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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 |
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 [
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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