for $|(\xi ,\mu )|\ge 1$, with appropriate remainder estimates).

These spaces $S^{m,d,s}$ are a generalization of the spaces $S^{m,d}$ introduced in [@GS1] (in fact, $S^{m,d}(\Gamma )$ equals the space of holomorphic symbols in $S^{m,d,0}(\Gamma )$); they satisfy $$\aligned
S^{m,d,s}&\subset S^{m+s,d,0}\cap
S^{m,d+s,0}\text{ if }s\le 0,\\
S^{m,d,s}&\subset S^{m+s,d,0}+
S^{m,d+s,0}\text{ if }s\ge 0,
\endaligned\tag3.3$$ by [@GS1; @Lemma; @1.13]. The notation with the third upper index spares us from keeping track of such intersections and sums in general calculations. (It was used systematically in [@G3], [@G5], [@G6].)

When $f\in S^{m,d,0}$, there is an expansion in decreasing powers of $\mu $ starting with $\mu ^d$: $$f(x,\xi ,\mu )=\sum_{0\le k<N}\mu ^{d-k}f_{(k)}(x,\xi)+O(\mu
^{d-N}\ang\xi ^{m+N}),\quad \text{any $N$};\tag3.4$$ in fact the $N$’th remainder is in $S^{m+N,d-N,0}$. Such expansions play an important role in the proof of the following fact: When $F=\operatorname{OP}(f(x,\xi ,\mu ))$ has a polyhomogeneous symbol in $S^{m,d,0}$ such that all terms are integrable with respect to $\xi $, then by , the kernel of $F$ has an asymptotic expansion on the diagonal: $$K(F,x,x,\mu )\sim \sum_{j\in\Bbb N}c_j(x)\mu ^{m+d+n'-j}+\sum_{k\in\Bbb
N}(c'_k(x)\log\mu +c''_k(x))\mu ^{d-k};\tag 3.5$$ here the coefficients $c_j(x)$ and $c'_k(x)$ with $k=-m+j-n'$ are local, determined from the strictly homogeneous version of the $j$’th homogeneous term in the symbol of $F$, 