Unimodular Fourier multipliers for modulation spaces

Abstract

We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol $e^{i|\xi {|}^{\alpha }}$, where $\alpha \in [0,2]$, 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 Schrödinger and wave equations are preserved. As a byproduct, we also obtain boundedness results on modulation spaces for singular multipliers $|\xi {|}^{-\delta }\sin (|\xi {|}^{\alpha })$ for $0\le \delta \le \alpha$.

18 pages

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