Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions

Graczyk, Piotr ; Kemp, Todd ; Loeb, Jean-Jacques

Canadian journal of mathematics, Tome 67 (2015) no. 6, pp. 1384-1410

Résumé

We prove an intrinsic equivalence between strong hypercontractivity and a strong logarithmic Sobolev inequality for the cone of logarithmically subharmonic ( $\text{LSH}$ ) functions. We introduce a new large class of measures, Euclidean regular and exponential type, in addition to all compactly-supported measures, for which this equivalence holds. We prove a Sobolev density theorem through $\text{LSH}$ functions and use it to prove the equivalence of strong hypercontractivity and the strong logarithmic Sobolev inequality for such log-subharmonic functions.

Détail

DOI : 10.4153/CJM-2015-015-8

Mots-clés : 47D06, logarithmic Sobolev inequalities

Bibliographie

[1] [1] Anderson, G., Guionnet, A., and Zeitouni, O., An introduction to random matrices. Cambridge Studies in Advanced Mathematics, 118, Cambridge University Press, Cambridge, 2010. Google Scholar

[2] [2] Bakry, D., L'hypercontractivité et son utilisation en théorie des semigroupes. In: Lectures on probability theory (Saint-Flour, 1992), Lecture Notes in Math., 1581, Springer, Berlin, 1994, pp. 1–114. Google Scholar

[3] [3] Bakry, D., On Sobolev and logarithmic Sobolev inequalities true for Markov semigroups. In: New trends in stochastic analysis (Charingworth, 1994), World Sci. Publ., River Edge, NJ, 1997, pp. 43–75. Google Scholar

[4] [4] Bakry, D. and Emery, M., Diffusions hypercontractives. Séminaire de Probabilités, XIX, Lecture Notes in Math. 1123, Springer, Berlin, 1985, pp. 177–206. Google Scholar

[5] [5] Bobkov, S. and Houdré, C., Some connections between isoperimetric and Sobolev-type inequalities. Mem. Amer. Math. Soc. 129(1997), no. 616. Google Scholar

[6] [6] Bobkov, S. and Ledoux, M., From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10(2000), no. 5, 1028–1052. Google Scholar | DOI

[7] [7] Bobkov, S. and Tetali, P., Modified logarithmic Sobolev inequalities in discrete settings. J. Theoret. Probab. 19(2006), no. 2, 289–336. Google Scholar | DOI

[8] [8] Davies, E. B., Explicit constants for Gaussian upper bounds on heat kernels. Amer. J. Math. 109(1987), no. 2, 319–333. Google Scholar | DOI

[9] [9] Davies, E. B., Heat kernels and spectral theory. Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1990. Google Scholar

[10] [10] Davies, E. B. and Simon, B., Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59(1984), no. 2, 335–395. Google Scholar | DOI

[11] [11] Diaconis, P. and Saloff-Coste, L., Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6(1996), no. 3, 695–750. Google Scholar | DOI

[12] [12] Galaz-Fontes, F., Gross, L., and Sontz, S. B., Reverse hypercontractivity over manifolds. Ark. Math. 39(2001), no. 2, 283–309. Google Scholar | DOI

[13] [13] Graczyk, P., Kemp, T., and Loeb, J.-J., Hypercontractivity for log-subharmonic functions. J. Funct. Anal. 258(2010), no. 6, 1785–1805. Google Scholar | DOI

[14] [14] Greene, R. and Krantz, S., Function theory of one complex variable. Third ed., Graduate Studies in Mathematics, 40, American Mathematical Society, Providence, RI, 2006. Google Scholar

[15] [15] Gross, L., Logarithmic Sobolev inequalities. Amer. J. Math. 97(1975), no. 4, 1061–1083. Google Scholar | DOI

[16] [16] Gross, L., Hypercontractivity over complex manifolds. Acta Math. 182(1999), no. 2, 159–206. Google Scholar | DOI

[17] [17] Gross, L. and Grothaus, M., Reverse hypercontractivity for subharmonic functions. Canad. J. Math. 57(2005), no. 3, 506–534. Google Scholar | DOI

[18] [18] Guionnet, A. and Zegarlinski, B., Lectures on logarithmic Sobolev inequalities. Séminaire de Probabilités, XXXVI, Lecture Notes in Math., Springer, Berlin, 2003, pp. 1–134. Google Scholar

[19] [19] Holley, R. and Stroock, D., Logarithmic Sobolev inequalities and stochastic Ising models. J. Statist. Phys. 46(1987), lno. 5–6, 159–1194. Google Scholar | DOI

[20] [20] Janson, S., On hypercontractivity for multipliers of orthogonal polynomials. Ark. Mat. 21(1983), no. 1, 97–110. Google Scholar | DOI

[21] [21] Janson, S., On complex hypercontractivity. J. Funct. Anal. 151(1997), no. 1, 270–280. Google Scholar | DOI

[22] [22] Ledoux, M., Isoperimetry and Gaussian analysis. In: Lectures on probability theory and statistics, Lecture Notes in Math 1648, Springer, Berlin, 1996, pp. 165–294. Google Scholar

[23] [23] Ledoux, M., Concentration of measure and logarithmic Sobolev inequalities. Séminaire de Probabilités, XXXIII, Lecture Notes in Math., 1709, Springer, Berlin, 1999, pp. 120–216. Google Scholar

[24] [24] Ledoux, M., The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse Math. (6) 9(2000), no. 2, 305–366. Google Scholar | DOI

[25] [25] Ledoux, M., The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89, American Mathematical Society, Providence, RI, 2001. Google Scholar

[26] [26] Ledoux, M., A remark on hypercontractivity and tail inequalities for the largest eigenvalues of random matrices. Séminaire de Probabilités XXXVII, Lecture Notes in Math., 1832, Springer, Berlin,2003, pp. 360–369. Google Scholar

[27] [27] Ledoux, M., Spectral gap, logarithmic Sobolev constant, and geometric bounds. In: Surveys in differential geometry. Vol. IX, Surv. Differ. Geom., IX, Int. Press, Somerville, MA, 2004, pp. 219–240. Google Scholar

[28] [28] Lelong, P. and Gruman, L., Entire functions of several complex variables. Grundlehren der Mathematischen Wissenschaften, 282, Springer-Verlag, Berlin, 1986. Google Scholar

[29] [29] Nelson, E., The free Markov field. J. Funct. Anal. 12(1973), 211–227. Google Scholar | DOI

[30] [30] Stam, A. J., Some inequalities satisfied by the quantities of information of Fisher and Shannon. Information and Control 2(1959), 101–112. Google Scholar | DOI

[31] [31] Villani, C., Topics in optimal transportation. Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. Google Scholar

[32] [32] Yau, H.-T., Logarithmic Sobolev inequality for the lattice gases with mixing conditions. Commun. Math. Phys. 181(1996), no. 2, 367–408. Google Scholar | DOI

[33] [33] Yau, H.-T., Logorithmic Sobolev inequality for generalized simple exclusion processes. Probab. Theory Related Fields 109(1997), no. 4, 507–538. Google Scholar | DOI

[34] [34] Zegarlinski, B., Dobrushin uniqueness theorem and logarithmic Sobolev inequalities. J. Funct. Anal. 105(1992), no. 1, 77–111. Google Scholar | DOI

[35] [35] Zimmermann, D., Logarithmic Sobolev inequalities for mollified compactly supported measures. J. Funct. Anal. 265(2013), no. 6, 1064–1083. Google Scholar | DOI