Sharp estimates of unimodular Fourier multipliers on Wiener amalgam spaces

We study the boundedness on the Wiener amalgam spaces $W^{p,q}_s$ of Fourier multipliers with symbols of the type $e^{imu(xi)}$, for some real-valued functions $mu(xi)$ whose prototype is $|xi|^{beta}$ with $betain (0,2]$. Under some suitable assumptions on $mu$, we give the characterization of $W^{p,q}_srightarrow W^{p,q}$ boundedness of $e^{imu(D)}$, for arbitrary pairs of $0< p,qleq infty$. Our results are an essential improvement of the previous known results, for both sides of sufficiency and necessity, even for the special case $mu(xi)=|xi|^{beta}$ with $1<beta<2$.