point is to show that the resulting symbols belong to suitable $S^{m,d,0}$-spaces such that (3.5) can be established.

Since we are particularly interested in the nonlocal coefficient $c_0''$, we have to be very careful with remainders and smoothing terms. Recall from that when an operator $F$ with symbol in $S^{m,d,0}$ is of order $-\infty $ (i.e., lies in $S^{-\infty ,d,0}$), then it has a $C^\infty $ kernel $K(F,x,y,\mu )$ with an expansion in $C^\infty $ terms$$K(F,x,y,\mu )=\sum_{0\le l<L}K_l(x,y)\mu ^{d-l}+K'_L(x,y,\mu ),\tag3.7$$ for all $L$, with $K'_L$ and its $(x,y)$-derivatives being $O(\ang \mu
^{d-L})$. Moreover, since $\partial _z^j(z^d F)=(-\mu ^2\partial _\mu )^j(\mu
^{-d}F)$ has symbol in $S^{-\infty ,0,0}$ for all $j$, there are expansions $$K((-\mu ^2\partial _\mu )^j(\mu
^{-d}F),x,y,\mu )=\sum_{0\le l<L}K^{(j)}_l(x,y)\mu ^{-l}+K^{(j)\prime}_L(x,y,\mu ),\tag3.8$$ for all $L$, with $K^{(j)\prime}_L$ and its $(x,y)$-derivatives being $O(\ang \mu
^{-L})$. Conversely, these kernel properties characterize the operators $F$ with symbol in $S^{-\infty ,d,0}$ (as is seen straightforwardly from the definition).

It follows from (3.3) that $S^{-\infty ,d,s}=S^{-\infty ,d+s,0}$. In the proof that a polyhomogeneous symbol lies in a symbol space $S^{m,d,s}$ one shows the symbol properties for the homogeneous terms and treats the remainders by establishing the kernel expansionss (3.8) (with $d$ replaced by $d+s$) with smoother 