by Árpád Bényi, Kasso A. Okoudjou, Karlheniz Gröchenig and Luke G. Rogers.

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 $$0\leq\delta\leq\alpha$$.

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