whereas the $c''_k(x)$ depend on the full operator (are global). If $m\notin\Bbb Z$, all $c'_k$ are zero. When $F$ is a $\psi $do family on $\widetilde X$ (or on $X'$), the trace expansion for $F$ is found by establishing expansions (3.5) for components of $F$ in local coordinate systems, carrying them back to $\widetilde X$ and integrating the fibrewise traces over $\widetilde X$; this leads to a similar expansion of $\Tr F$: $$\Tr(F )\sim \sum_{j\in\Bbb N}c_j\mu ^{m+d+n'-j}+\sum_{k\in\Bbb
N}(c'_k\log\mu +c''_k)\mu ^{d-k}.\tag 3.6$$ Of course, such a calculation depends on the choice of partitions of unity and local coordinates, but it is useful for qualitative information. The final result (3.6) is independent of the choices, in the sense that the collected coefficient of each power or log-power is so (for $k=j-m-n'\in\Bbb N$, $c_j+c''_k$ is independent of the choices).

This is the basis for the trace expansions we shall show. For purely $\psi $do terms on $X$, the expansions will be established relative to $\widetilde X$, but the integration of the kernels will be restricted to $X$. For all the other types of operators, the main idea is to reduce the considerations to $\psi $do’s over the boundary $X'$. In all cases the 