Highly oscillatory unimodular Fourier multipliers on modulation spaces

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 $\ge 2$ of the phase $\mu (\xi )$ are bounded or, more generally, that its second derivatives belong to the Sjöstrand 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.

19 pages, 1 figure