In this paper we establish $L^p-L^q$ estimates for the solution of the strictly hyperbolic first-order linear systems with bounded time-dependent coefficients. In the equation setting, Reissig and others obtained such estimates by using WKB representations of the solutions. Here the crucial point is to find minimal assumptions on the coefficients of the system so that Reissig's approach still works for systems.
"$L^p$-$L^q$ estimates for regularly linear hyperbolic systems." Adv. Differential Equations 14 (9/10) 801 - 834, September/October 2009.