Energy estimates for weakly hyperbolic systems of the first order

For a class of weakly hyperbolic systems of the form D_t - A(t,x,D_x), where A(t,x,D_x) is a first-order pseudodifferential operator whose principal symbol degenerates like t^{l_*} at time t=0, for some integer l_* \geq 1, well-posedness of the Cauchy problem is proved in an adapted scale of Sobolev spaces. In addition, an upper bound for the loss of regularity that occurs when passing from the Cauchy data to the solutions is established. In examples, this upper bound turns out to be sharp.