Dilation Properties for Weighted Modulation Spaces

Abstract

We give a sharp estimate on the norm of the scaling operator Uλf(x)=f(λx) acting on the weighted modulation spaces Ms,tp,q(ℝd). 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.