$x_0$ in ${\bf C}^n$. Here the weight $\Phi _0$ is smooth and strictly pluri-subharmonic. If $\Lambda _{\Phi _0}:=\{ (x,\frac{2}{i}\frac{\partial \Phi _0}{\partial x});\, x\in \mathrm{neigh\,}(x_0)\}$, then (in the sense of germs) $\Lambda _{\Phi _0}=\kappa _T(T^*X)$. The conjugated operator $\widetilde{P}=TPT^{-1}$ can be defined locally modulo ${\cal O}(h^\infty )$ (see also [@LaSj]) as a bounded operator from $H_\Phi \to H_\Phi $ provided that the weight $\Phi $ is smooth and satisfies $\Phi '-\Phi _0'={\cal O}(h^\delta) $ for some $\delta >0$. (In the analytic frame work this condition can be relaxed.) Egorov’s theorem applies in this situation, so the leading symbol $\widetilde{p}$ of $\widetilde{P}$ is given by $\widetilde{p}\circ \kappa _T=p$. Thus (under the assumptions of Theorem \[in2\]) we have ${{\Re \widetilde{p}}_\vert}_{\Lambda _{\Phi _0}}\ge 0$, which in turn can be used to see that for $0\le t\le h^\delta $, we have $e^{-t\widetilde{P}/h}={\cal O}(1)$: $H_{\Phi _0}\to H_{\Phi _t}$, where $\Phi _t\le \Phi _0$ is determined by the real Hamilton-Jacobi problem [$$\label{in.20}
\frac{\partial \Phi _t}{\partial t}+\Re \widetilde{p}(x,\frac{2}{i}\frac{\partial \Phi _t}{\partial x})=0,\ \Phi _{t=0}=\Phi _0.$$]{} Now the bound (\[in.15\]) follows from the estimate [$$\label{in.21}
\Phi _t\le \Phi _0-\frac{t^{k+1}}{C}$$]{} where $C>0$. An easy proof of (\[in.21\]) is to represent the I-Lagrangian manifold $\Lambda _{\Phi _t}$ as the image under $\kappa 