situations where one has some control over the local terms. Our formulas for $C_0(P_+,P_{1,\operatorname{D}})$ in Theorem 4.1 and for $C_0(G,P_{1,\operatorname{D}})$ in Theorem 3.6 moreover show how the values are related to formulas for the boundaryless manifolds $\widetilde X$ resp. $X'$.

2\. Preliminaries

Let $X$ be an $n$-dimensional compact $C^\infty $ manifold with boundary $\partial X=X'$, $X$ and $X'$ provided with $C^\infty $ vector bundles $E$ and $E'$. We can assume that $X$ is smoothly imbedded in an $n$-dimensional manifold $\widetilde X$ without boundary, provided with a vector bundle $\widetilde E$ such that $E=\widetilde E|_X$. We consider the algebra of (one-step) polyhomogeneous operators in the calculus of Boutet de Monvel $$\pmatrix P_++G & K\\ \quad&\quad \\T&S
\endpmatrix
\:
\matrix
C^\infty (E)&&C^\infty(E)\\
\oplus&\longrightarrow&\oplus\\
 C^\infty (E')&& C^\infty (E')
\endaligned.
 \tag2.1$$ Here $P$ is a pseudodifferential operator satisfying the transmission condition. The subscript ‘$+$’ indicates that we are taking the truncation of $P$ to $X$, i.e., the operator given by extending a function in $C^\infty (E)$ by zero to a function in $L^2(\widetilde X,\widetilde E)$, applying $P$, and restricting the result to $X^\circ$; the transmission property assures that this gives an element of $C^\infty (E)$. Moreover, $G$ is a singular Green operator (s.g.o.), $K$ 