Strong hypercontractivity and logarithmic Sobolev inequalities on stratified complex Lie groups

Eldredge, Nathaniel; Gross, Leonard; Saloff-Coste, Laurent

doi:10.1090/tran/7200

Strong hypercontractivity and logarithmic Sobolev inequalities on stratified complex Lie groups
HTML articles powered by AMS MathViewer

by Nathaniel Eldredge, Leonard Gross and Laurent Saloff-Coste PDF

Trans. Amer. Math. Soc. 370 (2018), 6651-6683 Request permission

Abstract:

We show that for a hypoelliptic Dirichlet form operator $A$ on a stratified complex Lie group, if the logarithmic Sobolev inequality holds, then a holomorphic projection of $A$ is strongly hypercontractive in the sense of Janson. This extends previous results of Gross to a setting in which the operator $A$ is not holomorphic.

References

D. Bakry, On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, New trends in stochastic analysis (Charingworth, 1994) World Sci. Publ., River Edge, NJ, 1997, pp. 43–75. MR 1654503
Dominique Bakry, L’hypercontractivité et son utilisation en théorie des semigroupes, Lectures on probability theory (Saint-Flour, 1992) Lecture Notes in Math., vol. 1581, Springer, Berlin, 1994, pp. 1–114 (French). MR 1307413, DOI 10.1007/BFb0073872
Dominique Bakry and Michel Émery, Hypercontractivité de semi-groupes de diffusion, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 15, 775–778 (French, with English summary). MR 772092
Dominique Bakry, Fabrice Baudoin, Michel Bonnefont, and Djalil Chafaï, On gradient bounds for the heat kernel on the Heisenberg group, J. Funct. Anal. 255 (2008), no. 8, 1905–1938. MR 2462581, DOI 10.1016/j.jfa.2008.09.002
Fabrice Baudoin, Martin Hairer, and Josef Teichmann, Ornstein-Uhlenbeck processes on Lie groups, J. Funct. Anal. 255 (2008), no. 4, 877–890. MR 2433956, DOI 10.1016/j.jfa.2008.05.004
A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin, 2007. MR 2363343
Eugenio Calabi, Extremal Kähler metrics, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 259–290. MR 645743
Eric A. Carlen, Some integral identities and inequalities for entire functions and their application to the coherent state transform, J. Funct. Anal. 97 (1991), no. 1, 231–249. MR 1105661, DOI 10.1016/0022-1236(91)90022-W
Eric A. Carlen, Superadditivity of Fisher’s information and logarithmic Sobolev inequalities, J. Funct. Anal. 101 (1991), no. 1, 194–211. MR 1132315, DOI 10.1016/0022-1236(91)90155-X
Bruce K. Driver and Leonard Gross, Hilbert spaces of holomorphic functions on complex Lie groups, New trends in stochastic analysis (Charingworth, 1994) World Sci. Publ., River Edge, NJ, 1997, pp. 76–106. MR 1654507
Bruce K. Driver, Leonard Gross, and Laurent Saloff-Coste, Holomorphic functions and subelliptic heat kernels over Lie groups, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 5, 941–978. MR 2538496, DOI 10.4171/JEMS/171
Bruce K. Driver, Leonard Gross, and Laurent Saloff-Coste, Surjectivity of the Taylor map for complex nilpotent Lie groups, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 1, 177–195. MR 2461876, DOI 10.1017/S0305004108001692
Bruce K. Driver, Leonard Gross, and Laurent Saloff-Coste, Growth of Taylor coefficients over complex homogeneous spaces, Tohoku Math. J. (2) 62 (2010), no. 3, 427–474. MR 2742018, DOI 10.2748/tmj/1287148621
Nathaniel Eldredge, Gradient estimates for the subelliptic heat kernel on $H$-type groups, J. Funct. Anal. 258 (2010), no. 2, 504–533. MR 2557945, DOI 10.1016/j.jfa.2009.08.012
Nathaniel Eldredge, On complex H-type Lie algebras, preprint, arXiv:1406.2396, 2014.
Nathaniel Eldredge, Strong hypercontractivity and strong logarithmic Sobolev inequalities for log-subharmonic functions on stratified Lie groups, preprint, arXiv:1706.07517, 2017.
Paul Federbush, Partially alternate derivation of a result of Nelson, J. Math. Physics 10 (1969), no. 1, 50–52.
G. B. Folland and Elias M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR 657581
Akito Futaki, Kähler-Einstein metrics and integral invariants, Lecture Notes in Mathematics, vol. 1314, Springer-Verlag, Berlin, 1988. MR 947341, DOI 10.1007/BFb0078084
Bernard Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math. 139 (1977), no. 1-2, 95–153. MR 461589, DOI 10.1007/BF02392235
