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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

 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.

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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