Local energy integrals for effectively hyperbolic operators. I (Q1068409)

An operator P is called strongly hyperbolic if the Cauchy problem is well-posed in the \(C^{\infty}\) sense, independent of lower order terms. An operator P is called effectively hyperbolic if the fundamental matrix of P has non-zero real eigenvalues at every critical point of P. V. Ya. Ivrij and V. M. Petkov have proved that strongly hyperbolic operators are effectively hyperbolic. The present paper proves the converse for second order operators. The technique used is the energy integral method and the paper is based on two previous ones by the author quite heavily [Equations aux dérivées partielles hyperboliques et holomorphes, Sémin. Paris, Année 1981-82, Exp. No.4 (1982; Zbl 0523.35070) and weakly hyperbolic Cauchy problem for second order operators, RIMS Kyoto, 21, 1985).