Sharp estimates of unimodular Fourier multipliers on Wiener amalgam spaces

Abstract

We study the boundedness on the Wiener amalgam spaces $W_s^{p,q}$ 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 $W_s^{p,q}\to W^{p,q}$ boundedness of $e^{i\mu (D)}$, for arbitrary pairs of $0<p,q\le \infty$. 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 $1<\beta <2$.