Logarithmic Sobolev and Shannon's inequalities and an application to the uncertainty principle

Takayoshi Ogawa; Kento Seraku

doi:10.3934/cpaa.2018079

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2018, Volume 17, Issue 4: 1651-1669. Doi: 10.3934/cpaa.2018079

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Logarithmic Sobolev and Shannon's inequalities and an application to the uncertainty principle

Takayoshi Ogawa^1, and
Kento Seraku²

1.
Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan
2.
Tamachi Branch, Risona Bank Co., Ltd., Tokyo 108-014, Japan

Received: December 31, 2016

Revised: November 30, 2017

Published: April 2018

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Abstract

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.

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Mathematics Subject Classification: Primary: 42B37, 49K20; Secondary: 26D10.

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References

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