Modulus of continuity and decay at infinity in evolution equations with real characteristics

In the hyperbolic Cauchy problem, the well-posedness in Sobolev spaces and the modulus of continuity of the coefficients are deeply connected. This holds true in the more general framework of p-evolution equations with real characteristics where a sharp scale of Hoelder continuity, with respect to the time variable has been established. We show that, for p>1, a lack of regularity in t can be compensated by a decay as the space variable x tends to infinity