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 $$|\Im (z-z_0)|<\frac{1}{C_0},\ \frac{-1}{C_0}<\Re
z<C_1h\ln\frac{1}{h},\ h<\frac{1}{C_2},$$ and satisfies the estimate [$$\label{in.13}
\| (z-P)^{-1}\|\le \cases{ \frac{C_0}{|\Re z|},\ \Re z\le -h,\cr
  \frac{C_0}{h}\exp (\frac{C_0}{h}\Re z), \Re z\ge -h.
 }$$]{}

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 [$$\label{in.13.5}|\Im (z-z_0)|<\frac{1}{C_0},\ \frac{-1}{C_0}<\Re
z<C_1(h\ln\frac{1}{h})^{\frac{k}{k+1}},\ h<\frac{1}{C_2},$$]{} and satisfies the estimate [$$\label{in.14}
\| (z-P)^{-1}\|\le \cases{ \frac{C_0}{|\Re z|},\ \Re z\le -h^{\frac{k}{k+1}},\cr
  \frac{C_0}{h^{\frac{k}{k+1}}}\exp (\frac{C_0}{h}(\Re z)_+^{^{\frac{k}{k+1}}}), \Re z\ge -h^{\frac{k}{k+1}}.
 }$$]{}

The case (A) in the theorems is practically identical with the corresponding results in [@DeSjZw] and can be obtained by inspection of the proof there, and from now on we concentrate on the case (B). Away from the set $p^{-1}(z_0)$ we can use ellipticity, so the problem is to obtain microlocal estimates near a point $\rho \in p^{-1}(z_0)$. After a standard factorization of $P-z$ in such a region, we can further reduce the proof of the first theorem to that of the second one.

The main (quite standard) idea of the proof of Theorem \[in2\] is to study $\exp 