_T$ of the I-Lagrangian manifold $\Lambda _{G_t}=\{ \rho +iH_{G_t}(\rho );\,\rho \in \mathrm{neigh\,}(\rho _0,T^*X)\}$, where $H_{G_t}$ denotes the Hamilton field of $G_t$. It turns out that the $G_t$ are given by the real Hamilton-Jacobi problem [$$\label{in.22}
\frac{\partial G_t}{\partial t}+\Re (p(\rho +iH_{G_t}(\rho )))=0,\ G_0=0,$$]{} and there is a simple minimax type formula expressing $\Phi _t$ in terms of $G_t$, so it suffices to show that [$$\label{in.23}
G_t\le -t^{k+1}/C.$$]{}

This estimate is quite simple to obtain: (\[in.22\]) first implies that $G_t\le 0$, so $(\nabla G_t)^2={\cal O}(G_t)$. Then if we Taylor expand (\[in.22\]), we get $$\frac{\partial G_t}{\partial t}+H_{\Im p}(G_t)+{\cal O}(G_t)+\Re p(\rho )=0$$ and we obtain (\[in.23\]) from a simple differential inequality and an estimate for certain integrals of $\Re p$.

The use of the representation with $G_t$ is here very much taken from the joint work [@HeSj86] with B. Helffer.

In Section \[ex\] we discuss some examples.

IR-manifolds close to ${\bf R}^{2n}$ and their FBI-representations {#ge}
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Much of this section is just an adaptation of the discussion in [@HeSj86] with the difference that we here use the simple FBI-transforms of generalized Bargmann type from [@Sj82], rather than the more complicated variant that was necessary to treat a neighborhood of infinity in the resonance 