The Cauchy problem for a class of Kovalevskian pseudo-differential operators

We prove the $H^{\infty}$ well-posedness of the forward Cauchy problem for a pseudo-differential operator $P$ of order $m\geq 2$ with the Log-Lipschitz continuous symbol in the time variable. The characteristic roots $\lambda_k$ of $P$ are distinct and satisfy the necessary Lax-Mizohata condition Im $\lambda_k\geq 0$. The Log-Lipschitz regularity has been tested as the optimal one for $H^{\infty}$ well-posedness in the case of second-order hyperbolic operators. Our main aim is to present a simple proof which needs only a little of the basic calculus of standard pseudo-differential operators.