Sharpness of some properties of Wiener amalgam and modulation spaces

TL;DR

This paper establishes sharp dilation estimates for Wiener amalgam spaces and confirms the optimality of certain convolution, pointwise, and Schrödinger propagator estimates on modulation spaces, enhancing understanding of their scaling properties.

Contribution

It provides the first sharp dilation bounds for Wiener amalgam spaces and verifies the optimality of key estimates on modulation spaces, including those related to the Schrödinger propagator.

Findings

Sharp dilation estimates for Wiener amalgam spaces $W(L^p,L^q)$.

Optimality of convolution and pointwise relations for modulation spaces $M^{p,q}$.

Best possible bounds for the Schrödinger propagator on modulation spaces.

Abstract

We prove sharp estimates for the dilation operator $f(x)\longmapsto f(\lambda x)$, when acting on Wiener amalgam spaces $W(L^p,L^q)$. Scaling arguments are also used to prove the sharpness of the known convolution and pointwise relations for modulation spaces $M^{p,q}$, as well as the optimality of an estimate for the Schr\"odinger propagator on modulation spaces.