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 and Continuous Dynamical Systems - S, 2014, 7 (4) : 695-724. doi: 10.3934/dcdss.2014.7.695
References:
[1]

A. Arnold, J.-P. Bartier and J. Dolbeault, Interpolation between logarithmic Sobolev and Poincaré inequalities, Commun. Math. Sci., 5 (2007), 971-979. doi: 10.4310/CMS.2007.v5.n4.a12.

[2]

A. Arnold and J. Dolbeault, Refined convex Sobolev inequalities, J. Funct. Anal., 225 (2005), 337-351. doi: 10.1016/j.jfa.2005.05.003.

[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), 43-100. doi: 10.1081/PDE-100002246.

[4]

A. Baernstein II and B. A. Taylor, Spherical rearrangements, subharmonic functions, and $*$-functions in $n$-space, Duke Math. J., 43 (1976), 245-268. doi: 10.1215/S0012-7094-76-04322-2.

[5]

D. Bakry, Une suite d'inégalités remarquables pour les opérateurs ultrasphériques, C. R. Acad. Sci. Paris Sér. I Math., 318 (1994), 161-164.

[6]

D. Bakry and M. Émery, Hypercontractivité de semi-groupes de diffusion, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 775-778.

[7]

D. Bakry and M. Émery, Diffusions hypercontractives, in Séminaire de Probabilités, XIX, (1983/84), Lecture Notes in Math., 1123, Springer, Berlin, 1985, 177-206. doi: 10.1007/BFb0075847.

[8]

D. Bakry and M. Émery, Inégalités de Sobolev pour un semi-groupe symétrique, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 411-413.

[9]

D. Bakry and M. Ledoux, Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator, Duke Math. J., 85 (1996), 253-270. doi: 10.1215/S0012-7094-96-08511-7.

[10]

W. Beckner, A generalized Poincaré inequality for Gaussian measures, Proc. Amer. Math. Soc., 105 (1989), 397-400. doi: 10.2307/2046956.

[11]

_______, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere $S^n$, Proc. Nat. Acad. Sci. U.S.A., 89 (1992), 4816-4819.

[12]

_______, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2), 138 (1993), 213-242.

[13]

A. Bentaleb, Inégalité de Sobolev pour l'opérateur ultrasphérique, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 187-190.

[14]

_______, Sur les fonctions extrémales des inégalités de Sobolev des opérateurs de diffusion, in Séminaire de Probabilités, XXXVI, Lecture Notes in Math., 1801, Springer, Berlin, 2003, 230-250.

[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), 55-59. doi: 10.3792/pjaa.86.55.

[16]

M. Berger, P. Gauduchon and E. Mazet, Le Spectre d'une Variété Riemannienne, Lecture Notes in Mathematics, 194, Springer-Verlag, Berlin, 1971.

[17]

G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. Funct. Anal., 100 (1991), 18-24. doi: 10.1016/0022-1236(91)90099-Q.

[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), 859-875. doi: 10.1137/S0036141092230593.

[19]

M.-F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., 106 (1991), 489-539. doi: 10.1007/BF01243922.

[20]

F. Bolley and I. Gentil, Phi-entropy inequalities and Fokker-Planck equations, in Progress in Analysis and Its Applications, World Sci. Publ., Hackensack, NJ, 2010, 463-469. doi: 10.1142/9789814313179_0060.

[21]

_______, Phi-entropy inequalities for diffusion semigroups, J. Math. Pures Appl., 93 (2010), 449-473.

[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), 117-146. doi: 10.1017/S0013091501000426.

[23]

_______, On the second best constant in logarithmic Sobolev inequalities on complete Riemannian manifolds, Bull. Sci. Math., 127 (2003), 292-312.

[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), 478-494. doi: 10.1137/S0036141001398435.

[25]

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

[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), 113-142. doi: 10.1512/iumj.2000.49.1756.

[27]

D. Chafai, Entropies, convexity, and functional inequalities: On $\Phi$-entropies and $\Phi$-Sobolev inequalities, J. Math. Kyoto Univ., 44 (2004), 325-363.

