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Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions

  Published:2015-09-15
 Printed: Dec 2015
  • Piotr Graczyk,
    Université d'Angers, 2 Boulevard Lavoisier , 49045 Angers Cedex 01, France
  • Todd Kemp,
    Department of Mathematics, University of California San Diego , 9500 Gilman Drive, La Jolla, CA 92093-0112, USA
  • Jean-Jacques Loeb,
    Université d'Angers, 2 Boulevard Lavoisier , 49045 Angers Cedex 01, France
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Abstract

We prove an intrinsic equivalence between strong hypercontractivity and a strong logarithmic Sobolev inequality for the cone of logarithmically subharmonic (LSH) functions. We introduce a new large class of measures, Euclidean regular and exponential type, in addition to all compactly-supported measures, for which this equivalence holds. We prove a Sobolev density theorem through LSH functions and use it to prove the equivalence of strong hypercontractivity and the strong logarithmic Sobolev inequality for such log-subharmonic functions.
Keywords: logarithmic Sobolev inequalities logarithmic Sobolev inequalities
MSC Classifications: 47D06 show english descriptions One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] 47D06 - One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]
 

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