Remarks on the Unimodular Fourier Multipliers on α-Modulation Spaces

Abstract

We study the boundedness properties of the Fourier multiplier operator eiμ(D) on α-modulation spaces Mp,qs,α (0≤α<1) and Besov spaces Bp,qs(Mp,qs,1). We improve the conditions for the boundedness of Fourier multipliers with compact supports and for the boundedness of eiμ(D) on Mp,qs,α. If μ is a radial function ϕ(|ξ|) and ϕ satisfies some size condition, we obtain the sufficient and necessary conditions for the boundedness of eiϕ(|D|) between Mp1,q1s1,α and Mp2,q2s2,α.