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Quantitative logarithmic Sobolev inequalities and stability estimates
1. | Université Pierre et Marie Curie, Paris, France |
2. | Carnegie Mellon University, Pittsburgh, United States |
3. | University of Toulouse and Institut Universitaire de France, Toulouse, France |
References:
[1] |
D. Bakry, L'hypercontractivité et son utilisation en théorie des semigroupes, École d'Été de Probabilités de Saint-Flour, Lecture Notes in Math., Springer, 1581 (1994), 1-114.
doi: 10.1007/BFb0073872. |
[2] |
D. Bakry, F. Barthe, P. Cattiaux and A. Guillin, A simple proof of the Poincaré inequality in a large class of probability measures including log-concave cases, Elec. Comm. Prob., 13 (2008), 60-66.
doi: 10.1214/ECP.v13-1352. |
[3] |
D. Bakry and M. Émery, Diffusions hypercontractives, Séminaire de Probabilités XIX, Lecture Notes in Math., Springe, 1123 (1985), 177-206.
doi: 10.1007/BFb0075847. |
[4] |
M. Barchiesi, A. Brancolini and V. Julin, Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality, to appear in Ann. Probab., 2015. |
[5] |
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. |
[6] |
D. Bakry, I. Gentil and M. Ledoux, Analysis and geometry of Markov diffusion operators, Grundlehren der mathematischen Wissenschaften, 348, Springer, Berlin, 2014.
doi: 10.1007/978-3-319-00227-9. |
[7] |
F. Barthe and A. V. Kolesnikov, Mass transport and variants of the logarithmic Sobolev inequality, J. Geom. Anal., 18 (2008), 921-979.
doi: 10.1007/s12220-008-9039-6. |
[8] |
S. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal., 163 (1999), 1-28.
doi: 10.1006/jfan.1998.3326. |
[9] |
S. Bobkov, N. Gozlan, C. Roberto and P.-M. Samson, Bounds on the deficit in the logarithmic Sobolev inequality, J. Funct. Anal., 267 (2014), 4110-4138.
doi: 10.1016/j.jfa.2014.09.016. |
[10] |
L. Caffarelli, A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. of Math., 131 (1990), 129-134.
doi: 10.2307/1971509. |
[11] |
L. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5 (1992), 99-104.
doi: 10.1090/S0894-0347-1992-1124980-8. |
[12] |
E. Carlen, Some integral identities and inequalities for entire functions and their application to the coherent state transform, J. Funct. Anal., 97 (1991), 231-249.
doi: 10.1016/0022-1236(91)90022-W. |
[13] |
E. Carlen, Superadditivity of Fisher's information and logarithmic Sobolev inequalities, J. Funct. Anal., 101 (1991), 194-211.
doi: 10.1016/0022-1236(91)90155-X. |
[14] |
M. Christ, A sharpened Hausdorff-Young inequality,, , ().
|
[15] |
A. Cianchi, N. Fusco, F. Maggi and A. Pratelli, The sharp Sobolev inequality in quantitative form, J. Eur. Math. Soc., 11 (2009), 1105-1139.
doi: 10.4171/JEMS/176. |
[16] |
D. Cordero-Erausquin, Some applications of mass transport to Gaussian type inequalities, Arch. Rational Mech. Anal., 161 (2002), 257-269.
doi: 10.1007/s002050100185. |
[17] |
G. De Philipis and A. Figalli, $W^{2,1}$ regularity of solutions to the Monge-Ampère equation, Invent. Math., 192 (2013), 55-69.
doi: 10.1007/s00222-012-0405-4. |
[18] |
R. Eldan, A two-sided estimate for the Gaussian noise stability deficit, Invent. Math., 201 (2015), 561-624.
doi: 10.1007/s00222-014-0556-6. |
[19] |
A. Figalli and E. Indrei, A sharp stability result for the relative isoperimetric inequality inside convex cones, J. Geom. Anal., 23 (2013), 938-969.
