Hokkaido Mathematical Journal Vol. 6 (1977) p. 313-344 Construction of a parametrix for the Cauchy problem of some weakly hyperbolic equation I. By Atsushi YOSHIKAWA (Received September 20, 1976) \S 0. Introduction Consider the partial differential operator (0. 0) P=D_{t}^{2}-t^{2} \sum_{f=1}^{n}D_{f}^{2}+a(t, x)D_{t}+\sum_{f=1}^{n}bj(t, x)D_{j}+c(t, x) in R^{n+1} . Here a , b_{1} , \cdots , b_{n} , c are C^{\infty} functions of (t, x)=(t, x_{1^{ }},\cdots, x_{n})\in R\cross R^{n} , and D_{t}=-i\partial/\partial t , D_{j}=-i\partial/\partial x_{j} , (j=1, \cdots, n) i^{2}=-1 as usual. We are going to construct a parametrix for the Cauchy problem associated to the operator P : (0. 1) (Pu) (t, x)=0 , t>0 , x\in R^{n} , (0. 2) u(0, x)=f(x) , D_{t}u(0, x)=g(x) , x\in R^{n} , f , g being distributions in \mathcal{E}’(R^{n}) . For simplicity, we shall assume that (0. 3) |{\rm Im} \sum_{f=1}^{n}b_{j}(0, x)\xi_{j}| be uniformly bounded for all x\in R^{n} and \xi=(\xi_{1^{ }},\cdots, \xi_{n}) on the unit sphere S^{n-1} . Let (0. 4) m( \sigma)=-\frac{1}{4}+\frac{1}{4} sup \{\sigma Im \sum_{f=1}^{n}b_{j}(0, x)\xi_{j}\} , \sigma^{2}=1 , the supremum being taken over (x, \xi)\in R^{n}\cross S^{n-1} . We then have the following THEOREM. There exist symbols (0. 5) \wedge p_{\sigma}(t, x, \xi)\in S^{m(\sigma)+}*\prime 2m(\sigma)+2* , \tilde{p}_{\sigma}(t, x, \xi)\in S^{m(\sigma)+e} , q_{\sigma}(t, x, \xi)\in S^{m(\sigma)\dagger*-1/22m(\sigma)+2*}, , \backslash \tilde{q}_{\sigma}(t, x, \xi)\in S^{m(\sigma)+\text{\’{e}}-1/2} , \sigma^{2}=1 , (t, x, \xi)\in\overline{R}_{+}\cross R^{n}\cross R^{n}