Highly oscillatory unimodular Fourier multipliers on modulation spaces

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 Sj"ostrand class $M^{\mathrm{\infty },1}$. 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 $\cos \mathrm{\mid }\xi {\mathrm{\mid }}^2$. 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.