doi: 10.1007/s12220-011-9270-4. |
[20] |
A. Figalli and D. Jerison, Quantitative stability for the Brunn-Minkowski inequality,, J. Eur. Math. Soc., ().
|
[21] |
A. Figalli, F. Maggi and A. Pratelli, A refined Brunn-Minkowski inequality for convex sets, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2511-2519.
doi: 10.1016/j.anihpc.2009.07.004. |
[22] |
A. Figalli, F. Maggi and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math., 182 (2010), 167-211.
doi: 10.1007/s00222-010-0261-z. |
[23] |
A. Figalli, F. Maggi and A. Pratelli, Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation, Adv. Math., 242 (2013), 80-101.
doi: 10.1016/j.aim.2013.04.007. |
[24] |
J. Fontbona, N. Gozlan and J.-F. Jabir, A variational approach to some transport inequalities, preprint (2015). |
[25] |
N. Fusco, F. Maggi and A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math., 168 (2008), 941-980.
doi: 10.4007/annals.2008.168.941. |
[26] |
A. Figalli and R. Neumayer, Gradient stability for the Sobolev inequality: the case $p \geq 2$, preprint, 2015. |
[27] |
L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083.
doi: 10.2307/2373688. |
[28] |
E. Indrei, A sharp lower bound on the polygonal isoperimetric deficit, Proc. Amer. Math. Soc., 144 (2016), 3115-3122.
doi: 10.1090/proc/12947. |
[29] |
E. Indrei and D. Marcon, A quantitative log-Sobolev inequality for a two parameter family of functions, Int. Math. Res. Not. IMRN, (2014), 5563-5580. |
[30] |
E. Indrei and L. Nurbekyan, On the stability of the polygonal isoperimetric inequality, Advances in Mathematics, 276 (2015), 62-86.
doi: 10.1016/j.aim.2015.02.013. |
[31] |
M. Ledoux, Logarithmic Sobolev inequalities for unbounded spin systems revisited, Séminaire de Probabilités XXXV, Lecture Notes in Math., Springer, 1755 (2001), 167-194.
doi: 10.1007/978-3-540-44671-2_13. |
[32] |
J. Lehec, Representation formula for the entropy and functional inequalities, Ann. IHP: Probab. Stat., 49 (2013), 885-899.
doi: 10.1214/11-AIHP464. |
[33] |
E. Lieb, Proof of an entropy conjecture of Wehrl, Comm. Math. Phys., 62 (1978), 35-41.
doi: 10.1007/BF01940328. |
[34] |
E. Lieb, Thomas-Fermi and related theories of atoms and molecules, Rev. Mod. Phys., 53 (1981), 603-641.
doi: 10.1103/RevModPhys.53.603. |
[35] |
R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J., 80 (1995), 309-323.
doi: 10.1215/S0012-7094-95-08013-2. |
[36] |
C. Mooney, Partial regularity for singular solutions to the Monge-Ampere equation, Comm. Pure Appl. Math., 68 (2015), 1066-1084.
doi: 10.1002/cpa.21534. |
[37] |
F. Otto and C. Villani, Generalization of an inequality by Talagrand, and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400.
doi: 10.1006/jfan.1999.3557. |
[38] |
I. Segal, Mathematical characterization of the physical vacuum of the physical vacuum for a linear Bose-Einstein field, Illinois J. Math., 6 (1962), 500-523. |
[39] |
I. Segal, Mathematical Problems in Relativistic Quantum Mechanics, American Mathematical Society, Providence, 1963. |
[40] |
I. Segal, Construction of non-linear local quantum processes I, Ann. of Math., 92 (1970), 462-481.
doi: 10.2307/1970628. |
[41] |
M. Talagrand, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal., 6 (1996), 587-600.
doi: 10.1007/BF02249265. |
[42] |
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematic,s 58, American Mathematical Society, Providence, 2003.
