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.
Highlights
The uncertainty principle of Heisenberg type can be generalized via the Boltzmann entropy functional
The logarithmic Sobolev inequality is a version of the Sobolev inequality in the Sobolev spaces
Stam [17] first obtained the logarithmic Sobolev inequality in H1(Rn) and later on Gross [7] reconsidered the inequality with the Gaussian measure and showed a relation with the hypercontractivity of the semi-group in the probability theory
Summary
We first show the L2 based inequality of the logarithmic Sobolev inequality due to Stam and Gross. The inequality is presented as the following form For any a > 0 and f ∈ H1(Rn), the following inequality holds. The inequality (1.3) and its proof is deeply related with various important inequalities of Young and Hardy-Littlewood-Sobolev type as well as the hypercontractivity of the heat and other semi-group with the sharp constants (cf Weissler [19]). One can optimize the parameter a > 0 appearing in Proposition 1.1 and the equivalent form of (1.3) is obtained as follows which is obtained by Weissler [19]: Corollary 1.2 (Sharp logarithmic Sobolev inequality). The equivalent form of the inequality (1.3) is
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