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Journal of Function Spaces and Applications
Volume 2012, Article ID 145491, 29 pages
http://dx.doi.org/10.1155/2012/145491
Research Article

Dilation Properties for Weighted Modulation Spaces

1Department of Mathematics, University of Torino, Via Carlo Alberto 10, 10123 Torino, Italy
2Department of Mathematics, University of Maryland, College Park, MD 20742, USA

Received 31 January 2011; Accepted 7 March 2011

Academic Editor: Hans G. Feichtinger

Copyright © 2012 Elena Cordero and Kasso A. Okoudjou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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(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.