on the transform side {#ev}
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Let $\widetilde{P}(x,\xi ;h)$ be a smooth symbol defined in $\mathrm{neigh\,}((x_0,\xi _0);\Lambda _{\Phi _0})$, with an asymptotic expansion $$\widetilde{P}(x,\xi ;h)
\sim \widetilde{p}(x,\xi )+h\widetilde{p}_1(x,\xi )+...\hbox{ in
}C^\infty (\mathrm{neigh\,}((x_0,\xi _0),\Lambda _{\Phi _0})).$$ By the same letter, we denote an almost holomorphic extension to a complex neighborhood of $(x_0,\xi _0)$: $$\widetilde{P}(x,\xi ;h)
\sim \widetilde{p}(x,\xi )+h\widetilde{p}_1(x,\xi )+...\hbox{ in
}C^\infty (\mathrm{neigh\,}((x_0,\xi _0),{\bf C}^{2n}),$$ where $\widetilde{p}$, $\widetilde{p}_j$ are smooth extensions such that $$\overline{\partial }\widetilde{p},\ \overline{\partial
}\widetilde{p}_j={\cal O}(\mathrm{dist\,}((x,\xi ),\Lambda _{\Phi
  _0})^\infty).$$ Then, as developed in [@LaSj] and later in [@MeSj02], if $u=u_h$ is holomorphic in a neighborhood $V$ of $x_0$ and belonging to $H_{\Phi _0}(V)$ in the sense that $\Vert u\Vert_{L^2(V,e^{-2\Phi _0/h}L(dx))}$ is finite and of temperate growth in $1/h$ when $h$ tends to zero, then we can define $\widetilde{P}u=\widetilde{P}(x,hD_x;h)u$ in any smaller neighborhood $W\Subset V$ by the formula, [$$\label{ev.0.1}
\widetilde{P}u(x)=\frac{1}{(2\pi h)^n}{\int\hskip -2mm\int}_{\Gamma
  (x)}e^{\frac{i}{h}(x-y)\cdot \theta
}\widetilde{P}(\frac{x+y}{2},\theta ;h)u(y)dyd\theta ,$$]{} where $\Gamma (x)$ is a good contour (in the sense of [@Sj82]) of the form $\theta =\frac{2}{i}\frac{\partial \Phi _0}{\partial
  x}(\frac{x+y}{2})+\frac{i}{C_1}(\overline{x-y})$, $
|x-y|\le 1/C_2$, $C_1,C_2>0$. Then $\overline{\partial }\widetilde{P}$ is negligible $H_{\Phi _0}(V)\to L^2_{\Phi _0}(W)$, i.e. of norm ${\cal O}(h^\infty
)$ and modulo such negligible operators, $\widetilde{P}$ is independent of the choice of good contour. By 