Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions. (1st December 2015)

• Record Type: Journal Article
• Authors: Graczyk, Piotr; Kemp, Todd; Loeb, Jean-Jacques
• Is Part Of: Canadian journal of mathematics. Volume 67:Number 6(2015)
• Page Start: 1384
• Page End: 1410
• Publication Date: 2015-12-01
• Subjects: 47D06; logarithmic Sobolev inequalities; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/CJM-2015-015-8
• Languages: English
• ISSNs: 0008-414X

Abstract

Abstract: 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.