Some remarks on ultracontractivity
type: Item
instance of: scholarly article
title: Some remarks on ultracontractivity
zbMATH Open document ID: 0807.47027
author: Otared Kavian
author: Gérard Kerkyacharian
author: Bernard Roynette
published in: Journal of Functional Analysis
publication date: 1993-06-29T00:00:00Z
review text: The authors study the semigroup \(T^ \mu_ t\) generated by the operator \({1\over 2}(\Delta f- \nabla u\cdot\nabla f)\) on the Lebesgue space \(L^ 2(\mathbb{R}^ d;\mu)\) with the measure \(\mu:= e^{-u}\). They prove, via different methods using probabilistic techniques or PDE arguments, that \(T^ \mu_ t\) is ultracontractive, i.e., for \(t>0\) it maps \(L^ 1(\mu)\) into \(L^ \infty\), when the function \(u\) satisfies a growth condition at infinity, which is essentially (for instance when the dimension \(d=1\)) the integrability of \(1/u'\) at infinity. Also, they consider the analogous properties of the semigroup generated by the fractional powers of the above operator.
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 47A60
zbMATH DE Number: 220690
zbMATH Keywords: semigroup
zbMATH Keywords: probabilistic techniques
zbMATH Keywords: PDE arguments
zbMATH Keywords: ultracontractive
zbMATH Keywords: fractional powers
MaRDI profile type: Publication
full work available at URL: jfan.1993.1008
OpenAlex ID: W2066355253
DOI: 10.1006/JFAN.1993.1008
description: scientific article; zbMATH DE number 220690