Construction of a parametrix for a weakly hyperbolic differential operator (Q1113013)

The paper is about construction of a microlocal parametrix for the Cauchy problem: \[ (D^ 2_ t-A(t,x,D_ x))u(t,x)=0,\quad (t,x)\in [0,T]\times {\mathbb{R}}^ N;\quad u(0,x)=g_ 0(x),\quad u_ t(0,x)=g_ 1(x),\quad x\in {\mathbb{R}}^ N, \] for small \(T>0\), where \(A(t,x,D_ x)\) is a second order differential operator whose principal part \(A_ 2(t,x,\xi)\) satisfies certain conditions, which permit the factorization \(A_ 2(t,x,\xi)=(t+B(x,\xi))X(t,x,\xi).\) When \(B=0\) and when B does not depend on x, a parametrix for the problem was given by \textit{M. Imai} [Hokkaido Math. J. 8, 126-143 (1979; Zbl 0415.35058) and 9, 190-216 (1980; Zbl 0471.35048)], respectively. Other cases were treated by \textit{G. Eskin} [Commun. Partial Differ. Equations 1, 521-560 (1976; Zbl 0355.35053)] and \textit{R. B. Melrose} [Duke Math. J. 42, 583-604 (1975; Zbl 0368.35054)].