An optimal logarithmic Sobolev inequality with Lipschitz constants |
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Authors: | Yasuhiro Fujita |
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Affiliation: | Department of Mathematics, University of Toyama, Toyama 930-8555, Japan |
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Abstract: | 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. |
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Keywords: | Lipschitz constants Logarithmic Sobolev inequality Heat equation Lipschitz regularizing effect |
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