Unimodular Multipliers on Modulation Spaces.

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