Dilations properties for weighted modulation spaces

We give a sharp estimate on the norm of the scaling operator U_\lambda f(x) = f(\lambda x) acting on the weighted modulation spaces M(s,t)(p,q) (R^d). In particular, we recover and extend recent results by Sugimoto and Tomita in the unweighted case. As an application of our results, we estimate the growth in time of solutions of the wave and vibrating plate equations, which is of interest when considering the well-posedness of the Cauchy problem for these equations. Finally, we provide new embedding results between modulation and Besov spaces.