doi: 10.1007/b12016. |
[43] |
C. Villani, Optimal transport. Old and new, Grundlehren der mathematischen Wissenschaften, 338, Springer, Berlin, 2009.
doi: 10.1007/978-3-540-71050-9. |
[44] |
A. Wehrl, On the relation between classical and quantum mechanical entropy, Rep. Mat. Phys., 16 (1979), 353-358.
doi: 10.1016/0034-4877(79)90070-3. |
show all references
References:
[1] |
D. Bakry, L'hypercontractivité et son utilisation en théorie des semigroupes, École d'Été de Probabilités de Saint-Flour, Lecture Notes in Math., Springer, 1581 (1994), 1-114.
doi: 10.1007/BFb0073872. |
[2] |
D. Bakry, F. Barthe, P. Cattiaux and A. Guillin, A simple proof of the Poincaré inequality in a large class of probability measures including log-concave cases, Elec. Comm. Prob., 13 (2008), 60-66.
doi: 10.1214/ECP.v13-1352. |
[3] |
D. Bakry and M. Émery, Diffusions hypercontractives, Séminaire de Probabilités XIX, Lecture Notes in Math., Springe, 1123 (1985), 177-206.
doi: 10.1007/BFb0075847. |
[4] |
M. Barchiesi, A. Brancolini and V. Julin, Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality, to appear in Ann. Probab., 2015. |
[5] |
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. |
[6] |
D. Bakry, I. Gentil and M. Ledoux, Analysis and geometry of Markov diffusion operators, Grundlehren der mathematischen Wissenschaften, 348, Springer, Berlin, 2014.
doi: 10.1007/978-3-319-00227-9. |
[7] |
F. Barthe and A. V. Kolesnikov, Mass transport and variants of the logarithmic Sobolev inequality, J. Geom. Anal., 18 (2008), 921-979.
doi: 10.1007/s12220-008-9039-6. |
[8] |
S. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal., 163 (1999), 1-28.
doi: 10.1006/jfan.1998.3326. |
[9] |
S. Bobkov, N. Gozlan, C. Roberto and P.-M. Samson, Bounds on the deficit in the logarithmic Sobolev inequality, J. Funct. Anal., 267 (2014), 4110-4138.
doi: 10.1016/j.jfa.2014.09.016. |
[10] |
L. Caffarelli, A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. of Math., 131 (1990), 129-134.
doi: 10.2307/1971509. |
[11] |
L. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5 (1992), 99-104.
doi: 10.1090/S0894-0347-1992-1124980-8. |
[12] |
E. Carlen, Some integral identities and inequalities for entire functions and their application to the coherent state transform, J. Funct. Anal., 97 (1991), 231-249.
doi: 10.1016/0022-1236(91)90022-W. |
[13] |
E. Carlen, Superadditivity of Fisher's information and logarithmic Sobolev inequalities, J. Funct. Anal., 101 (1991), 194-211.
doi: 10.1016/0022-1236(91)90155-X. |
[14] |
M. Christ, A sharpened Hausdorff-Young inequality,, , ().
|
[15] |
A. Cianchi, N. Fusco, F. Maggi and A. Pratelli, The sharp Sobolev inequality in quantitative form, J. Eur. Math. Soc., 11 (2009), 1105-1139.
doi: 10.4171/JEMS/176. |
[16] |
D. Cordero-Erausquin, Some applications of mass transport to Gaussian type inequalities, Arch. Rational Mech. Anal., 161 (2002), 257-269.
doi: 10.1007/s002050100185. |
[17] |
G. De Philipis and A. Figalli, $W^{2,1}$ regularity of solutions to the Monge-Ampère equation, Invent. Math., 192 (2013), 55-69.
doi: 10.1007/s00222-012-0405-4. |
[18] |
R. Eldan, A two-sided estimate for the Gaussian noise stability deficit, Invent. Math., 201 (2015), 561-624.
doi: 10.1007/s00222-014-0556-6. |
[19] |
A. Figalli and E. Indrei, A sharp stability result for the relative isoperimetric inequality inside convex cones, J. Geom. Anal., 23 (2013), 938-969.
doi: 10.1007/s12220-011-9270-4. |
[20] |
A. Figalli and D. Jerison, Quantitative stability for the Brunn-Minkowski inequality,, J. Eur. Math. Soc., ().
|
[21] |
A. Figalli, F. Maggi and A. Pratelli, A refined Brunn-Minkowski inequality for convex sets, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2511-2519.
