V. Bruneau, V. Petkov
Semiclassical resolvent estimates for trapping perturbations
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ABSTRACT.  We study the semiclassical estimates of the resolvent 
$R(\lambda + i\tau),\:\:\lambda \in J \subset\subset{\RR}^{+},\: \tau \in ]0,1]$ 
 of a self-adjoint operator $L(h)$ in the space of bounded operators 
${\cal L}({\cal H}^{0,s},{\cal H}^{0,-s}),\:s > 1/2$. 
In the general case of long-range trapping "black-box" perturbations 
we prove that the estimate of the cut-off resolvent 
$\|\chi(x)R(\lambda + i0)\chi(x)\|_{{\cal H} \to {\cal H}} \leq C\exp(Ch^{-p}),\:\chi(x) \in C^{\infty}_0({\RR^n}),\:p \geq 1$ 
implies the estimate 
$\|R(\lambda + i\tau)\|_{s,-s} \leq C_1\exp(C_1 h^{-p})$.