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

Abstract

The uncertainty principle of Heisenberg type can be generalized via the Boltzmann entropy functional. After reviewing the Lp 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.