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

Abstract

We prove the H ∞ H^{\infty } well-posedness of the forward Cauchy problem for a pseudo-differential operator P P of order m ≥ 2 m\geq 2 with the Log-Lipschitz continuous symbol in the time variable. The characteristic roots λ k \lambda _k of P P are distinct and satisfy the necessary Lax-Mizohata condition Im λ k ≥ 0 \lambda _k\geq 0 . The Log-Lipschitz regularity has been tested as the optimal one for H ∞ 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.