[28]

S.-Y. A. Chang and P. C. Yang, Prescribing Gaussian curvature on $\mathbb S^2$, Acta Math., 159 (1987), 215-259. doi: 10.1007/BF02392560.

[29]

A. Cianchi, N. Fusco, F. Maggi and A. Pratelli, The sharp Sobolev inequality in quantitative form, J. Eur. Math. Soc. (JEMS), 11 (2009), 1105-1139. doi: 10.4171/JEMS/176.

[30]

I. Csiszár, Information-type measures of difference of probability distributions and indirect observations, Studia Sci. Math. Hungar., 2 (1967), 299-318

[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), 847-875. doi: 10.1016/S0021-7824(02)01266-7.

[32]

J. 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.

[33]

______, Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature, J. Funct. Anal., 254 (2008), 593-611.

[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, Series B, 34 (2013), 99-112. doi: 10.1007/s11401-012-0756-6.

[35]

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

[36]

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

[37]

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

[38]

J. Dolbeault and G. Karch, Large time behaviour of solutions to nonhomogeneous diffusion equations, in Self-Similar Solutions of Nonlinear PDE, Banach Center Publ., 74, Polish Acad. Sci., Warsaw, 2006, 133-147. doi: 10.4064/bc74-0-8.

[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), 477-494. doi: 10.4310/CMS.2008.v6.n2.a10.

[40]

P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, New York, 1983.

[41]

É. Fontenas, Sur les constantes de Sobolev des variétés riemanniennes compactes et les fonctions extrémales des sphères, Bull. Sci. Math., 121 (1997), 71-96.

[42]

_______, Sur les minorations des constantes de Sobolev et de Sobolev logarithmiques pour les opérateurs de Jacobi et de Laguerre, in Séminaire de Probabilités, XXXII, Lecture Notes in Math., 1686, Springer, Berlin, 1998, 14-29.

[43]

N. Ghoussoub and C.-S. Lin, On the best constant in the Moser-Onofri-Aubin inequality, Comm. Math. Phys., 298 (2010), 869-878. doi: 10.1007/s00220-010-1079-7.

[44]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.

[45]

L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083. doi: 10.2307/2373688.

[46]

S. Kullback, On the convergence of discrimination information, IEEE Trans. Information Theory, IT-14 (1968), 765-766.

[47]

R. Latała and K. Oleszkiewicz, Between Sobolev and Poincaré, in Geometric Aspects of Functional Analysis, Lecture Notes in Math., 1745, Springer, Berlin, 2000, 147-168. doi: 10.1007/BFb0107213.

[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), 1337-1342.

[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), 249-283.

[50]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349-374. doi: 10.2307/2007032.

[51]

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

[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), 252-283. doi: 10.1016/0022-1236(82)90069-6.

[53]

E. Onofri, On the positivity of the effective action in a theory of random surfaces, Comm. Math. Phys., 86 (1982), 321-326. doi: 10.1007/BF01212171.

[54]

B. Osgood, R. Phillips and P. Sarnak, Compact isospectral sets of plane domains, Proc. Nat. Acad. Sci. U.S.A., 85 (1988), 5359-5361. doi: 10.1073/pnas.85.15.5359.

[55]

______, Compact isospectral sets of surfaces, J. Funct. Anal., 80 (1988), 212-234.

[56]

______, Extremals of determinants of Laplacians, J. Funct. Anal., 80 (1988), 148-211.

[57]

M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Translated and edited by Amiel Feinstein, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964.

[58]

A. Unterreiter, A. Arnold, P. Markowich and G. Toscani, On generalized Csiszár-Kullback inequalities, Monatsh. Math., 131 (2000), 235-253. doi: 10.1007/s006050070013.

[59]

F. B. Weissler, Logarithmic Sobolev inequalities and hypercontractive estimates on the circle, J. Funct. Anal., 37 (1980), 218-234. doi: 10.1016/0022-1236(80)90042-7.

