here is $S^{m,d,s}(\Bbb
R^{n'}\stimes\Bbb R^{n'},\Gamma  )$, for $m\in\Bbb
R$, $d$ and $s\in\Bbb Z$, and with $\Gamma $ denoting a sector in $\Bbb C\setminus \{0\}$; it was introduced in [@G3], where a detailed account can be found. Here $S^{m,d,s}(\Bbb
R^{n'}\stimes\Bbb R^{n'},\rp )$ is defined as the space of $C^\infty
$ functions $f(x,\xi ,\mu )$ on $\Bbb R^{n'}\stimes\Bbb
R^{n'}\stimes\rp $ such that, with $z=1/\mu $,$$\multline
\partial _z^j[z^{d}|(\xi ,1/z)|^{-s}f(x,\xi ,1/z)]\in
S^{m+j}(\Bbb R^{n'}\stimes \Bbb R^{n'}),\\
\text{with symbol estimates uniform in $z$ for }z\le 1.
\endmultline$$ For more general $\Gamma $, the estimates have to hold on each ray in $\Gamma $, uniformly in closed subsectors of $\Gamma $. When the symbols moreover are holomorphic in $\mu \in\Gamma ^\circ$ (just for $|(\mu ,\xi
')|\ge \varepsilon $ with some $\varepsilon $ depending on the closed subsector), we speak of holomorphic symbols. The indication $\Bbb
R^{n'}\stimes \Bbb R^{n'}$ (and $\Gamma $) will often be omitted. We say that $f(x,\xi
,\mu )$ is (weakly) polyhomogeneous in $S^{m,d,s}(\Gamma  )$, when there is a sequence of symbols $f_{j}$ in $S^{m-j,d,s}(\Gamma )$, homogeneous in $(\xi ,\mu )$ of degree $m-j+d+s$ for $|\xi |\ge 1$, such that $f-\sum_{j<J}f_j$ is in $S^{m-J,d,s}(\Gamma )$ for all $J\in\Bbb N$ (strongly polyhomogeneous if the homogeneity holds 