Maximally hyperbolic operators (Q1340379)

The author studies well-posedness of Cauchy problem for operators of the form \[ P = D^ m_ t + \sum^ m_{j=1} A_ j (x,D) D_ t^{m-j}, \quad (t,x) \in I \times \Omega \subset \mathbb{R} \times \mathbb{R}^ n, \] where \(A_ j\) are pseudo-differential operators in the classes \(N^ j (\Omega, \Sigma)\) of \textit{L. Boutet de Monvel}, \textit{A. Grigis} and \textit{B. Helffer} [Asterisque 34-35, 93-121 (1976; Zbl 0344.32009)], that is, essentially: \(A_ j (x, \xi) \sim \sum^ \infty_{k = 0} a^{(j)}_{j-k} (x, \xi)\) with \(a_{j-k}^{(j)}\) vanishing of order \(j - 2k\) at a given smooth submanifold \(\Sigma\) of \(T^* \Omega\). The operator \(P\) is assumed strictly hyperbolic for \((x, \xi)\) outside \(\Sigma\). Under additional hypotheses, and in particular supposing injectivity for \[ P_{x, \xi, \tau} : {\mathcal S} (\mathbb{R}^ n) \to {\mathcal S} (\mathbb{R}^ n), \] a suitable partial differential operator with polynomial coefficients associated to \(P\) at every \((x, \xi) \in \Sigma\), \(\tau \in \mathbb{C}\), the author proves \(P\) is maximally hyperbolic, that gives well- posedness for the Cauchy problem. Some examples are presented, but, as the author observes, it is difficult to check the injectivity of \(P_{x, \xi, \tau}\) for a given operator. A comparison with other known results on weakly hyperbolic Cauchy problem would be interesting.