Hermite Multipliers on Modulation Spaces

TL;DR

This paper investigates the boundedness of Hermite multipliers on modulation spaces, demonstrating their stability for solutions of wave and Schrödinger equations and the boundedness of related Riesz transforms.

Contribution

It establishes boundedness conditions for Hermite multipliers on modulation spaces and applies these results to evolution equations and Riesz transforms.

Findings

Hermite multipliers are bounded on modulation spaces under standard conditions.

Solutions to wave and Schrödinger equations with initial data in modulation spaces remain in the same space.

Riesz transforms associated with the Hermite operator are bounded on certain modulation spaces.

Abstract

We study multipliers associated to the Hermite operator $H=-\Delta + |x|^2$ on modulation spaces $M^{p,q}(\mathbb R^d)$. We prove that the operator $m(H)$ is bounded on $M^{p,q}(\mathbb R^d)$ under standard conditions on $m,$ for suitable choice of $p$ and $q$. As an application, we point out that the solutions to the free wave and Schr\"odinger equations associated to $H$ with initial data in a modulation space will remain in the same modulation space for all times. We also point out that Riesz transforms associated to $H$ are bounded on some modulation spaces.