An optimal logarithmic Sobolev inequality with Lipschitz constants

In this paper, we give an optimal logarithmic Sobolev inequality on Rn with Lipschitz constants. This inequality is a limit case of the Lp-logarithmic Sobolev inequality of Gentil (2003) [7] as p→∞. As a result of our inequality, we show that if a Lipschitz continuous function f on Rn fulfills some condition, then its Lipschitz constant can be expressed by using the entropy of f . We also show that a hypercontractivity of exponential type occurs in the heat equation on Rn. This is due to the Lipschitz regularizing effect of the heat equation. © 2011 Elsevier Inc. All rights reserved.