• Previous Article
    Global well-posedness of 2D incompressible Magnetohydrodynamic equations with horizontal dissipation
  • DCDS Home
  • This Issue
  • Next Article
    Asymptotic behavior of spreading fronts in an anisotropic multi-stable equation on $ \mathit{\boldsymbol{\mathbb{R}^N}} $
doi: 10.3934/dcds.2022053
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Instability results for the logarithmic Sobolev inequality and its application to related inequalities

University of Illinois at Urbana–Champaign, USA

Received  October 2021 Revised  February 2022 Early access April 2022

Fund Project: The author was supported in part by NSF grant #1403417-DMS; Rodrigo Bañuelos PI

We show that there are no general stability results for the logarithmic Sobolev inequality in terms of the Wasserstein distances and $ L^{p}(d\gamma) $ distance for $ p>1 $. To this end, we construct a sequence of centered probability measures such that the deficit of the logarithmic Sobolev inequality converges to zero but the relative entropy and the moments do not, which leads to instability for the logarithmic Sobolev inequality. As an application, we prove instability results for Talagrand's transportation inequality and the Beckner–Hirschman inequality.

Citation: Daesung Kim. Instability results for the logarithmic Sobolev inequality and its application to related inequalities. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022053
References:
[1]

K. I. Babenko, An inequality in the theory of Fourier integrals, Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 531-542. 

[2]

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.

[3]

W. Beckner, Inequalities in Fourier analysis, Ann. of Math., 102 (1975), 159-182.  doi: 10.2307/1970980.

[4]

S. G. BobkovN. GozlanC. 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.

[5]

F. BolleyI. Gentil and A. Guillin, Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp-Lieb inequalities, Ann. Probab., 46 (2018), 261-301.  doi: 10.1214/17-AOP1184.

[6]

L. A. Caffarelli, Monotonicity properties of optimal transportation and the FKG and related inequalities, Comm. Math. Phys., 214 (2000), 547-563.  doi: 10.1007/s002200000257.

[7]

E. A. 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.

[8]

M. Christ, A sharpened hausdorff-young inequality, arXiv e-print (2014).

[9]

D. Cordero-Erausquin, Transport inequalities for log-concave measures, quantitative forms, and applications, Canad. J. Math., 69 (2017), 481-501.  doi: 10.4153/CJM-2016-046-3.

[10]

J. Dolbeault and G. Toscani, Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities, Int. Math. Res. Not. IMRN, 2016 (2016), 473-498.  doi: 10.1093/imrn/rnv131.

[11]

R. EldanJ. Lehec and Y. Shenfeld, Stability of the logarithmic Sobolev inequality via the Föllmer process, Ann. Inst. Henri Poincaré Probab. Stat., 56 (2020), 2253-2269.  doi: 10.1214/19-AIHP1038.

[12]

M. FathiE. Indrei and M. Ledoux, Quantitative logarithmic Sobolev inequalities and stability estimates, Discrete Contin. Dyn. Syst., 36 (2016), 6835-6853.  doi: 10.3934/dcds.2016097.

[13]

F. FeoE. IndreiM. R. Posteraro and C. Roberto, Some remarks on the stability of the log-Sobolev inequality for the Gaussian measure, Potential Anal., 47 (2017), 37-52.  doi: 10.1007/s11118-016-9607-5.

[14]

N. Gozlan, The deficit in the gaussian log-sobolev inequality and inverse santalo inequalities, International Mathematics Research Notices, (2021). doi: 10.1093/imrn/rnab087.

[15]

I. I. Hirschman and Jr ., A note on entropy, Amer. J. Math., 79 (1957), 152-156.  doi: 10.2307/2372390.

[16]

E. Indrei and D. Kim, Deficit estimates for the logarithmic Sobolev inequality, Differential Integral Equations, 34 (2021), 437-466. 

