Note on strongly hyperbolic systems with involutive characteristics

We consider the Cauchy problem in L 2 for first order system. A necessary condition is that the system must be uniformly diagonaliz-able, or equivalently that it admits a bounded symmetrizer. A sufficient condition is that it admits a smooth (Lipschtitz) symmetrizer, which is true when the system is hyperbolic, diagonalizable with eigen-values of constant multiplicities. Counterexamples show that uniform diagonalizability is not sufficient in general for systems with variable coefficients and indicate that the symplectic properties of the set Σ of the singular points of the characteristic variety are important. In this paper, give a new class of systems for which the Cauchy problem is well posed in L 2. The main assumption is that Σ is a smooth involutive manifold and the system is transversally strictly hyperbolic.