coefficients and a higher number of terms, the higher the index of the remainder is taken.

The sector $\Gamma $ used in the present work is$$\Gamma =\{\,\mu \in\Bbb C\setminus \{0\}\mid |\arg \mu |\le\tfrac \pi
4+\tfrac
\varepsilon 2
\,\},\tag3.9$$ for some $\varepsilon>0$, since $\lambda =-\mu ^2$ runs in $\Lambda $ defined in (2.3). [*All symbols to be considered in this paper will be holomorphic on $\Gamma ^\circ$*]{}; this fact will not be mentioned explicitly each time.

One of the technical points in [@GSc] was that in compositions of two operators, the right-hand factor is taken in $y$-form and the resulting symbol is considered in $(x,y)$-form so that one does not need to consider the asymptotic expansion in the usual symbol composition formula (cf. e.g.  (3.31) below). However, the passage from one of these forms to another can induce low-order errors. These have to be dealt with carefully in the present situation, where nonlocal contributions are in focus. Also for this reason, the following analysis of the effect of smoothing operators is important.

Smoothing parts of $Q_\lambda $ and $G_\lambda $ are easy to deal with, since they have smooth kernels that are $O(\lambda ^{-M})$ for any $M$. For compositions containing smoothing 