The dilation property of modulation spaces and their inclusion relation with Besov spaces

摘要

We consider the dilation property of the modulation spaces M-p,M-q. Let D-lambda : f (t) -> (lambda t) be the dilation operator, and we consider the behavior of the operator norm parallel to D-lambda parallel to M-p,M-q -> M-p,M-q with respect to lambda. Our result determines the best order for it, and as an application, we establish the optimality of the inclusion relation between the modulation spaces and Besov spaces, which was proved by Toft [J. Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus, 1, J. Funct. Anal. 207 (2004) 399-429]. (c) 2007 Elsevier Inc. All rights reserved.