On Gaussian interpolation inequalities

Giovanni Brigati; Jean Dolbeault; Nikita Simonov

doi:10.5802/crmath.488

Research article - Functional analysis, Partial differential equations

On Gaussian interpolation inequalities

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

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 21-44.

Abstract
Résumé

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.

Received: 2023-02-08
Revised: 2023-02-23
Accepted: 2023-03-05
Published online: 2024-02-02

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

Author's affiliations:

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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     author = {Giovanni Brigati and Jean Dolbeault and Nikita Simonov},
     title = {On {Gaussian} interpolation inequalities},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {21--44},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.488},
     language = {en},
}

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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/

References
Cited by

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

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

[3] Anton Arnold; Peter Markowich; Giuseppe Toscani; Andreas Unterreiter 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] Dominique Bakry; Michel Émery Hypercontractivité de semi-groupes de diffusion, C. R. Math. Acad. Sci. Paris, Volume 299 (1984) no. 15, pp. 775-778 | MR | Zbl

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

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

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

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

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

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

[15] Marie-Françoise Bidaut-Véron; Laurent Véron 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] Marie-Françoise Bidaut-Véron; Laurent Véron 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] Adrien Blanchet; Matteo Bonforte; Jean Dolbeault; Gabriele Grillo; Juan Luis Vázquez 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] Giovanni Brigati; Jean Dolbeault; Nikita Simonov 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] José A. Carrillo; Ansgar Jüngel; Peter A. Markowich; Giuseppe Toscani; Andreas Unterreiter Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., Volume 133 (2001) no. 1, pp. 1-82 | DOI | MR

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

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

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

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

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

[29] Ugo Gianazza; Giuseppe Savaré; Giuseppe Toscani 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] Nicola Gigli; Christian Ketterer; Kazumasa Kuwada; Shin-ichi Ohta Rigidity for the spectral gap on $RCD (K, \infty)$ -spaces, Am. J. Math., Volume 142 (2020) no. 5, pp. 1559-1594 | DOI | MR | Zbl

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

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

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

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

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

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

[38] Matthias Liero; Alexander Mielke 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] Henry P. McKean Geometry of differential space, Ann. Probab., Volume 1 (1973), pp. 197-206 | DOI | MR | Zbl

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

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

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

[43] Anatoli M. Vershik 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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