August  2014, 7(4): 695-724. doi: 10.3934/dcdss.2014.7.695

Improved interpolation inequalities on the sphere

1. 

Ceremade (UMR CNRS 7534), Université Paris Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris, Cédex 16

2. 

CEREMADE - UMR C.N.R.S. 7534, Université Paris IX-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France

3. 

Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago

4. 

School of Mathematics, Skiles Building, Georgia Institute of Technology, Atlanta GA 30332-0160, United States

Received  September 2013 Revised  December 2013 Published  February 2014

This paper contains a review of available methods for establishing improved interpolation inequalities on the sphere for subcritical exponents. Pushing further these techniques we also establish some new results, clarify the range of applicability of the various existing methods and state several explicit estimates.
Citation: Jean Dolbeault, Maria J. Esteban, Michał Kowalczyk, Michael Loss. Improved interpolation inequalities on the sphere. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 695-724. doi: 10.3934/dcdss.2014.7.695
References:
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A. Arnold and J. Dolbeault, Refined convex Sobolev inequalities,, J. Funct. Anal., 225 (2005), 337.  doi: 10.1016/j.jfa.2005.05.003.  Google Scholar

[3]

A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations,, Comm. Partial Differential Equations, 26 (2001), 43.  doi: 10.1081/PDE-100002246.  Google Scholar

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[6]

D. Bakry and M. Émery, Hypercontractivité de semi-groupes de diffusion,, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 775.   Google Scholar

[7]

D. Bakry and M. Émery, Diffusions hypercontractives,, in Séminaire de Probabilités, (1123), 177.  doi: 10.1007/BFb0075847.  Google Scholar

[8]

D. Bakry and M. Émery, Inégalités de Sobolev pour un semi-groupe symétrique,, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 411.   Google Scholar

[9]

D. Bakry and M. Ledoux, Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator,, Duke Math. J., 85 (1996), 253.  doi: 10.1215/S0012-7094-96-08511-7.  Google Scholar

[10]

W. Beckner, A generalized Poincaré inequality for Gaussian measures,, Proc. Amer. Math. Soc., 105 (1989), 397.  doi: 10.2307/2046956.  Google Scholar

[11]

_______, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere $S^n$,, Proc. Nat. Acad. Sci. U.S.A., 89 (1992), 4816.   Google Scholar

[12]

_______, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality,, Ann. of Math. (2), 138 (1993), 213.   Google Scholar

[13]

A. Bentaleb, Inégalité de Sobolev pour l'opérateur ultrasphérique,, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 187.   Google Scholar

[14]

_______, Sur les fonctions extrémales des inégalités de Sobolev des opérateurs de diffusion,, in Séminaire de Probabilités, (1801), 230.   Google Scholar

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M.-F. Bidaut-Véron and M. Bouhar, On characterization of solutions of some nonlinear differential equations and applications,, SIAM J. Math. Anal., 25 (1994), 859.  doi: 10.1137/S0036141092230593.  Google Scholar

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F. Bolley and I. Gentil, Phi-entropy inequalities and Fokker-Planck equations,, in Progress in Analysis and Its Applications, (2010), 463.  doi: 10.1142/9789814313179_0060.  Google Scholar

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_______, Phi-entropy inequalities for diffusion semigroups,, J. Math. Pures Appl., 93 (2010), 449.   Google Scholar

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C. Brouttelande, The best-constant problem for a family of Gagliardo-Nirenberg inequalities on a compact Riemannian manifold,, Proc. Edinb. Math. Soc. (2), 46 (2003), 117.  doi: 10.1017/S0013091501000426.  Google Scholar

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_______, On the second best constant in logarithmic Sobolev inequalities on complete Riemannian manifolds,, Bull. Sci. Math., 127 (2003), 292.   Google Scholar

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M. J. Cáceres, J. A. Carrillo and J. Dolbeault, Nonlinear stability in $L^p$ for a confined system of charged particles,, SIAM J. Math. Anal., 34 (2002), 478.  doi: 10.1137/S0036141001398435.  Google Scholar

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E. A. Carlen, R. Frank and E. H. Lieb, Stability estimates for the lowest eigenvalue of a Schrödinger operator,, Geom. Funct. Anal., (2013).   Google Scholar

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J. A. Carrillo and G. Toscani, Asymptotic $L^1$-decay of solutions of the porous medium equation to self-similarity,, Indiana Univ. Math. J., 49 (2000), 113.  doi: 10.1512/iumj.2000.49.1756.  Google Scholar

