR | Zbl

[35] Lichnerowicz, André Géométrie des groupes de transformations, Travaux et Recherches Mathématiques, 3, Dunod, Paris, 1958, ix+193 pages | MR

[36] Licois, Jean Rene; Véron, Laurent A class of nonlinear conservative elliptic equations in cylinders, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 26 (1998) no. 2, pp. 249-283 | MR | Numdam | Zbl

[37] Lieb, Elliott H. Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. Math., Volume 118 (1983) no. 2, pp. 349-374 | DOI | MR | Zbl

[38] Liero, Matthias; Mielke, Alexander Gradient structures and geodesic convexity for reaction-diffusion systems, Philos. Trans. R. Soc. Lond., Ser. A, Volume 371 (2013) no. 2005, 20120346, 28 pages | DOI | MR | Zbl

[39] McKean, Henry P. Geometry of differential space, Ann. Probab., Volume 1 (1973), pp. 197-206 | DOI | MR | Zbl

[40] Nelson, Edward The free Markoff field, J. Funct. Anal., Volume 12 (1973), pp. 211-227 | DOI | MR | Zbl

[41] Obata, Morio The conjectures on conformal transformations of Riemannian manifolds, J. Differ. Geom., Volume 6 (1971), pp. 247-258 | MR | Zbl

[42] Vázquez, Juan Luis The porous medium equation. Mathematical theory, Oxford Mathematical Monographs, Oxford University Press, 2007 | Zbl | Numdam

[43] Vershik, Anatoli M. Does there exist a Lebesgue measure in the infinite-dimensional space?, Proc. Steklov Inst. Math., Volume 259 (2007) no. 1, pp. 248-272 | DOI | Zbl

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