[17]

E. Indrei and D. Marcon, A quantitative log-Sobolev inequality for a two parameter family of functions, Int. Math. Res. Not. IMRN, 2014 (2014), 5563-5580.  doi: 10.1093/imrn/rnt138.

[18]

A. V. Kolesnikov, On Sobolev regularity of mass transport and transportation inequalities, Theory Probab. Appl., 57 (2013), 243-264.  doi: 10.1137/S0040585X97985947.

[19]

M. LedouxI. Nourdin and G. Peccati, A Stein deficit for the logarithmic Sobolev inequality, Sci. China Math., 60 (2017), 1163-1180.  doi: 10.1007/s11425-016-0134-7.

[20]

E. H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math., 102 (1990), 179-208.  doi: 10.1007/BF01233426.

[21]

E. H. Lieb and M. Loss, Analysis, 2$^{nd}$ edition, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.

[22]

D. Mikulincer, Stability of Talagrand's Gaussian transport-entropy inequality via the Föllmer process, Israel J. Math., 242 (2021), 215-241.  doi: 10.1007/s11856-021-2129-x.

[23]

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.

[24]

M. Talagrand, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal., 6 (1996), 587-600.  doi: 10.1007/BF02249265.

show all references

References:
[1]

K. I. Babenko, An inequality in the theory of Fourier integrals, Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 531-542. 

[2]

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.

[3]

W. Beckner, Inequalities in Fourier analysis, Ann. of Math., 102 (1975), 159-182.  doi: 10.2307/1970980.

[4]

S. G. BobkovN. GozlanC. 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.

[5]

F. BolleyI. Gentil and A. Guillin, Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp-Lieb inequalities, Ann. Probab., 46 (2018), 261-301.  doi: 10.1214/17-AOP1184.

[6]

L. A. Caffarelli, Monotonicity properties of optimal transportation and the FKG and related inequalities, Comm. Math. Phys., 214 (2000), 547-563.  doi: 10.1007/s002200000257.

[7]

E. A. 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.

[8]

M. Christ, A sharpened hausdorff-young inequality, arXiv e-print (2014).

[9]

D. Cordero-Erausquin, Transport inequalities for log-concave measures, quantitative forms, and applications, Canad. J. Math., 69 (2017), 481-501.  doi: 10.4153/CJM-2016-046-3.

[10]

J. Dolbeault and G. Toscani, Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities, Int. Math. Res. Not. IMRN, 2016 (2016), 473-498.  doi: 10.1093/imrn/rnv131.

[11]

R. EldanJ. Lehec and Y. Shenfeld, Stability of the logarithmic Sobolev inequality via the Föllmer process, Ann. Inst. Henri Poincaré Probab. Stat., 56 (2020), 2253-2269.  doi: 10.1214/19-AIHP1038.

[12]

M. FathiE. Indrei and M. Ledoux, Quantitative logarithmic Sobolev inequalities and stability estimates, Discrete Contin. Dyn. Syst., 36 (2016), 6835-6853.  doi: 10.3934/dcds.2016097.

[13]

F. FeoE. IndreiM. R. Posteraro and C. Roberto, Some remarks on the stability of the log-Sobolev inequality for the Gaussian measure, Potential Anal., 47 (2017), 37-52.  doi: 10.1007/s11118-016-9607-5.

[14]

N. Gozlan, The deficit in the gaussian log-sobolev inequality and inverse santalo inequalities, International Mathematics Research Notices, (2021). doi: 10.1093/imrn/rnab087.

[15]

I. I. Hirschman and Jr ., A note on entropy, Amer. J. Math., 79 (1957), 152-156.  doi: 10.2307/2372390.

[16]

E. Indrei and D. Kim, Deficit estimates for the logarithmic Sobolev inequality, Differential Integral Equations, 34 (2021), 437-466. 

[17]

E. Indrei and D. Marcon, A quantitative log-Sobolev inequality for a two parameter family of functions, Int. Math. Res. Not. IMRN, 2014 (2014), 5563-5580.  doi: 10.1093/imrn/rnt138.

