Unimodular Fourier multipliers for modulation spaces

Details Unimodular Fourier multipliers for modulation spaces Unimodular Fourier multipliers for modulation spaces

2006-09-04

arxiv.org/abs/math/0609097v1

J.~Functional Anal.~246 (2007), 366 -- 384

Mathematics

Functional Analysis

18 pages

Scientific paper

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 Schr\"odinger 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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