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[28]

S.-Y. A. Chang and P. C. Yang, Prescribing Gaussian curvature on $\mathbb S^2$,, Acta Math., 159 (1987), 215.  doi: 10.1007/BF02392560.  Google Scholar

[29]

A. Cianchi, N. Fusco, F. Maggi and A. Pratelli, The sharp Sobolev inequality in quantitative form,, J. Eur. Math. Soc. (JEMS), 11 (2009), 1105.  doi: 10.4171/JEMS/176.  Google Scholar

[30]

I. Csiszár, Information-type measures of difference of probability distributions and indirect observations,, Studia Sci. Math. Hungar., 2 (1967), 299.   Google Scholar

[31]

M. Del Pino and J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions,, J. Math. Pures Appl. (9), 81 (2002), 847.  doi: 10.1016/S0021-7824(02)01266-7.  Google Scholar

[32]

J. Demange, Des équations à Diffusion Rapide aux Inégalités de Sobolev sur les Modèles de la Géométrie,, Ph.D thesis, (2005).   Google Scholar

[33]

______, Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature,, J. Funct. Anal., 254 (2008), 593.   Google Scholar

[34]

J. Dolbeault, M. J. Esteban, M. Kowalczyk and M. Loss, Sharp interpolation inequalities on the sphere: New methods and consequences,, Chinese Annals of Mathematics, 34 (2013), 99.  doi: 10.1007/s11401-012-0756-6.  Google Scholar

[35]

J. Dolbeault, M. J. Esteban and A. Laptev, Spectral estimates on the sphere,, to appear in Analysis & PDE, (2013).   Google Scholar

[36]

J. Dolbeault, M. J. Esteban, A. Laptev and M. Loss, One-Dimensional Gagliardo-Nirenberg-Sobolev Inequalities: Remarks on Duality and Flows,, Tech. Rep., (2013).   Google Scholar

[37]

J. Dolbeault, M. J. Esteban and M. Loss, Nonlinear Flows and Rigidity Results on Compact Manifolds,, Tech. Rep., (2013).   Google Scholar

[38]

J. Dolbeault and G. Karch, Large time behaviour of solutions to nonhomogeneous diffusion equations,, in Self-Similar Solutions of Nonlinear PDE, (2006), 133.  doi: 10.4064/bc74-0-8.  Google Scholar

[39]

J. Dolbeault, B. Nazaret and G. Savaré, On the Bakry-Emery criterion for linear diffusions and weighted porous media equations,, Commun. Math. Sci., 6 (2008), 477.  doi: 10.4310/CMS.2008.v6.n2.a10.  Google Scholar

[40]

P. L. Duren, Univalent Functions,, Grundlehren der Mathematischen Wissenschaften, (1983).   Google Scholar

[41]

É. Fontenas, Sur les constantes de Sobolev des variétés riemanniennes compactes et les fonctions extrémales des sphères,, Bull. Sci. Math., 121 (1997), 71.   Google Scholar

[42]

_______, Sur les minorations des constantes de Sobolev et de Sobolev logarithmiques pour les opérateurs de Jacobi et de Laguerre,, in Séminaire de Probabilités, (1686), 14.   Google Scholar

[43]

N. Ghoussoub and C.-S. Lin, On the best constant in the Moser-Onofri-Aubin inequality,, Comm. Math. Phys., 298 (2010), 869.  doi: 10.1007/s00220-010-1079-7.  Google Scholar

[44]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Comm. Pure Appl. Math., 34 (1981), 525.  doi: 10.1002/cpa.3160340406.  Google Scholar

[45]

L. Gross, Logarithmic Sobolev inequalities,, Amer. J. Math., 97 (1975), 1061.  doi: 10.2307/2373688.  Google Scholar

[46]

S. Kullback, On the convergence of discrimination information,, IEEE Trans. Information Theory, (1968), 765.   Google Scholar

[47]

R. Latała and K. Oleszkiewicz, Between Sobolev and Poincaré,, in Geometric Aspects of Functional Analysis, (1745), 147.  doi: 10.1007/BFb0107213.  Google Scholar

[48]

J. R. Licois and L. Véron, Un théorème d'annulation pour des équations elliptiques non linéaires sur des variétés riemanniennes compactes,, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 1337.   Google Scholar

[49]

J. R. Licois and L. Véron, A class of nonlinear conservative elliptic equations in cylinders,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 249.   Google Scholar

[50]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math. (2), 118 (1983), 349.  doi: 10.2307/2007032.  Google Scholar