James Glimm, Boson fields with nonlinear self-interaction in two dimensions, Comm. Math. Phys. 8 (1968), 12–25.
Piotr Graczyk, Todd Kemp, and Jean-Jacques Loeb, Hypercontractivity for log-subharmonic functions, J. Funct. Anal. 258 (2010), no. 6, 1785–1805. MR 2578455, DOI 10.1016/j.jfa.2009.08.014
Piotr Graczyk, Todd Kemp, and Jean-Jacques Loeb, Strong logarithmic Sobolev inequalities for log-subharmonic functions, Canad. J. Math. 67 (2015), no. 6, 1384–1410. MR 3415657, DOI 10.4153/CJM-2015-015-8
Leonard Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), no. 4, 1061–1083. MR 420249, DOI 10.2307/2373688
Leonard Gross, Logarithmic Sobolev inequalities and contractivity properties of semigroups, Dirichlet forms (Varenna, 1992) Lecture Notes in Math., vol. 1563, Springer, Berlin, 1993, pp. 54–88. MR 1292277, DOI 10.1007/BFb0074091
Leonard Gross, Hypercontractivity over complex manifolds, Acta Math. 182 (1999), no. 2, 159–206. MR 1710181, DOI 10.1007/BF02392573
Leonard Gross, Strong hypercontractivity and relative subharmonicity, J. Funct. Anal. 190 (2002), no. 1, 38–92. Special issue dedicated to the memory of I. E. Segal. MR 1895529, DOI 10.1006/jfan.2001.3883
Leonard Gross, Hypercontractivity, logarithmic Sobolev inequalities, and applications: a survey of surveys, Diffusion, quantum theory, and radically elementary mathematics, Math. Notes, vol. 47, Princeton Univ. Press, Princeton, NJ, 2006, pp. 45–73. MR 2325763
Leonard Gross and Zhongmin Qian, Holomorphic Dirichlet forms on complex manifolds, Math. Z. 246 (2004), no. 3, 521–561. MR 2073455, DOI 10.1007/s00209-003-0588-x
Jun-Qi Hu and Hong-Quan Li, Gradient estimates for the heat semigroup on H-type groups, Potential Anal. 33 (2010), no. 4, 355–386. MR 2726903, DOI 10.1007/s11118-010-9173-1
Svante Janson, On hypercontractivity for multipliers on orthogonal polynomials, Ark. Mat. 21 (1983), no. 1, 97–110. MR 706641, DOI 10.1007/BF02384302
Svante Janson, On complex hypercontractivity, J. Funct. Anal. 151 (1997), no. 1, 270–280. MR 1487778, DOI 10.1006/jfan.1997.3144
Françoise Lust-Piquard, Ornstein-Uhlenbeck semi-groups on stratified groups, J. Funct. Anal. 258 (2010), no. 6, 1883–1908. MR 2578458, DOI 10.1016/j.jfa.2009.11.012
Tai Melcher, Hypoelliptic heat kernel inequalities on Lie groups, Stochastic Process. Appl. 118 (2008), no. 3, 368–388. MR 2389050, DOI 10.1016/j.spa.2007.04.012
Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. MR 1867362, DOI 10.1090/surv/091
Alexander Nagel, Elias M. Stein, and Stephen Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147. MR 793239, DOI 10.1007/BF02392539
Edward Nelson, A quartic interaction in two dimensions, Mathematical Theory of Elementary Particles (Proc. Conf., Dedham, Mass., 1965) M.I.T. Press, Cambridge, Mass., 1966, pp. 69–73. MR 0210416
Edward Nelson, The free Markoff field, J. Functional Analysis 12 (1973), 211–227. MR 0343816, DOI 10.1016/0022-1236(73)90025-6
Barry Simon, Harmonic analysis, A Comprehensive Course in Analysis, Part 3, American Mathematical Society, Providence, RI, 2015. MR 3410783, DOI 10.1090/simon/003
A. J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Information and Control 2 (1959), 101–112. MR 109101, DOI 10.1016/S0019-9958(59)90348-1
Thomas Taylor, A parametrix for step-two hypoelliptic diffusion equations, Trans. Amer. Math. Soc. 296 (1986), no. 1, 191–215. MR 837807, DOI 10.1090/S0002-9947-1986-0837807-X
E. C. Titchmarsh, The theory of functions, Oxford University Press, Oxford, 1958. Reprint of the second (1939) edition. MR 3155290
N. Th. Varopoulos, Small time Gaussian estimates of heat diffusion kernels. II. The theory of large deviations, J. Funct. Anal. 93 (1990), no. 1, 1–33. MR 1070036, DOI 10.1016/0022-1236(90)90136-9
N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and geometry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press, Cambridge, 1992. MR 1218884
Robert Wallstén, The $S^p$-criterion for Hankel forms on the Fock space, $0<p<1$, Math. Scand. 64 (1989), no. 1, 123–132. MR 1036432, DOI 10.7146/math.scand.a-12251
Zheng-Fang Zhou, The contractivity of the free Hamiltonian semigroup in the $L_p$ space of entire functions, J. Funct. Anal. 96 (1991), no. 2, 407–425. MR 1101263, DOI 10.1016/0022-1236(91)90067-F