Upper bounds for symmetric Markov transition functions
type: Item
instance of: scholarly article
title: Upper bounds for symmetric Markov transition functions
zbMATH Open document ID: 0634.60066
author: Daniel W. Stroock
author: Eric Carlen
author: Shigeo Kusuoka
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77309
review text: Let (E,\({\mathcal B},m)\) be a locally compact separable metric space with a locally finite measure m, \(\{\bar P{}_ t\), \(t\geq 0\}\) be a strongly continuous semigroup of self-adjoint contractions on \(L^ 2(m)\) and (\({\mathcal E},{\mathcal D}({\mathcal E}))\) be an associated Dirichlet form. Nash- type inequalities are proved. Denote \(\| f\|_ p\) the \(L^ p(m)\)- norm of f and \(\| \bar P_ t\|_{p\to q}=\sup \{\| \bar P_ tf\|_ q:\| f\|_ p=1\), f is Borel with compact support\(\}\). Let \(\nu\in (0,\infty)\) and \(\delta\in [0,\infty)\) be fixed. If \[ (*)\quad \| f\|_ 2^{2+4\nu}\leq A[{\mathcal E}(f,f)+\delta \| f\|^ 2_ 2]\| f\|_ 1^{4/\nu},\quad f\in L^ 2(m), \] for some \(A\in (0,\infty)\), then \[ \| \bar P_ t\|_{1\to \infty}\leq Be^{\delta t}/t^{\nu /2},\quad t>0, \] for some \(B\in (0,\infty)\) depending only on \(\nu\) and A and vice versa. If \(\nu\in (2,\infty)\), these inequalities are equivalent to the Sobolev-type inequality \[ \| f\|^ 2_ p\leq A'({\mathcal E}(f,f)+\delta \| f\|^ 2_ 2);\quad p=2\nu /\nu -2. \] Cases when \(\delta =0\) or \(\nu =2\) as well as discrete-time analogues are also considered. Under assumption (*) the semigroup \(\{\bar P{}_ t\), \(t\geq 0\}\) possesses a kernel p(t,x,y) with respect to m. E. B. Davies' method is extended for obtaining diagonal estimates of p(t,x,y).
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 60J99
zbMATH DE Number: 4032743
zbMATH Keywords: locally compact separable metric space
zbMATH Keywords: strongly continuous semigroup
zbMATH Keywords: Nash-type inequalities
zbMATH Keywords: Sobolev-type inequality
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4032743