first $\nu +\nu '+n+1$ homogeneity degrees in the symbols of $P_++G $, $P'_++G'$ and $P_1$.

(Actually, the number of homogeneous terms entering from $G$ is one step lower; see the statements in Section 3.)

The operators of the form $A=P_++G$ of integer order with $G$ of class zero and $P$ of normal order $\le 0$ form an algebra; Theorem 2.1 shows that $C_0(A,P_{1,\operatorname{D}})$ is a quasi-trace on this algebra as well as on the singular Green operators of class zero.

Similarly as in [@GSc], our analysis is based on precise information about the resolvent. We therefore recall the structure of $R_\lambda  =(P_{1,\operatorname{D}}-\lambda)^{-1}$: The Dirichlet problem $$(P_1-\lambda )u=f\text{ on }X,\quad \gamma _0u=\varphi \text{ on
}X',\tag2.9$$ is solved by a a row matrix $$\pmatrix P_1-\lambda \\ \gamma _0
\endpmatrix^{-1}=
\pmatrix R_\lambda   & K_\lambda
\endpmatrix = \pmatrix Q_{\lambda ,+}+G_\lambda  & K_\lambda
\endpmatrix ,\text{ with }G_\lambda  =- K_\lambda  \gamma
_0Q_{\lambda ,+}.\tag2.10$$ This is seen as follows: We denote by $K_\lambda $ the Poisson operator solving the semi-homogeneous problem with $f=0$, i.e., $K_\lambda \varphi$ is the solution $u$ of the equations $$(P_1-\lambda )u=0\text{ on }X,\quad \gamma _0u=\varphi \text{ on }X'.$$ The resolvent $R_\lambda $ of $P_{1,\operatorname{D}}$ should solve the other semi-homogeneous problem, with $\varphi =0$. 