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Proceedings of the American Mathematical Society

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Beckner type of the logarithmic Sobolev and a new type of Shannon's inequalities and an application to the uncertainty principle


Authors: Hideo Kubo, Takayoshi Ogawa and Takeshi Suguro
Journal: Proc. Amer. Math. Soc. 147 (2019), 1511-1518
MSC (2010): Primary 26D10, 39B62
DOI: https://doi.org/10.1090/proc/14350
Published electronically: December 19, 2018
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Abstract: We consider the inequality which has a ``dual'' relation with Beckner's logarithmic Sobolev inequality. By using the relative entropy, we identify the sharp constant and the extremal of this inequality. Moreover, we derive the logarithmic uncertainty principle like Beckner's one.


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Additional Information

Hideo Kubo
Affiliation: Faculty of Science, Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

Takayoshi Ogawa
Affiliation: Mathematical Institute/Research Alliance Center of Mathematical Science, Tohoku University, Sendai 980-8578, Japan

Takeshi Suguro
Affiliation: Mathematical Institue, Tohoku University, Sendai 980-8578, Japan

DOI: https://doi.org/10.1090/proc/14350
Received by editor(s): May 31, 2018
Published electronically: December 19, 2018
Additional Notes: The work of the first author was partially supported by JSPS Grant-in-aid for Scientific Research S #16H06339.
The work of the second author was partially supported by JSPS Grant-in-aid for Scientific Research S #25220702 and Challenging Research (Pioneering) #17H06199.
Communicated by: Joachim Krieger
Article copyright: © Copyright 2018 American Mathematical Society