American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement
  • CPAA Home
  • This Issue
  • Next Article
    Steady-state mode interactions of radially symmetric modes for the Lugiato-Lefever equation on a disk
July  2018, 17(4): 1651-1669. doi: 10.3934/cpaa.2018079

Logarithmic Sobolev and Shannon's inequalities and an application to the uncertainty principle

Takayoshi Ogawa 1, and  Kento Seraku 2,

1. 

Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan
2. 

Tamachi Branch, Risona Bank Co., Ltd., Tokyo 108-014, Japan

Received  January 2017 Revised  December 2017 Published  April 2018

The uncertainty principle of Heisenberg type can be generalized via the Boltzmann entropy functional. After reviewing the $L^p$ generalization of the logarithmic Sobolev inequality by Del Pino-Dolbeault [6], we introduce a generalized version of Shannon's inequality for the Boltzmann entropy functional which may regarded as a counter part of the logarithmic Sobolev inequality. Obtaining best possible constants of both inequalities, we connect both the inequalities to show a generalization of uncertainty principle of the Heisenberg type.

Keywords: Logarithmic Sobolev inequality, Shannon's inequality, uncertainty principle, the best possible constants.
Mathematics Subject Classification: Primary: 42B37, 49K20; Secondary: 26D10.
Citation: Takayoshi Ogawa, Kento Seraku. Logarithmic Sobolev and Shannon's inequalities and an application to the uncertainty principle. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1651-1669. doi: 10.3934/cpaa.2018079
References:
[1]

W. Beckner, Pitt's inequality and the uncertainty principle, Proc. Ameri. Math. Soc., 123 (1995), 1897-1905.   Google Scholar

[2]

W. Beckner and M. Pearson, On sharp Sobolev embeddings and the logarithmic Sobolev inequality, Bull. London Math. Soc., 30 (1998), 80-84.   Google Scholar

[3]

J.-F. Bercher, On a (β, q)-generalized Fisher information and inequalities involving q-Gaussian distributions, J. Math. Phys., 53 (2012), 82B03. Google Scholar

[4]

J.-F. Bercher, On generalized Cramér-Rao inequalities, generalized Fisher information and characterizations of generalized q-Gaussian distributions, J. Phys. A, 45 (2012), 82B30. Google Scholar

[5]

M. Del Pino and J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl., 81 (2002), 847-875.   Google Scholar

[6]

M. Del Pino and J. Dolbeault, The optimal Euclidean Lp-Sobolev logarithmic inequality, J. Funct. Anal., 197 (2003), 151-161.   Google Scholar

[7]

L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083.   Google Scholar

[8]

M. Ledoux, Isoperimetry and Gaussian Analysis, Lectures on Probability Theory and Statistics (Saint-Flour 1994), Lecture Notes in Mathematics, Vol. 1648, Springer, Berlin, (1996), 165-294. Google Scholar

[9]

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

[10]

E. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Mathematics, 14, American Mathematical Society, 2001. Google Scholar

[11]

M. Kurokiba and T. Ogawa, Finite time blow up for a solution to system of the drift-diffusion equations in higher dimensions, Math. Z., 284 (2016), 231-253.   Google Scholar

[12]

T. Ogawa and H. Wakui, Non-uniform and finite time blow up for solutions to a drift-diffusion equation in higher dimensions, Anal. Appl., 14 (2016), 145-183.   Google Scholar

[13]

G. Rosen, Minimum value for c in the Sobolev inequality, SIAM J. Appl. Math., 21 (1971), 30-32.   Google Scholar

[14]

J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923.   Google Scholar

[15]

C. E. Shannon, A mathematical theory of communication, Bell System Tech. J., 27 (1948), 379-423, 623-656.   Google Scholar

[16]

C. E. Shannon and W. Weaver, The Mathematical Theory of Communication, The University of Illinois Press, Urbana, 1949. Google Scholar

[17]

A. J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Inform. Control, 2 (1959), 255-269.   Google Scholar

[18]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.   Google Scholar

[19]

F. B. Weissler, Logarithmic Sobolev inequalities for the heat-diffusion semigroup, Trans. Amer. Math. Soc., 237 (1978), 255-269.   Google Scholar

show all references

References:
[1]

W. Beckner, Pitt's inequality and the uncertainty principle, Proc. Ameri. Math. Soc., 123 (1995), 1897-1905.   Google Scholar

[2]

W. Beckner and M. Pearson, On sharp Sobolev embeddings and the logarithmic Sobolev inequality, Bull. London Math. Soc., 30 (1998), 80-84.   Google Scholar

[3]

