An Operator-Theoretic Proof of an Estimate on the Transfer Operator

• Published in: Journal of functional analysis Vol. 165; no. 2; pp. 240 - 257
• Main Authors: Descombes, Stéphane, Dia, Boun Oumar
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 10.07.1999
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1006/jfan.1999.3412

Let ρ be a nonnegative number; denote ⌈1/ρ⌉ as the smallest integer which is larger than 1/ρ. Let k=max(2⌈1/ρ⌉, 4), and let V be a positive potential on RN of class Ck such that for all multi-index α satisfying |α|⩽k we have |∂αV(x)|⩽Cα(1+V(x))(1−ρ|α|)+. We prove that‖e−tVe2tΔe−tV−e2t(−V+Δ)‖L(L2)=O(t1+2inf(ρ, 1/2)) as t decreases to zero. Our techniques rely on estimates on commutators. Copyright 1999 Academic Press. Soit ρ un réel strictement positif, notons ⌈1/ρ⌉ le plus petit entier supérieur ou égal à 1/ρ. Soit V un potentiel positif sur RN de classe Ck tel que pour tout multiindice α satisfaisant |α|⩽k on ait |∂αV(x)|⩽Cα(1+V(x))(1−ρ|α|)+. Nous prouvons que‖e−tVe2tΔe−tV−e2t(−V+Δ)‖L(L2)=O(t1+2inf(ρ, 1/2))quand t decroı̂t vers zéro. Nos techniques utilisent des estimations sur des commutateurs.