2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/113806We consider the dilation property of the modulation spaces $M^{p,q}$. Let $D_λ:f(t)\mapsto f(λt)$ be the dilation operator, and we consider the behavior of the operator norm $\|D_λ\|_{M^{p,q}\to M^{p,q}}$ with respect to $λ$. 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 space, which was proved by Toft.23 pagesFunctional AnalysisAnalysis of PDEs42B35text