Sharp estimates of unimodular multipliers on frequency decomposition spaces

This paper is devoted to give a general method for the study of unimodular multipliers on certain function spaces defined by decomposition on the frequency plane. These spaces include modulation spaces, αα-modulation spaces and (homogeneous and inhomogeneous) Besov spaces. We give a complete characterization of the Fourier multipliers on these function spaces, and also give a characterization of unimodular multipliers under some mild assumptions. As applications, we obtain some sharp boundedness properties of unimodular Fourier multipliers between these function spaces. We also obtain the asymptotic estimates for certain free dispersive semigroups.