[60]

H. Widom, On an inequality of Osgood, Phillips and Sarnak, Proc. Amer. Math. Soc., 102 (1988), 773-774. doi: 10.2307/2047262.

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), 971-979. doi: 10.4310/CMS.2007.v5.n4.a12.

[2]

A. Arnold and J. Dolbeault, Refined convex Sobolev inequalities, J. Funct. Anal., 225 (2005), 337-351. doi: 10.1016/j.jfa.2005.05.003.

[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), 43-100. doi: 10.1081/PDE-100002246.

[4]

A. Baernstein II and B. A. Taylor, Spherical rearrangements, subharmonic functions, and $*$-functions in $n$-space, Duke Math. J., 43 (1976), 245-268. doi: 10.1215/S0012-7094-76-04322-2.

[5]

D. Bakry, Une suite d'inégalités remarquables pour les opérateurs ultrasphériques, C. R. Acad. Sci. Paris Sér. I Math., 318 (1994), 161-164.

[6]

D. Bakry and M. Émery, Hypercontractivité de semi-groupes de diffusion, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 775-778.

[7]

D. Bakry and M. Émery, Diffusions hypercontractives, in Séminaire de Probabilités, XIX, (1983/84), Lecture Notes in Math., 1123, Springer, Berlin, 1985, 177-206. doi: 10.1007/BFb0075847.

[8]

D. Bakry and M. Émery, Inégalités de Sobolev pour un semi-groupe symétrique, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 411-413.

[9]

D. Bakry and M. Ledoux, Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator, Duke Math. J., 85 (1996), 253-270. doi: 10.1215/S0012-7094-96-08511-7.

[10]

W. Beckner, A generalized Poincaré inequality for Gaussian measures, Proc. Amer. Math. Soc., 105 (1989), 397-400. doi: 10.2307/2046956.

[11]

_______, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere $S^n$, Proc. Nat. Acad. Sci. U.S.A., 89 (1992), 4816-4819.

[12]

_______, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2), 138 (1993), 213-242.

[13]

A. Bentaleb, Inégalité de Sobolev pour l'opérateur ultrasphérique, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 187-190.

[14]

_______, Sur les fonctions extrémales des inégalités de Sobolev des opérateurs de diffusion, in Séminaire de Probabilités, XXXVI, Lecture Notes in Math., 1801, Springer, Berlin, 2003, 230-250.

[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), 55-59. doi: 10.3792/pjaa.86.55.

[16]

M. Berger, P. Gauduchon and E. Mazet, Le Spectre d'une Variété Riemannienne, Lecture Notes in Mathematics, 194, Springer-Verlag, Berlin, 1971.

[17]

G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. Funct. Anal., 100 (1991), 18-24. doi: 10.1016/0022-1236(91)90099-Q.

[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), 859-875. doi: 10.1137/S0036141092230593.

[19]

M.-F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., 106 (1991), 489-539. doi: 10.1007/BF01243922.

[20]

F. Bolley and I. Gentil, Phi-entropy inequalities and Fokker-Planck equations, in Progress in Analysis and Its Applications, World Sci. Publ., Hackensack, NJ, 2010, 463-469. doi: 10.1142/9789814313179_0060.

[21]

_______, Phi-entropy inequalities for diffusion semigroups, J. Math. Pures Appl., 93 (2010), 449-473.

[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), 117-146. doi: 10.1017/S0013091501000426.

[23]

_______, On the second best constant in logarithmic Sobolev inequalities on complete Riemannian manifolds, Bull. Sci. Math., 127 (2003), 292-312.

[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), 478-494. doi: 10.1137/S0036141001398435.

[25]

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

[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), 113-142. doi: 10.1512/iumj.2000.49.1756.

[27]

D. Chafai, Entropies, convexity, and functional inequalities: On $\Phi$-entropies and $\Phi$-Sobolev inequalities, J. Math. Kyoto Univ., 44 (2004), 325-363.

[28]

S.-Y. A. Chang and P. C. Yang, Prescribing Gaussian curvature on $\mathbb S^2$, Acta Math., 159 (1987), 215-259. doi: 10.1007/BF02392560.

