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 R n of degree α ≥ 2 , we show the boundedness of the operator e i μ ( D ) between the weighted Wiener amalgam space W s p , q and W p , 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 e i μ ( D ) on W p , q for α ∈ ( 0 , 2 ) .