Highly oscillatory unimodular Fourier multipliers on modulation spaces

TL;DR

This paper investigates the boundedness of highly oscillatory Fourier multipliers with phase functions having unbounded second derivatives on modulation spaces, extending previous results to broader classes of symbols with stronger oscillations.

Contribution

It extends known boundedness results for Fourier multipliers to cases where the phase's second derivatives are unbounded, using Wiener amalgam spaces.

Findings

Boundedness on weighted modulation spaces with derivative loss

Extension to phases with unbounded second derivatives

Inclusion of oscillatory symbols like cos |ξ|^2

Abstract

We study the continuity on the modulation spaces $M^{p,q}$ of Fourier multipliers with symbols of the type $e^{i\mu(\xi)}$, for some real-valued function $\mu(\xi)$. A number of results are known, assuming that the derivatives of order $\geq 2$ of the phase $\mu(\xi)$ are bounded or, more generally, that its second derivatives belong to the Sj\"ostrand class $M^{\infty,1}$. Here we extend those results, by assuming that the second derivatives lie in the bigger Wiener amalgam space $W(\mathcal{F} L^1,L^\infty)$; in particular they could have stronger oscillations at infinity such as $\cos |\xi|^2$. Actually our main result deals with the more general case of possibly unbounded second derivatives. In that case we have boundedness on weighted modulation spaces with a sharp loss of derivatives.