[29]

A. Cianchi, N. Fusco, F. Maggi and A. Pratelli, The sharp Sobolev inequality in quantitative form, J. Eur. Math. Soc. (JEMS), 11 (2009), 1105-1139. doi: 10.4171/JEMS/176.

[30]

I. Csiszár, Information-type measures of difference of probability distributions and indirect observations, Studia Sci. Math. Hungar., 2 (1967), 299-318

[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), 847-875. doi: 10.1016/S0021-7824(02)01266-7.

[32]

J. 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.

[33]

______, Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature, J. Funct. Anal., 254 (2008), 593-611.

[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, Series B, 34 (2013), 99-112. doi: 10.1007/s11401-012-0756-6.

[35]

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

[36]

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

[37]

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

[38]

J. Dolbeault and G. Karch, Large time behaviour of solutions to nonhomogeneous diffusion equations, in Self-Similar Solutions of Nonlinear PDE, Banach Center Publ., 74, Polish Acad. Sci., Warsaw, 2006, 133-147. doi: 10.4064/bc74-0-8.

[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), 477-494. doi: 10.4310/CMS.2008.v6.n2.a10.

[40]

P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, New York, 1983.

[41]

É. Fontenas, Sur les constantes de Sobolev des variétés riemanniennes compactes et les fonctions extrémales des sphères, Bull. Sci. Math., 121 (1997), 71-96.

[42]

_______, Sur les minorations des constantes de Sobolev et de Sobolev logarithmiques pour les opérateurs de Jacobi et de Laguerre, in Séminaire de Probabilités, XXXII, Lecture Notes in Math., 1686, Springer, Berlin, 1998, 14-29.

[43]

N. Ghoussoub and C.-S. Lin, On the best constant in the Moser-Onofri-Aubin inequality, Comm. Math. Phys., 298 (2010), 869-878. doi: 10.1007/s00220-010-1079-7.

[44]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.

[45]

L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083. doi: 10.2307/2373688.

[46]

S. Kullback, On the convergence of discrimination information, IEEE Trans. Information Theory, IT-14 (1968), 765-766.

[47]

R. Latała and K. Oleszkiewicz, Between Sobolev and Poincaré, in Geometric Aspects of Functional Analysis, Lecture Notes in Math., 1745, Springer, Berlin, 2000, 147-168. doi: 10.1007/BFb0107213.

[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), 1337-1342.

[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), 249-283.

[50]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118 (1983), 349-374. doi: 10.2307/2007032.

[51]

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

[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), 252-283. doi: 10.1016/0022-1236(82)90069-6.

[53]

E. Onofri, On the positivity of the effective action in a theory of random surfaces, Comm. Math. Phys., 86 (1982), 321-326. doi: 10.1007/BF01212171.

[54]

B. Osgood, R. Phillips and P. Sarnak, Compact isospectral sets of plane domains, Proc. Nat. Acad. Sci. U.S.A., 85 (1988), 5359-5361. doi: 10.1073/pnas.85.15.5359.

[55]

______, Compact isospectral sets of surfaces, J. Funct. Anal., 80 (1988), 212-234.

[56]

______, Extremals of determinants of Laplacians, J. Funct. Anal., 80 (1988), 148-211.

[57]

M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Translated and edited by Amiel Feinstein, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964.

[58]

A. Unterreiter, A. Arnold, P. Markowich and G. Toscani, On generalized Csiszár-Kullback inequalities, Monatsh. Math., 131 (2000), 235-253. doi: 10.1007/s006050070013.

[59]

F. B. Weissler, Logarithmic Sobolev inequalities and hypercontractive estimates on the circle, J. Funct. Anal., 37 (1980), 218-234. doi: 10.1016/0022-1236(80)90042-7.

[60]

H. Widom, On an inequality of Osgood, Phillips and Sarnak, Proc. Amer. Math. Soc., 102 (1988), 773-774. doi: 10.2307/2047262.

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