Estimates for unimodular Fourier multipliers on modulation spaces

We study the action on modulation spaces of Fourier multipliers with symbols e^iμ(ξ), for real-valued functions μ having unbounded second derivatives. In a simplified form our result reads as follows: if μ satisfies the usual symbol estimates of order α ≥ 2, or if μ is a positively homogeneous function of degree α, 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 ≤ p, q ≤ ∞ and s ≥ (α − 2)n|1/p − 1/2|. Here s represents the loss of derivatives. The above threshold is shown to be sharp for any homogeneous function μ whose Hessian matrix is non-degenerate at some point.