Pseudodifferential operators with multiple characteristics and Gevrey singularities.

Let $P(x,D)$ be a classical analytic pseudodifferential operator with principal symbol $p_m(x,\xi)=q_{m-k}(x,\xi) a_1(x,\xi)^k$ in some cone $\Gamma$ for a fixed $k\ge1$, where $q_{m-k}(x,\xi)$ is an elliptic symbol, homogeneous of order $m-k$, and the first order symbol $a_1(x,\xi)$ is real-valued and of principal type, i.e., $d_{x,\xi}a_1(x,\xi)$ never vanishes and is not parallel to $\xi\,dx$. Suppose that $P=\sum^k_{j=0}Q_jA^{k-j}$, where $A$ is a classical analytic pseudodifferential operator with principal symbol $a_1(x,\xi)$ and $Q_j$ are classical analytic pseudodifferential operators of order $\le m-k+\rho j$, $0<\rho<1$. Let $G^s$, $1<s<\infty$, be the Gevrey class, let $G^s_0(\Omega)=G^s(\Omega)\cap C^\infty_0(\Omega)$, and let $\text{WF}_sf$ be the $s$-singular support of $f$. If $f,g\in G^s_0(\Omega)'$, then $f\sim g$ means that $\Gamma\cap\text{WF}_s(f-g)=\varnothing$ and $M^s(\Gamma)$ is the factor space $G^s_0(\Omega)'/\sim$