Resolvent estimates for non-self-adjoint operators via semi-groups
نویسنده
چکیده
We consider a non-self-adjoint h-pseudodifferential operator P in the semi-classical limit (h → 0). If p is the leading symbol, then under suitable assumptions about the behaviour of p at infinity, we know that the resolvent (z − P )−1 is uniformly bounded for z in any compact set not intersecting the closure of the range of p. Under a subellipticity condition, we show that the resolvent extends locally inside the range up to a distance O(1)((h ln 1 h)) from certain boundary points, where k ∈ {2, 4, ...}. This is a slight improvement of a result by Dencker, Zworski and the author, and it has recently been obtained by W. Bordeaux Montrieux in a model situation where k = 2. The method of proof is different from the one of Dencker et al, and is based on estimates of an associated semi-group. Résumé Nous considérons un opérateur h-pseudodifférentiel non-autoadjoint P dans la limite semi-classique (h → 0). Si p désigne le symbole principal, alors sous des hypothèses convenables sur le comportment de p à l’infini nous savons que la résolvante (z − P )−1 est uniformément bornée pour z dans un compact qui ne rencontre pas l’adhérence ∗Ce travail a bénéficié d’une aide de l’Agence Nationale de la Recherche portant la référence ANR-08-BLAN-0228-01
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