On weakly hyperbolic equations with analytic principal part

In this paper we show how to include low order terms in the $C^{\mathrm{\infty }}$ well-posedness results for weakly hyperbolic equations with analytic time-dependent coefficients. This is achieved by doing a different reduction to a system from the previously used one. We find the Levi conditions such that the $C^{\mathrm{\infty }}$ well-posedness continues to hold.