Sharp estimates of unimodular Fourier multipliers on Wiener amalgam spaces

Abstract: We study the boundedness on the Wiener amalgam spaces $W^{p,q}_s$ of Fourier multipliers with symbols of the type $e^{i\mu(\xi)}$, for some real-valued functions $\mu(\xi)$ whose prototype is $|\xi|^{\beta}$ with $\beta\in (0,2]$. Under some suitable assumptions on $\mu$, we give the characterization of $W^{p,q}_s\rightarrow W^{p,q}$ boundedness of $e^{i\mu(D)}$, for arbitrary pairs of $0< p,q\leq \infty$. 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 $1<\beta<2$.