J. Math. Kyoto Univ. (JMKYAZ) 38-3 (1998) 415-418 Note on a paper of N. Iwasaki To the memory of Professor N . Iwasaki By Tatsuo NISHITANI 1. Introduction Let P(x, D) be a differential operator of order in defined in an open set Q in R n w ith principal symbol p(x,) which can be factorized as p(x,) = q/(x, gi(x, = - 1-1 where q l (x, ) are real valued pseudodifferential symbol of order 1 and x = (x 1 x') = (xi x2, • • • , xn), — — • • .t7)• W e assume that the characteristics of q j intersects normally and non-involutively each other, that is {q„ q i } 0 on q, = qj = 0 for i j where { q„ q i } denotes the Poisson bracket of q, and q j . According to Iwasaki [2], we define the signature of a triplet (q,.qj ,qk) at z ° where qi(z°) _ c z o) _ qk ( z o) _ O. Let us say that three real numbers a, b, c have the same sign if they are simultaneously positive or simultaneously negative. We say sgn(q„ q j ,qk)(z ° ) = ± if {g o q j }(?), qj, q k yz0) , qk. yzO‘ ) have the same sign and sgn(q„ qi,qk)(z ° ) = — otherwise. When m = 3, in [2], Iwasaki proved that in order that the Cauchy problem of P(x, D) is well posed the lower order terms must verify additional conditions at z ° more than Ivrii-Petkov condition if sgn(q], q2, q3)(z ° ) + . On the other hand, in [3] we proved that if the propagation cone at every triple characteristic is transversal to the doubly characteristic set and the lower order terms verify the Ivrii-Petkov condition then the Cauchy problem is well posed. Here we recall that the localization pz o of p is the first non-trivial term in the Taylor expansion of p at z ° which is a hyperbolic polynomial on T .: 0( P Q ) with respect to e = E T z 0(T*Q) (see [1]) where Hf denotes the Hamilton vector field of f E C'(T*S2). The propagation cone at z ° is the dual cone of the hyperbolic cone F(p z o, 0) (for the definition, see [1]) with respect to the canonical symplectic structure on 77,0 ( T*S2) induced by the 2-form ck A dx. Communicated by K. Ueno, December 17, 1996