On a class of nonhyperbolic microdifferential equations with involutory double characteristics
Let M be a real analytic manifold with a complexification X. We consider the microdifferential equation defined in a neighborhood of \(\rho_ 0\in \dot T^*_ MX\) of the form \[ (1)\quad Pu=\{(P_ 1+\sqrt{-1}P_ 2)P_ 3+Q\}u=0. \] Here we set \(p_ j=\sigma (P_ j)\) and we assume the following conditions: (2) \(ord(P_ 1)=ord(P_ 2)=m_ 1\), \(ord(P_ 3=m_ 2\), \(ord(Q)=m_ 1+m_ 2-1\), (3) \(p_ 1\), \(p_ 2\) and \(p_ 3\) are real valued on \(T^*_ MX\), (4) \(dp_ 1\), \(dp_ 2\), \(dp_ 3\) and the canonical 1-form \(\omega\) are linearly independent at \(\rho_ 0\), (5) \(\{p_ i,p_ j\}=0\) if \(p_ i=p_ j=0\) \(\{\) \(\cdot,\cdot \}\) denotes the Poisson bracket on \(T^*_ MX.\) In the above situation, we study propagation of singularities of solutions to (1) on the regular involutory submanifold \(\Sigma =\{\rho \in \dot T^*_ MX\); \(p_ j(\rho)=0\) (1\(\leq j\leq 3)\}\). More precisely, we transform second microlocally the equation (1) into \((D_ 1+\sqrt{-1}D_ 2)u=0\) or \(D_ 3u=0\) or \(u=0\).
scientific article; zbMATH DE number 4088108