)}p)}$ through the point $\rho $ is not contained in $p^{-1}(z_0)$.

-   \(B) There exists an integer $k\ge 1$ such that for every $\rho
      \in p^{-1}(z_0)$, there exists $j\in \{ 1,2,..,k\}$ such that $$p^*(\exp tH_p(\rho ))=at^j+{\cal O}(t^{j+1}),\ t\to 0,$$ where $a=a(\rho )\ne 0$. Here $p$ also denotes an almost holomorphic extesnion to a complex neighborhood of $\rho $ and we put $p^*(\mu )=\overline{p(\overline{\mu })}$. Equivalently, $H_p^j(\overline{p})(\rho )/(j!)=a\ne 0$.

Then, in case (A), there exists a constant $C_0>0$ such that for every constant $C_1>0$ there is a constant $C_2>0$ such that the resolvent $(z-P)^{-1}$ is well-defined for $|z-z_0|<C_1h\ln
\frac{1}{h}$, $h<\frac{1}{C_2}$, and satisfies the estimate [$$\label{in.9}
\| (z-P)^{-1}\|\le \frac{C_0}{h}\exp ( \frac{C_0}{h}|z-z_0|).$$]{}

In case (B), there exists a constant $C_0>0$ such that for every constant $C_1>0$ there is a constant $C_2>0$ such that the resolvent $(z-P)^{-1}$ is well-defined for $|z-z_0|<C_1(h\ln
\frac{1}{h})^{k/(k+1)}$, $h<\frac{1}{C_2}$ and satisfies the estimate [$$\label{in.10}
\| (z-P)^{-1}\|\le \frac{C_0}{h^{\frac{k}{k+1}}}
\exp (\frac{C_0}{h}|z-z_0|^{\frac{k+1}{k}}).$$]{}

In [@DeSjZw] we obtained (\[in.9\]), (\[in.10\]) for $z=z_0$, implying that the resolvent exists and satisfies the same bound for $|z-z_0|\le h^{k/(k+1)}/{\cal O}(1)$ in case (B) and with $k/(k+1)$ replaced by 1 in case (A). In case (A) we also showed that the resolvent exists with norm bounded by a negative power 