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Logarithmic Sobolev inequalities on non-isotropic Heisenberg groups. (English) Zbl 1487.58023

Summary: We study logarithmic Sobolev inequalities with respect to a heat kernel measure on finite-dimensional and infinite-dimensional Heisenberg groups. Such a group is the simplest non-trivial example of a sub-Riemannian manifold. First we consider logarithmic Sobolev inequalities on non-isotropic Heisenberg groups. These inequalities are considered with respect to the hypoelliptic heat kernel measure, and we show that the logarithmic Sobolev constants can be chosen to be independent of the dimension of the underlying space. In this setting, a natural Laplacian is not an elliptic but a hypoelliptic operator. The argument relies on comparing logarithmic Sobolev constants for the three-dimensional non-isotropic and isotropic Heisenberg groups, and tensorization of logarithmic Sobolev inequalities in the sub-Riemannian setting. Furthermore, we apply these results in an infinite-dimensional setting and prove a logarithmic Sobolev inequality on an infinite-dimensional Heisenberg group modeled on an abstract Wiener space.

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
22E30 Analysis on real and complex Lie groups
22E66 Analysis on and representations of infinite-dimensional Lie groups
35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
35K08 Heat kernel
35R03 PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.
60J65 Brownian motion
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[1] Bakry, Dominique; Baudoin, Fabrice; Bonnefont, Michel; Chafaï, Djalil, On gradient bounds for the heat kernel on the Heisenberg group, J. Funct. Anal., 255, 8, 1905-1938 (2008), MR 2462581 (2010m:35534) · Zbl 1156.58009
[2] Bakry, Dominique; Gentil, Ivan; Ledoux, Michel, Analysis and Geometry of Markov Diffusion Operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 348 (2014), Springer: Springer Cham, MR 3155209 · Zbl 1376.60002
[3] Baudoin, Fabrice; Bonnefont, Michel, Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality, J. Funct. Anal., 262, 6, 2646-2676 (2012), MR 2885961 · Zbl 1254.58004
[4] Baudoin, Fabrice; Feng, Qi, Log-Sobolev inequalities on the horizontal path space of a totally geodesic foliation (2015), arxiv preprint
[5] Baudoin, Fabrice; Garofalo, Nicola, Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries, J. Eur. Math. Soc., 19, 1, 151-219 (2017), MR 3584561 · Zbl 1359.53018
[6] Baudoin, Fabrice; Gordina, Maria; Melcher, Tai, Quasi-invariance for heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg groups, Trans. Am. Math. Soc., 365, 8, 4313-4350 (2013), MR 3055697 · Zbl 1294.35182
[7] Beals, Richard; Gaveau, Bernard; Greiner, Peter C., Hamilton-Jacobi theory and the heat kernel on Heisenberg groups, J. Math. Pures Appl. (9), 79, 7, 633-689 (2000), MR 1776501 (2001g:35047) · Zbl 0959.35035
[8] Bogachev, Vladimir I., Gaussian Measures, Mathematical Surveys and Monographs, vol. 62 (1998), American Mathematical Society: American Mathematical Society Providence, RI, MR MR1642391 (2000a:60004) · Zbl 0913.60035
[9] Bonfiglioli, A.; Lanconelli, E.; Uguzzoni, F., Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer Monographs in Mathematics (2007), Springer: Springer Berlin, MR 2363343 · Zbl 1128.43001
[10] Bonnefont, Michel; Chafaï, Djalil; Herry, Ronan, On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group, Ann. Fac. Sci. Toulouse Math. (6), 29, 2, 335-355 (2020), MR 4150544 · Zbl 1451.35009
[11] Corwin, Lawrence J.; Greenleaf, Frederick P., Basic theory and examples, (Representations of Nilpotent Lie Groups and Their Applications. Part I. Representations of Nilpotent Lie Groups and Their Applications. Part I, Cambridge Studies in Advanced Mathematics, vol. 18 (1990), Cambridge University Press: Cambridge University Press Cambridge), MR 1070979 (92b:22007) · Zbl 0704.22007
[12] Dagher, Esther Bou; Zegarliński, Bogusław, Coercive inequalities in higher-dimensional anisotropic Heisenberg group, Anal. Math. Phys., 12, 3 (2022) · Zbl 1489.53043
[13] Driver, Bruce K.; Eldredge, Nathaniel; Melcher, Tai, Hypoelliptic heat kernels on infinite-dimensional Heisenberg groups, Trans. Am. Math. Soc., 368, 2, 989-1022 (2016), MR 3430356 · Zbl 1341.58017
[14] Driver, Bruce K.; Gordina, Maria, Heat kernel analysis on infinite-dimensional Heisenberg groups, J. Funct. Anal., 255, 9, 2395-2461 (2008), MR MR2473262 · Zbl 1163.43005
[15] Driver, Bruce K.; Gross, Leonard; Saloff-Coste, Laurent, Holomorphic functions and subelliptic heat kernels over Lie groups, J. Eur. Math. Soc., 11, 5, 941-978 (2009), MR 2538496 (2010h:32052) · Zbl 1193.32023