[51]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077.   Google Scholar

[52]

C. E. Mueller and F. B. Weissler, Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the $n$-sphere,, J. Funct. Anal., 48 (1982), 252.  doi: 10.1016/0022-1236(82)90069-6.  Google Scholar

[53]

E. Onofri, On the positivity of the effective action in a theory of random surfaces,, Comm. Math. Phys., 86 (1982), 321.  doi: 10.1007/BF01212171.  Google Scholar

[54]

B. Osgood, R. Phillips and P. Sarnak, Compact isospectral sets of plane domains,, Proc. Nat. Acad. Sci. U.S.A., 85 (1988), 5359.  doi: 10.1073/pnas.85.15.5359.  Google Scholar

[55]

______, Compact isospectral sets of surfaces,, J. Funct. Anal., 80 (1988), 212.   Google Scholar

[56]

______, Extremals of determinants of Laplacians,, J. Funct. Anal., 80 (1988), 148.   Google Scholar

[57]

M. S. Pinsker, Information and Information Stability of Random Variables and Processes,, Translated and edited by Amiel Feinstein, (1964).   Google Scholar

[58]

A. Unterreiter, A. Arnold, P. Markowich and G. Toscani, On generalized Csiszár-Kullback inequalities,, Monatsh. Math., 131 (2000), 235.  doi: 10.1007/s006050070013.  Google Scholar

[59]

F. B. Weissler, Logarithmic Sobolev inequalities and hypercontractive estimates on the circle,, J. Funct. Anal., 37 (1980), 218.  doi: 10.1016/0022-1236(80)90042-7.  Google Scholar

[60]

H. Widom, On an inequality of Osgood, Phillips and Sarnak,, Proc. Amer. Math. Soc., 102 (1988), 773.  doi: 10.2307/2047262.  Google Scholar

show all references

References:
[1]

A. Arnold, J.-P. Bartier and J. Dolbeault, Interpolation between logarithmic Sobolev and Poincaré inequalities,, Commun. Math. Sci., 5 (2007), 971.  doi: 10.4310/CMS.2007.v5.n4.a12.  Google Scholar

[2]

A. Arnold and J. Dolbeault, Refined convex Sobolev inequalities,, J. Funct. Anal., 225 (2005), 337.  doi: 10.1016/j.jfa.2005.05.003.  Google Scholar

[3]

A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations,, Comm. Partial Differential Equations, 26 (2001), 43.  doi: 10.1081/PDE-100002246.  Google Scholar

[4]

A. Baernstein II and B. A. Taylor, Spherical rearrangements, subharmonic functions, and $*$-functions in $n$-space,, Duke Math. J., 43 (1976), 245.  doi: 10.1215/S0012-7094-76-04322-2.  Google Scholar

[5]

D. Bakry, Une suite d'inégalités remarquables pour les opérateurs ultrasphériques,, C. R. Acad. Sci. Paris Sér. I Math., 318 (1994), 161.   Google Scholar

[6]

D. Bakry and M. Émery, Hypercontractivité de semi-groupes de diffusion,, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 775.   Google Scholar

[7]

D. Bakry and M. Émery, Diffusions hypercontractives,, in Séminaire de Probabilités, (1123), 177.  doi: 10.1007/BFb0075847.  Google Scholar

[8]

D. Bakry and M. Émery, Inégalités de Sobolev pour un semi-groupe symétrique,, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 411.   Google Scholar

[9]

D. Bakry and M. Ledoux, Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator,, Duke Math. J., 85 (1996), 253.  doi: 10.1215/S0012-7094-96-08511-7.  Google Scholar

[10]

W. Beckner, A generalized Poincaré inequality for Gaussian measures,, Proc. Amer. Math. Soc., 105 (1989), 397.  doi: 10.2307/2046956.  Google Scholar

[11]

_______, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere $S^n$,, Proc. Nat. Acad. Sci. U.S.A., 89 (1992), 4816.   Google Scholar

[12]

_______, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality,, Ann. of Math. (2), 138 (1993), 213.   Google Scholar

[13]

A. Bentaleb, Inégalité de Sobolev pour l'opérateur ultrasphérique,, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 187.   Google Scholar

[14]

_______, Sur les fonctions extrémales des inégalités de Sobolev des opérateurs de diffusion,, in Séminaire de Probabilités, (1801), 230.   Google Scholar

[15]

A. Bentaleb and S. Fahlaoui, A family of integral inequalities on the circle $S^1$,, Proc. Japan Acad. Ser. A Math. Sci., 86 (2010), 55.  doi: 10.3792/pjaa.86.55.  Google Scholar