[18]

A. V. Kolesnikov, On Sobolev regularity of mass transport and transportation inequalities, Theory Probab. Appl., 57 (2013), 243-264.  doi: 10.1137/S0040585X97985947.

[19]

M. LedouxI. Nourdin and G. Peccati, A Stein deficit for the logarithmic Sobolev inequality, Sci. China Math., 60 (2017), 1163-1180.  doi: 10.1007/s11425-016-0134-7.

[20]

E. H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math., 102 (1990), 179-208.  doi: 10.1007/BF01233426.

[21]

E. H. Lieb and M. Loss, Analysis, 2$^{nd}$ edition, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.

[22]

D. Mikulincer, Stability of Talagrand's Gaussian transport-entropy inequality via the Föllmer process, Israel J. Math., 242 (2021), 215-241.  doi: 10.1007/s11856-021-2129-x.

[23]

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.

[24]

M. Talagrand, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal., 6 (1996), 587-600.  doi: 10.1007/BF02249265.

Figure 1.  The graph of $ f_k(x)\gamma(x) $ constructed in the proof Lemma 1.7
[1]

Takayoshi Ogawa, Kento Seraku. Logarithmic Sobolev and Shannon's inequalities and an application to the uncertainty principle. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1651-1669. doi: 10.3934/cpaa.2018079

[2]

YanYan Li, Tonia Ricciardi. A sharp Sobolev inequality on Riemannian manifolds. Communications on Pure and Applied Analysis, 2003, 2 (1) : 1-31. doi: 10.3934/cpaa.2003.2.1

[3]

Igor E. Verbitsky. The Hessian Sobolev inequality and its extensions. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 6165-6179. doi: 10.3934/dcds.2015.35.6165

[4]

Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171

[5]

Felipe Riquelme. Ruelle's inequality in negative curvature. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2809-2825. doi: 10.3934/dcds.2018119

[6]

S. S. Dragomir, C. E. M. Pearce. Jensen's inequality for quasiconvex functions. Numerical Algebra, Control and Optimization, 2012, 2 (2) : 279-291. doi: 10.3934/naco.2012.2.279

[7]

Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951

[8]

Pablo Raúl Stinga, Chao Zhang. Harnack's inequality for fractional nonlocal equations. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3153-3170. doi: 10.3934/dcds.2013.33.3153

[9]

Prosenjit Roy. On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5207-5222. doi: 10.3934/dcds.2019212

[10]

Jianqing Chen. Best constant of 3D Anisotropic Sobolev inequality and its applications. Communications on Pure and Applied Analysis, 2010, 9 (3) : 655-666. doi: 10.3934/cpaa.2010.9.655

[11]

José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138

[12]

Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935

[13]

Frank Jochmann. A variational inequality in Bean's model for superconductors with displacement current. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 545-565. doi: 10.3934/dcds.2009.25.545

[14]

Anat Amir. Sharpness of Zapolsky's inequality for quasi-states and Poisson brackets. Electronic Research Announcements, 2011, 18: 61-68. doi: 10.3934/era.2011.18.61

[15]

Pierdomenico Pepe. A nonlinear version of Halanay's inequality for the uniform convergence to the origin. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021045

[16]

Jorge A. Becerril, Javier F. Rosenblueth. Necessity for isoperimetric inequality constraints. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1129-1158. doi: 10.3934/dcds.2017047

[17]

Gisella Croce, Bernard Dacorogna. On a generalized Wirtinger inequality. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1329-1341. doi: 10.3934/dcds.2003.9.1329

[18]

Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043

[19]

Hubert L. Bray, Marcus A. Khuri. A Jang equation approach to the Penrose inequality. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 741-766. doi: 10.3934/dcds.2010.27.741

[20]

Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (110)
  • HTML views (53)
  • Cited by (0)

Other articles
by authors

[Back to Top]