solving a $\overline{\partial }$-problem (assuming, as we may, that our neighborhoods are pseudoconvex) we can always correct $\widetilde{P}$ with a negligible operator such that (after an arbitrarily small decrease of $W$) $\widetilde{P}={\cal O}(1):H_{\Phi _0}(V)\to H_{\Phi
  _0}(W)$. Also, if $\Phi =\Phi _0+{\cal O}(h\ln \frac{1}{h})$ in $C^2$, then clearly $\widetilde{P}={\cal O}(h^{-N_0}): H_{\Phi }(V)\to H_{\Phi
}(W)$, for some $N_0$. Using Stokes’ formula, we can show that $\widetilde{P}$ will change only by a negligible term if we replace $\Phi _0$ by $\Phi $ in the definition of $\Gamma (x)$, and then it follows that $\widetilde{P}={\cal O}(1):H_{\Phi }(V)\to 
H_{\Phi}(W)$.

Before discussing evolution equations, let us recall ([@MeSj02]) that the identity operator $H_{\Phi_0} (V)\to H_{\Phi_0}
 (W)$ is up to a negligible operator of the form [$$\label{ev.0.2}
Iu(x)=h^{-n}{\int\hskip -2mm\int}e^{\frac{2}{h}\Psi _0(x,\overline{y})}a(x,\overline{y};h)u(y)e^{-\frac{2}{h}\Phi _0(y)}dyd\overline{y},$$]{} where $\Psi _0(x,y)$, $a(x,y;h)$ are almost holomorphic on the antidiagonal $y=\overline{x}$ with $\Psi _0(x,\overline{x})=\Phi
_0(x)$, $a(x,y;h)\sim a_0(x,y)+ha_1(x,y)+...$, $a_0(x,\overline{x})\ne
0$. More generally a pseudodifferential operator like $\widetilde{P}$ takes the form [$$\begin{aligned}
\label{ev.0.3}
\widetilde{P}u(x)&=&h^{-n}{\int\hskip -2mm\int}e^{\frac{2}{h}\Psi _0(x,\overline{y})}q
(x,\overline{y};h)u(y)e^{-\frac{2}{h}\Phi _0(y)}dyd\overline{y}
 \\ q_0(x,\overline{x})&=&\widetilde{p}(x,\frac{2}{i}\frac{\partial \Phi _0}{\partial
    x}(x))a_0(x,\overline{x}),
\nonumber\end{aligned}$$]{} and where $q_0$ denotes the first term in the asymptotic expansion of the symbol $q$. In this discussion, $\Phi _0$ can be replaced by any other smooth exponent $\Phi$ which is ${\cal O}(h^\delta ) 