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

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+2 inf(ρ, 1/2)) as t decreases to zero. Our techniques rely on estimates on commutators. Copyright 1999 Academic Press. 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+2 inf(ρ, 1/2))quand t decroı̂t vers zéro. Nos techniques utilisent des estimations sur des commutateurs.