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Some local questions for hyperbolic systems with non-regular time dependent coefficients

In this note we investigate local properties for microlocally symmetrizable hyperbolic systems with just time dependent coefficients. Thanks to Paley-Wiener theorem, we establish finite propagation speed by showing precise estimates on the evolution of the support of the solution in terms of suitable norms of the coefficients of the operator and of the symmetrizer. From this result, local existence and uniqueness follow by quite standard methods. Our argument relies on the use of Fourier transform, and it cannot be extended to operators whose coefficients depend also on the space variables. On the other hand, it works under very mild regularity assumptions on the coefficients of the operator and of the symmetrizer.