[16]

M. Berger, P. Gauduchon and E. Mazet, Le Spectre d'une Variété Riemannienne,, Lecture Notes in Mathematics, (1971).   Google Scholar

[17]

G. Bianchi and H. Egnell, A note on the Sobolev inequality,, J. Funct. Anal., 100 (1991), 18.  doi: 10.1016/0022-1236(91)90099-Q.  Google Scholar

[18]

M.-F. Bidaut-Véron and M. Bouhar, On characterization of solutions of some nonlinear differential equations and applications,, SIAM J. Math. Anal., 25 (1994), 859.  doi: 10.1137/S0036141092230593.  Google Scholar

[19]

M.-F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations,, Invent. Math., 106 (1991), 489.  doi: 10.1007/BF01243922.  Google Scholar

[20]

F. Bolley and I. Gentil, Phi-entropy inequalities and Fokker-Planck equations,, in Progress in Analysis and Its Applications, (2010), 463.  doi: 10.1142/9789814313179_0060.  Google Scholar

[21]

_______, Phi-entropy inequalities for diffusion semigroups,, J. Math. Pures Appl., 93 (2010), 449.   Google Scholar

[22]

C. Brouttelande, The best-constant problem for a family of Gagliardo-Nirenberg inequalities on a compact Riemannian manifold,, Proc. Edinb. Math. Soc. (2), 46 (2003), 117.  doi: 10.1017/S0013091501000426.  Google Scholar

[23]

_______, On the second best constant in logarithmic Sobolev inequalities on complete Riemannian manifolds,, Bull. Sci. Math., 127 (2003), 292.   Google Scholar

[24]

M. J. Cáceres, J. A. Carrillo and J. Dolbeault, Nonlinear stability in $L^p$ for a confined system of charged particles,, SIAM J. Math. Anal., 34 (2002), 478.  doi: 10.1137/S0036141001398435.  Google Scholar

[25]

E. A. Carlen, R. Frank and E. H. Lieb, Stability estimates for the lowest eigenvalue of a Schrödinger operator,, Geom. Funct. Anal., (2013).   Google Scholar

[26]

J. A. Carrillo and G. Toscani, Asymptotic $L^1$-decay of solutions of the porous medium equation to self-similarity,, Indiana Univ. Math. J., 49 (2000), 113.  doi: 10.1512/iumj.2000.49.1756.  Google Scholar

[27]

D. Chafai, Entropies, convexity, and functional inequalities: On $\Phi$-entropies and $\Phi$-Sobolev inequalities,, J. Math. Kyoto Univ., 44 (2004), 325.   Google Scholar

[28]

S.-Y. A. Chang and P. C. Yang, Prescribing Gaussian curvature on $\mathbb S^2$,, Acta Math., 159 (1987), 215.  doi: 10.1007/BF02392560.  Google Scholar

[29]

A. Cianchi, N. Fusco, F. Maggi and A. Pratelli, The sharp Sobolev inequality in quantitative form,, J. Eur. Math. Soc. (JEMS), 11 (2009), 1105.  doi: 10.4171/JEMS/176.  Google Scholar

[30]

I. Csiszár, Information-type measures of difference of probability distributions and indirect observations,, Studia Sci. Math. Hungar., 2 (1967), 299.   Google Scholar

[31]

M. Del Pino and J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions,, J. Math. Pures Appl. (9), 81 (2002), 847.  doi: 10.1016/S0021-7824(02)01266-7.  Google Scholar

[32]

J. Demange, Des équations à Diffusion Rapide aux Inégalités de Sobolev sur les Modèles de la Géométrie,, Ph.D thesis, (2005).   Google Scholar

[33]

______, Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature,, J. Funct. Anal., 254 (2008), 593.   Google Scholar

[34]

J. Dolbeault, M. J. Esteban, M. Kowalczyk and M. Loss, Sharp interpolation inequalities on the sphere: New methods and consequences,, Chinese Annals of Mathematics, 34 (2013), 99.  doi: 10.1007/s11401-012-0756-6.  Google Scholar

[35]

J. Dolbeault, M. J. Esteban and A. Laptev, Spectral estimates on the sphere,, to appear in Analysis & PDE, (2013).   Google Scholar

[36]

J. Dolbeault, M. J. Esteban, A. Laptev and M. Loss, One-Dimensional Gagliardo-Nirenberg-Sobolev Inequalities: Remarks on Duality and Flows,, Tech. Rep., (2013).   Google Scholar