doi: 10.1016/j.anihpc.2009.07.004. |
[22] |
A. Figalli, F. Maggi and A. Pratelli, A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math., 182 (2010), 167-211.
doi: 10.1007/s00222-010-0261-z. |
[23] |
A. Figalli, F. Maggi and A. Pratelli, Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation, Adv. Math., 242 (2013), 80-101.
doi: 10.1016/j.aim.2013.04.007. |
[24] |
J. Fontbona, N. Gozlan and J.-F. Jabir, A variational approach to some transport inequalities, preprint (2015). |
[25] |
N. Fusco, F. Maggi and A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math., 168 (2008), 941-980.
doi: 10.4007/annals.2008.168.941. |
[26] |
A. Figalli and R. Neumayer, Gradient stability for the Sobolev inequality: the case $p \geq 2$, preprint, 2015. |
[27] |
L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083.
doi: 10.2307/2373688. |
[28] |
E. Indrei, A sharp lower bound on the polygonal isoperimetric deficit, Proc. Amer. Math. Soc., 144 (2016), 3115-3122.
doi: 10.1090/proc/12947. |
[29] |
E. Indrei and D. Marcon, A quantitative log-Sobolev inequality for a two parameter family of functions, Int. Math. Res. Not. IMRN, (2014), 5563-5580. |
[30] |
E. Indrei and L. Nurbekyan, On the stability of the polygonal isoperimetric inequality, Advances in Mathematics, 276 (2015), 62-86.
doi: 10.1016/j.aim.2015.02.013. |
[31] |
M. Ledoux, Logarithmic Sobolev inequalities for unbounded spin systems revisited, Séminaire de Probabilités XXXV, Lecture Notes in Math., Springer, 1755 (2001), 167-194.
doi: 10.1007/978-3-540-44671-2_13. |
[32] |
J. Lehec, Representation formula for the entropy and functional inequalities, Ann. IHP: Probab. Stat., 49 (2013), 885-899.
doi: 10.1214/11-AIHP464. |
[33] |
E. Lieb, Proof of an entropy conjecture of Wehrl, Comm. Math. Phys., 62 (1978), 35-41.
doi: 10.1007/BF01940328. |
[34] |
E. Lieb, Thomas-Fermi and related theories of atoms and molecules, Rev. Mod. Phys., 53 (1981), 603-641.
doi: 10.1103/RevModPhys.53.603. |
[35] |
R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J., 80 (1995), 309-323.
doi: 10.1215/S0012-7094-95-08013-2. |
[36] |
C. Mooney, Partial regularity for singular solutions to the Monge-Ampere equation, Comm. Pure Appl. Math., 68 (2015), 1066-1084.
doi: 10.1002/cpa.21534. |
[37] |
F. Otto and C. Villani, Generalization of an inequality by Talagrand, and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400.
doi: 10.1006/jfan.1999.3557. |
[38] |
I. Segal, Mathematical characterization of the physical vacuum of the physical vacuum for a linear Bose-Einstein field, Illinois J. Math., 6 (1962), 500-523. |
[39] |
I. Segal, Mathematical Problems in Relativistic Quantum Mechanics, American Mathematical Society, Providence, 1963. |
[40] |
I. Segal, Construction of non-linear local quantum processes I, Ann. of Math., 92 (1970), 462-481.
doi: 10.2307/1970628. |
[41] |
M. Talagrand, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal., 6 (1996), 587-600.
doi: 10.1007/BF02249265. |
[42] |
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematic,s 58, American Mathematical Society, Providence, 2003.
doi: 10.1007/b12016. |
[43] |
C. Villani, Optimal transport. Old and new, Grundlehren der mathematischen Wissenschaften, 338, Springer, Berlin, 2009.
doi: 10.1007/978-3-540-71050-9. |
[44] |
A. Wehrl, On the relation between classical and quantum mechanical entropy, Rep. Mat. Phys., 16 (1979), 353-358.
doi: 10.1016/0034-4877(79)90070-3. |
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