J.-F. Bercher, On a (β, q)-generalized Fisher information and inequalities involving q-Gaussian distributions, J. Math. Phys., 53 (2012), 82B03. Google Scholar

[4]

J.-F. Bercher, On generalized Cramér-Rao inequalities, generalized Fisher information and characterizations of generalized q-Gaussian distributions, J. Phys. A, 45 (2012), 82B30. Google Scholar

[5]

M. Del Pino and J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl., 81 (2002), 847-875.   Google Scholar

[6]

M. Del Pino and J. Dolbeault, The optimal Euclidean Lp-Sobolev logarithmic inequality, J. Funct. Anal., 197 (2003), 151-161.   Google Scholar

[7]

L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 97 (1975), 1061-1083.   Google Scholar

[8]

M. Ledoux, Isoperimetry and Gaussian Analysis, Lectures on Probability Theory and Statistics (Saint-Flour 1994), Lecture Notes in Mathematics, Vol. 1648, Springer, Berlin, (1996), 165-294. Google Scholar

[9]

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

[10]

E. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Mathematics, 14, American Mathematical Society, 2001. Google Scholar

[11]

M. Kurokiba and T. Ogawa, Finite time blow up for a solution to system of the drift-diffusion equations in higher dimensions, Math. Z., 284 (2016), 231-253.   Google Scholar

[12]

T. Ogawa and H. Wakui, Non-uniform and finite time blow up for solutions to a drift-diffusion equation in higher dimensions, Anal. Appl., 14 (2016), 145-183.   Google Scholar

[13]

G. Rosen, Minimum value for c in the Sobolev inequality, SIAM J. Appl. Math., 21 (1971), 30-32.   Google Scholar

[14]

J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923.   Google Scholar

[15]

C. E. Shannon, A mathematical theory of communication, Bell System Tech. J., 27 (1948), 379-423, 623-656.   Google Scholar

[16]

C. E. Shannon and W. Weaver, The Mathematical Theory of Communication, The University of Illinois Press, Urbana, 1949. Google Scholar

[17]

A. J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Inform. Control, 2 (1959), 255-269.   Google Scholar

[18]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.   Google Scholar

[19]

F. B. Weissler, Logarithmic Sobolev inequalities for the heat-diffusion semigroup, Trans. Amer. Math. Soc., 237 (1978), 255-269.   Google Scholar

Figure 1.1.  Young Functions
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Jianqing Chen. Best constant of 3D Anisotropic Sobolev inequality and its applications. Communications on Pure & Applied Analysis, 2010, 9 (3) : 655-666. doi: 10.3934/cpaa.2010.9.655
[2]

Ezequiel R. Barbosa, Marcos Montenegro. On the geometric dependence of Riemannian Sobolev best constants. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1759-1777. doi: 10.3934/cpaa.2009.8.1759
[3]

Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171
[4]

Bernhard Kawohl. Symmetry results for functions yielding best constants in Sobolev-type inequalities. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 683-690. doi: 10.3934/dcds.2000.6.683
[5]

YanYan Li, Tonia Ricciardi. A sharp Sobolev inequality on Riemannian manifolds. Communications on Pure & Applied Analysis, 2003, 2 (1) : 1-31. doi: 10.3934/cpaa.2003.2.1
[6]

Igor E. Verbitsky. The Hessian Sobolev inequality and its extensions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6165-6179. doi: 10.3934/dcds.2015.35.6165
[7]

Carlo Morosi, Livio Pizzocchero. On the constants in a Kato inequality for the Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 557-586. doi: 10.3934/cpaa.2012.11.557
[8]

Felipe Riquelme. Ruelle's inequality in negative curvature. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2809-2825. doi: 10.3934/dcds.2018119
[9]

S. S. Dragomir, C. E. M. Pearce. Jensen's inequality for quasiconvex functions. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 279-291. doi: 10.3934/naco.2012.2.279
[10]

Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951
[11]

Prosenjit Roy. On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5207-5222. doi: 10.3934/dcds.2019212
[12]

Pablo Raúl Stinga, Chao Zhang. Harnack's inequality for fractional nonlocal equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3153-3170. doi: 10.3934/dcds.2013.33.3153
[13]

Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935
[14]

José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138
[15]

Frank Jochmann. A variational inequality in Bean's model for superconductors with displacement current. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 545-565. doi: 10.3934/dcds.2009.25.545
[16]

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
[17]

Dirk Pauly. On Maxwell's and Poincaré's constants. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 607-618. doi: 10.3934/dcdss.2015.8.607
[18]

Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043
[19]

Jochen Merker. Generalizations of logarithmic Sobolev inequalities. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 329-338. doi: 10.3934/dcdss.2008.1.329
[20]

Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (70)
  • HTML views (126)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences