On the Cauchy problem for hyperbolic operators with nearly constant coefficient principal part

In this paper we shall deal with hyperbolic operators whose principal symbols can be microlocally transformed to symbols depending only on the fiber variables by homogeneous canonical transformations. We call such operators "hyperbolic operators with nearly constant coefficient principal part." Operators with constant coefficient hyperbolic principal part and hyperbolic operators with involutive characteristics belong to this class of operators. We shall give a necessary and sufficient condition for the Cauchy problem to be C∞ well-posed under some additional assumptions. Namely, we shall generalize "Levi condition" and prove that the generalized Levi condition is necessary and sufficient for the Cauchy problem to be C∞ well-posed.