Unimodular Fourier multipliers on Wiener amalgam spaces

We study the boundedness of unimodular Fourier multipliers on Wiener amalgam spaces. For a real-valued homogeneous function μ on Rn of degree α≥2, we show the boundedness of the operator eiμ(D) between the weighted Wiener amalgam space Wsp,q and Wp,q for all 1≤p,q≤∞ and s>n(α−2)|1/p−1/2|+n|1/p−1/q|. This threshold is shown to be optimal for regions max(1/q,1/2)≤1/p and min(1/q,1/2)≥1/p. Moreover, we give sufficient conditions for the boundedness of eiμ(D) on Wp,q for α∈(0,2).