The 2-microlocal canonical form for a class of microdifferential equations and propagation of singularities (Q1812715)

The author generalizes his results from an earlier paper [J. Fac. Sci., Univ. Tokyo, Sect. I A 33, 619-634 (1986; Zbl 0637.35089)] to a class of microdifferential equations with microlocal canonical form of the type (*) \(P_ 1u = D_ 1^{m_ 1}D_ 2^{m_ 2} + Qu = 0\), defined in a neighborhood of \((0,dz_ 3)\) in \(T^*\mathbb{C}^ n\). Here \(Q\) is a lower- order term. He assumes that the operator \(P_ 1\) has regular singularities along \(\Lambda = \{(z,\xi) \in T^*\mathbb{C}^ n: \xi_ 1 = \xi_ 2 = 0\}\) in the sense of \textit{M. Kashiwara} and \textit{T. Ōshima} [Ann. Math., II. Ser. 106, 145-200 (1977; Zbl 0358.35073)]. Then the equation (*) is 2-microlocally equivalent to one of the following: \[ \begin{aligned}\left\{\begin{matrix} D_ 1u_ 1 = 0\\\vdots\\D_ 1u_{m_ 1} = 0\end{matrix}\right.\text{ or } \left\{\begin{matrix} D_ 2u_ 1 = 0\\\vdots\\D_ 2u_{m_ 2} = 0\end{matrix}\right. \text{ or } u=0.\end{aligned} \] {}.