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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 & Applied Analysis, 2018, 17 (4) : 1651-1669. doi: 10.3934/cpaa.2018079
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