J. Math. Kyoto Univ. (JMKYAZ) 18-3 (1978) 509-521 On the Lax-Mizohata theorem in the analytic and Gevrey classes By Tatsuo NISHITANI (Received July 12, 1977) 1. Introduction In this paper, we consider the non-characteristic Cauchy problem for the differ- ential operators with Gevrey or analytic coefficients. L. Boutet de Monvel and P. Krée [2] have showed some fundamental proper- ties of analytic and Gevrey symbols of pseudo-differential operators. In [1], L. Hârmander has localized the pseudo-differential operators with analytic symbols in a suitable way on the dual space to extend the regularity and uniqueness theorems and to study the propagation of the singularities. Let L(x, t; D, Dt )=P(x, t; D x , D t )Q(x. t; D. D t ) be a differential operator of order in with Gevrey or analytic coefficients, and Lu =0. If P is a elliptic differential operator of order v, then the analytic-hypoellipticity means that Qu is a Gevrey or analytic function. Therefore, 13.114(x, 0) (v + it = in, j < m — l) are also Gevrey or analytic functions provided that t=0 is non characteristic for Q and Mu(x, 0) (O j p —J) are in Gevrey or analytic class. This show s that, for the Cauchy problem of L, we cannot give the first /I+ I initial data arbitrarily in C class. Here, using the above localized differential operator, we shall generalize this simple example and give the same necessary relation between the admissible initial data and the number of real roots of the characteristic equation. And, as applica- tion of this relation, we extend the Lax-Mizohata theorem to the analytic and Gevrey classes. 2. Definitions and Results Definition 2.1. Let V be an open set in Rrn, we shall denote by y(s)(V) (s the set of allle C(V) such that for every compact set Kc V, there are constants C, A with (2.1) 1Dmf(x)I<CA1.11aMs, xe K, for all multi-indexes Œ.