[16] Driver, Bruce K.; Gross, Leonard; Saloff-Coste, Laurent, Growth of Taylor coefficients over complex homogeneous spaces, Tohoku Math. J. (2), 62, 3, 427-474 (2010), MR 2742018 · Zbl 1208.43002
[17] Driver, Bruce K.; Melcher, Tai, Hypoelliptic heat kernel inequalities on the Heisenberg group, J. Funct. Anal., 221, 340-365 (2005) · Zbl 1071.22005
[18] Eldredge, Nathaniel, Gradient estimates for the subelliptic heat kernel on H-type groups, J. Funct. Anal., 258, 2, 504-533 (2010), MR 2557945 (2011d:35217) · Zbl 1185.43004
[19] Frank, Rupert L.; Lieb, Elliott H., Sharp constants in several inequalities on the Heisenberg group, Ann. Math. (2), 176, 1, 349-381 (2012), MR 2925386 · Zbl 1252.42023
[20] Fraser, A. J., An \((n + 1)\)-fold Marcinkiewicz multiplier theorem on the Heisenberg group, Bull. Aust. Math. Soc., 63, 1, 35-58 (2001), MR 1812307 · Zbl 0979.43004
[21] Gordina, M.; Melcher, T., A subelliptic Taylor isomorphism on infinite-dimensional Heisenberg groups, Probab. Theory Relat. Fields, 155, 379-426 (2013) · Zbl 1260.43005
[22] Gordina, Maria, An Application of a Functional Inequality to Quasi-Invariance in Infinite Dimensions, 251-266 (2017), Springer New York: Springer New York New York, NY · Zbl 1384.39016
[23] Gordina, Maria; Laetsch, Thomas, Sub-Laplacians on sub-Riemannian manifolds, Potential Anal., 44, 4, 811-837 (2016), MR 3490551 · Zbl 1338.35447
[24] Gross, Leonard, Logarithmic Sobolev inequalities, Am. J. Math., 97, 4, 1061-1083 (1975), MR MR0420249 (54 #8263) · Zbl 0318.46049
[25] Gross, Leonard, Logarithmic Sobolev inequalities on Lie groups, Ill. J. Math., 36, 3, 447-490 (1992), MR 1161977 (93i:22012) · Zbl 0761.46019
[26] Gross, Leonard, Logarithmic Sobolev inequalities and contractivity properties of semigroups, (Dirichlet Forms. Dirichlet Forms, Varenna 1992. Dirichlet Forms. Dirichlet Forms, Varenna 1992, Lecture Notes in Math., vol. 1563 (1993), Springer: Springer Berlin), 54-88, MR MR1292277 (95h:47061) · Zbl 0812.47037
[27] Guionnet, A.; Zegarlinski, B., Lectures on logarithmic Sobolev inequalities, (Séminaire de Probabilités, XXXVI. Séminaire de Probabilités, XXXVI, Lecture Notes in Math., vol. 1801 (2003), Springer: Springer Berlin), 1-134, MR 1971582 · Zbl 1125.60111
[28] Hall, Brian C., An elementary introduction, (Lie Groups, Lie Algebras, and Representations. Lie Groups, Lie Algebras, and Representations, Graduate Texts in Mathematics, vol. 222 (2003), Springer-Verlag: Springer-Verlag New York), MR 1997306 (2004i:22001) · Zbl 1026.22001
[29] Hebisch, W.; Zegarliński, B., Coercive inequalities on metric measure spaces, J. Funct. Anal., 258, 3, 814-851 (2010), MR 2558178 · Zbl 1189.26032
[30] Hörmander, Lars, Hypoelliptic second order differential equations, Acta Math., 119, 147-171 (1967), MR 0222474 (36 #5526) · Zbl 0156.10701
[31] Hu, Jun-Qi; Li, Hong-Quan, Gradient estimates for the heat semigroup on H-type groups, Potential Anal., 33, 4, 355-386 (2010), MR 2726903 (2012c:43010) · Zbl 1206.35241
[32] Kuo, Hui Hsiung, Gaussian Measures in Banach Spaces, Lecture Notes in Mathematics, vol. 463 (1975), Springer-Verlag: Springer-Verlag Berlin, MR MR0461643 (57 #1628) · Zbl 0306.28010
[33] Li, Hong-Quan, Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg, J. Funct. Anal., 236, 2, 369-394 (2006), MR MR2240167 (2007d:58045) · Zbl 1106.22009
[34] Li, Hong-Quan, Estimations asymptotiques du noyau de la chaleur sur les groupes de Heisenberg, C. R. Math. Acad. Sci. Paris, 344, 8, 497-502 (2007), MR MR2324485 · Zbl 1114.22007
[35] Li, Hong-Quan; Zhang, Ye, Revisiting the heat kernel on isotropic and nonisotropic Heisenberg groups, Commun. Partial Differ. Equ., 44, 6, 467-503 (2019), MR 3946611 · Zbl 1471.58029
[36] Ming Ma, Zhi; Röckner, Michael, Introduction to the Theory of (Nonsymmetric) Dirichlet Forms, Universitext (1992), Springer-Verlag: Springer-Verlag Berlin, MR 1214375 · Zbl 0826.31001
[37] McDuff, Dusa; Salamon, Dietmar, Introduction to Symplectic Topology, Oxford Mathematical Monographs (1998), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York, MR 1698616 · Zbl 1066.53137
[38] Reed, Michael; Simon, Barry, Functional Analysis, Methods of Modern Mathematical Physics. I (1980), Academic Press Inc. [Harcourt Brace Jovanovich Publishers]: Academic Press Inc. [Harcourt Brace Jovanovich Publishers] New York, MR MR751959 (85e:46002) · Zbl 0459.46001
[39] Röckner, Michael; Schmuland, Byron, Tightness of general \(C_{1 , p}\) capacities on Banach space, J. Funct. Anal., 108, 1, 1-12 (1992), MR 1174156 · Zbl 0757.60058
[40] Schechtman, Gideon, Concentration results and applications, (Handbook of the Geometry of Banach Spaces, vol. 2 (2003), North-Holland: North-Holland Amsterdam), 1603-1634, MR 1999604 · Zbl 1057.46011
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