Decay rate of solutions to hyperbolic system of first order

In this paper we generalize the global Sobolev inequality introduced by Klainerman in studying wave equation to the hyperbolic system case. We obtain several decay estimates of solutions of a hyperbolic system of first order by different norms of initial data. In particular, the result mentioned in Theorem 1.5 offers an optimal decay rate of solutions, if the initial data belongs to the assigned weighted Sobolev space. In the proof of the theorem we reduce the estimate of solutions of a hyperbolic system to the corresponding case for a scalar pseudodifferential equation of the first order, and then establish the required estimate by using microlocal analysis.