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(1) fPu=f in (0, T) x Rn Dju(0, x) = u j, j=0,1, • • for given f e C°°(Q) and uj e C~(Rn). It is well-known that the characteristic roots are real if the Cauchy problem is well-posed in C°° (cf. [3]). In the present note we study a sufficient conditions for the problem (1) to be well posed when charateristics are real and have constant multiplicity. Concerning this problem, S. Mizohata and Y. Ohya [4], [5] obtained a necessary and sufficient condition when the multiplicity is less than 2, and Y. Ohya [6] studied a sufficient condition when the multiplicity is less than 3. Recently, J. Chazarain [1] discusses the case of the arbitrary multiplicity by making use of the theory of Fourier integral operators. Our arguments seem to be different from his. 1. E. E. Levi's condition and the main theorem. Let the principal part Pm of P be written as Pm(t, x ; z,) _ (v-2(t, x ; C))l', and j=0 assume that 2j(t, x ; ), 1 < j < r, are real for e Rn {0} and inf 12j (t, x ; C) Ak (t, x ; C) J >_ d > 0, j ~ k. Moreover, without loss (t, x) E Q,i E I=1 of generality we may assume that l1= l2 • • • = lri > lrl+1= lrz > • • • > • • • = lr, and put 1=11. Let 0(C) be a C°°-f unction such that 0(C) = 0 for 1, and 0(C) =1 for C J > 1, and define 2j (t, x ; Dx) by -4 2

Energy inequalities and finite propagation speed of the Cauchy problem for hyperbolic equations withconstantly multiple characteristics