Estimates for unimodular Fourier multipliers on modulation spaces

We study the action on modulation spaces of Fourier multipliers with symbols $e^{i\mu(\xi)}$, for real-valued functions $\mu$ having unbounded second derivatives. In a simplified form our result reads as follows: if $\mu$ satisfies the usual symbol estimates of order $\alpha\geq2$, or if $\mu$ is a positively homogeneous function of degree $\alpha$, then the corresponding Fourier multiplier is bounded as an operator between the weighted modulation spaces $M^{p,q}_s$ and $M^{p,q}$, for all $1\leq p,q\leq\infty$ and $s\geq (\alpha-2)n\vert{1/p}-1/2\vert$. Here $s$ represents the loss of derivatives. The above threshold is shown to be sharp for any homogeneous function $\mu$ whose Hessian matrix is non-degenerate at some point.