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Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions

Published online by Cambridge University Press:  20 November 2018

Piotr Graczyk ,
Todd Kemp  and
Jean-Jacques Loeb
Piotr Graczyk
Affiliation:
Université d'Angers, 2 Boulevard Lavoisier, 49045 Angers Cedex 01, France. e-mail: piotr.graczyk@univ-angers.fr
Todd Kemp
Affiliation:
Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0112, USA. e-mail: tkemp@math.ucsd.edu
Jean-Jacques Loeb
Affiliation:
Université d'Angers, 2 Boulevard Lavoisier, 49045 Angers Cedex 01, France. e-mail: jean-jacques.loeb@univ-angers.fr
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Abstract

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We prove an intrinsic equivalence between strong hypercontractivity and a strong logarithmic Sobolev inequality for the cone of logarithmically subharmonic ($\text{LSH}$) 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 $\text{LSH}$ functions and use it to prove the equivalence of strong hypercontractivity and the strong logarithmic Sobolev inequality for such log-subharmonic functions.

Keywords

47D06logarithmic Sobolev inequalities
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 6 , 01 December 2015 , pp. 1384 - 1410
Copyright
Copyright © Canadian Mathematical Society 2015

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