Precise Finite Speed and Uniqueness in the Cauchy Problem for Symmetrizable Hyperbolic Systems

Precise finite speed, in the sense of that the domain of influence is a subset of the union of influence curves through the support of the initial data is proved for hyperbolic systems symmetrized by pseudodifferential operators in the spatial variables. From this, uniqueness in the Cauchy problem at spacelike hypersurfaces is derived by a Hölmgren style duality argument. Sharp finite speed is derived from an estimate for propagation in each direction. Propagation in a fixed direction is proved by regularizing the problem in the orthogonal directions. Uniform estimates for the regularized equations is proved using pseudodiffential techniques of Beals-Fefferman type.