Dilation properties for weighted modulation spaces

TL;DR

This paper provides sharp estimates for the scaling operator on weighted modulation spaces, extending previous results, and applies these to analyze solution growth in wave and plate equations, also establishing new embeddings between modulation and Besov spaces.

Contribution

It offers a precise norm estimate for the dilation operator on weighted modulation spaces, extending prior work and applying it to PDE solution growth and space embeddings.

Findings

Sharp dilation norm estimates for weighted modulation spaces

Extended results to the unweighted case by Sugimoto and Tomita

Estimated solution growth for wave and vibrating plate equations

Abstract

In this paper 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 $\M{p,q}{s,t}(\R^{d})$. 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 posedeness of the Cauchy problem for these equations. Finally, we provide new embedding results between modulation and Besov spaces.