[37]

J. Dolbeault, M. J. Esteban and M. Loss, Nonlinear Flows and Rigidity Results on Compact Manifolds,, Tech. Rep., (2013).   Google Scholar

[38]

J. Dolbeault and G. Karch, Large time behaviour of solutions to nonhomogeneous diffusion equations,, in Self-Similar Solutions of Nonlinear PDE, (2006), 133.  doi: 10.4064/bc74-0-8.  Google Scholar

[39]

J. Dolbeault, B. Nazaret and G. Savaré, On the Bakry-Emery criterion for linear diffusions and weighted porous media equations,, Commun. Math. Sci., 6 (2008), 477.  doi: 10.4310/CMS.2008.v6.n2.a10.  Google Scholar

[40]

P. L. Duren, Univalent Functions,, Grundlehren der Mathematischen Wissenschaften, (1983).   Google Scholar

[41]

É. Fontenas, Sur les constantes de Sobolev des variétés riemanniennes compactes et les fonctions extrémales des sphères,, Bull. Sci. Math., 121 (1997), 71.   Google Scholar

[42]

_______, Sur les minorations des constantes de Sobolev et de Sobolev logarithmiques pour les opérateurs de Jacobi et de Laguerre,, in Séminaire de Probabilités, (1686), 14.   Google Scholar

[43]

N. Ghoussoub and C.-S. Lin, On the best constant in the Moser-Onofri-Aubin inequality,, Comm. Math. Phys., 298 (2010), 869.  doi: 10.1007/s00220-010-1079-7.  Google Scholar

[44]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Comm. Pure Appl. Math., 34 (1981), 525.  doi: 10.1002/cpa.3160340406.  Google Scholar

[45]

L. Gross, Logarithmic Sobolev inequalities,, Amer. J. Math., 97 (1975), 1061.  doi: 10.2307/2373688.  Google Scholar

[46]

S. Kullback, On the convergence of discrimination information,, IEEE Trans. Information Theory, (1968), 765.   Google Scholar

[47]

R. Latała and K. Oleszkiewicz, Between Sobolev and Poincaré,, in Geometric Aspects of Functional Analysis, (1745), 147.  doi: 10.1007/BFb0107213.  Google Scholar

[48]

J. R. Licois and L. Véron, Un théorème d'annulation pour des équations elliptiques non linéaires sur des variétés riemanniennes compactes,, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 1337.   Google Scholar

[49]

J. R. Licois and L. Véron, A class of nonlinear conservative elliptic equations in cylinders,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 249.   Google Scholar

[50]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math. (2), 118 (1983), 349.  doi: 10.2307/2007032.  Google Scholar

[51]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077.   Google Scholar

[52]

C. E. Mueller and F. B. Weissler, Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the $n$-sphere,, J. Funct. Anal., 48 (1982), 252.  doi: 10.1016/0022-1236(82)90069-6.  Google Scholar

[53]

E. Onofri, On the positivity of the effective action in a theory of random surfaces,, Comm. Math. Phys., 86 (1982), 321.  doi: 10.1007/BF01212171.  Google Scholar

[54]

B. Osgood, R. Phillips and P. Sarnak, Compact isospectral sets of plane domains,, Proc. Nat. Acad. Sci. U.S.A., 85 (1988), 5359.  doi: 10.1073/pnas.85.15.5359.  Google Scholar

[55]

______, Compact isospectral sets of surfaces,, J. Funct. Anal., 80 (1988), 212.   Google Scholar

[56]

______, Extremals of determinants of Laplacians,, J. Funct. Anal., 80 (1988), 148.   Google Scholar

[57]

M. S. Pinsker, Information and Information Stability of Random Variables and Processes,, Translated and edited by Amiel Feinstein, (1964).   Google Scholar

[58]

A. Unterreiter, A. Arnold, P. Markowich and G. Toscani, On generalized Csiszár-Kullback inequalities,, Monatsh. Math., 131 (2000), 235.  doi: 10.1007/s006050070013.  Google Scholar

[59]

F. B. Weissler, Logarithmic Sobolev inequalities and hypercontractive estimates on the circle,, J. Funct. Anal., 37 (1980), 218.  doi: 10.1016/0022-1236(80)90042-7.  Google Scholar

[60]

H. Widom, On an inequality of Osgood, Phillips and Sarnak,, Proc. Amer. Math. Soc., 102 (1988), 773.  doi: 10.2307/2047262.  Google Scholar

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