The Cauchy Problem for properly hyperbolic equations in one space variable

In this paper we consider the Cauchy problem for higher order weakly hyperbolic equations. We assume that the principal symbol depends only on one space variable and the characteristic roots $\tau_j$ verify the inequality $$\tau_j^2(x) + \tau_k^2(x) \le M \bigl(\tau_j(x)-\tau_k(x)\bigr)^2$$ for some constant $M$ independent of $x$. We prove that if the lower order terms verify a suitable Levi condition, the Cauchy problem is well-posed in $\mathcal{C}$-infinity.