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Unimodular Fourier multipliers for modulation spaces. (English) Zbl 1120.42010

A Fourier multiplier is a linear operator \(H_\sigma\) whose action on a test function \(f\) on \(\mathbb{R}^d\) formally is defined by \[ H_\sigma f(x)= \int_{\mathbb{R}^d} \sigma(\xi)\widehat{f}(\xi)2^{2\pi i \xi \cdot x} \,d\xi. \] The function \(\sigma\) is called the symbol. It is shown that the Fourier multipliers with symbols \(e^{i| \xi| ^\alpha}\) for \(\alpha\in [0,2]\) are bounded on all modulation spaces, but, in general, fail to be bounded on the \(L^p\)-spaces. The result has consequences for the phase-space concentration of the solutions to the free Schrödinger equation and the wave equation.

MSC:

42B15 Multipliers for harmonic analysis in several variables
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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