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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 and 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. 

[2]

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

[3]

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

[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.

[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. 

[6]

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

[7]

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

[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.

[9]

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

[10]

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

[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. 

[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. 

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[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.

[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. 

[6]

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

[7]

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

[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.

[9]

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

[10]

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

[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. 

[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. 

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

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