Unimodular Fourier multipliers for modulation spaces

We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol e(i)vertical bar xi vertical bar(alpha), where alpha is an element of [0, 2], are bounded on all modulation spaces, but, in general, fail to be bounded on the usual L-P-spaces. As a consequence, the phase-space concentration of the solutions to the free Schrodinger and wave equations are preserved. As a byproduct, we also obtain boundedness results on modulation spaces for singular multipliers vertical bar xi vertical bar(-delta) sin(vertical bar xi vertical bar(alpha)) for 0 <= delta <= alpha. (c) 2007 Elsevier Inc. All rights reserved.