of $h$ in any disc $D(z_0,C_1h\ln (1/h))$. (The condition in case (B) was formulated a little differently in [@DeSjZw], but as we shall see later on the two conditions leed to the same microlocal models and hence they are equivalent.) Actually the proof in [@DeSjZw] also gives (\[in.9\]), so even if the methods of the present paper also most likely lead to that bound, we shall not not elaborate the details in that case.

Let us now consider the special situation of potential interest for evolution equations, namely the case when [$$\label{in.11}z_0\in i{\bf R},$$]{} [$$\label{in.12}
\Re p(\rho )\ge 0\hbox{ in }\mathrm{neigh\,}(p^{-1}(z_0),T^*X).$$]{}

\[in2\] We adopt the general assumptions above. Let $z_0\in \partial \Sigma
(p)\setminus \Sigma _\infty (p)$ and assume (\[in.11\]), (\[in.12\]). Also assume that $dp\ne 0$ on $p^{-1}(z_0)$, so that $d\Im p\ne 0$, $d\Re p=0$ on that set. Consider the two cases of Theorem \[in1\]:

-   \(A) For every $\rho \in p^{-1}(z_0)$, the maximal integral curve of $H_{\Im p}$ through the point $\rho $ contains a point where $\Re
      p>0$.

-   \(B) There exists an integer $k\ge 1$ such that for every $\rho
      \in p^{-1}(z_0)$, we have $H_{\Im p}^j\Re p(\rho )\ne 0$ for some $j\in \{ 1,2,...,k\}$.

Then, in case (A), 