Unimodular Fourier multipliers for modulation spaces

We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol ei|ξ |α, where α ∈ [0, 2], are bounded on all modulation spaces, but, in general, fail to be bounded on the usual Lp-spaces. As a consequence, the phase-space concentration of the solutions to the free Schrödinger and wave equations are preserved. As a byproduct, we also obtain boundedness results on modulation spaces for singular multipliers |ξ |−δ sin(|ξ |α) for 0 δ α. © 2007 Elsevier Inc. All rights reserved