DagSemProc.08492.10.pdf
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Time-Frequency Analysis and PDE's
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Dagstuhl Seminar Proceedings, Volume 8492
Series: Dagstuhl Seminar Proceedings (DagSemProc) - License:
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- Publication Date: 2009-02-24
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Anita Tabacco. Time-Frequency Analysis and PDE's. In Structured Decompositions and Efficient Algorithms. Dagstuhl Seminar Proceedings, Volume 8492, pp. 1-4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)
https://doi.org/10.4230/DagSemProc.08492.10
https://doi.org/10.4230/DagSemProc.08492.10
Abstract
We study the action on modulation spaces of Fourier multipliers with symbols
$e^{imu(xi)}$, for real-valued functions $mu$ having unbounded second
derivatives. We show that if $mu$ satisfies the usual symbol estimates of order
$alphageq2$, or if $mu$ is a positively homogeneous function of degree $alpha$,
the corresponding Fourier multiplier is bounded as an operator between the weighted modulation spaces $mathcal{M}^{p,q}_delta$ and $mathcal{M}^{p,q}$,
for every $1leq p,qleqinfty$ and $deltageq d(alpha-2)|frac{1}{p}-frac{1}{2}|$.
Here $delta$ represents the loss of derivatives. The above threshold is shown to
be sharp for {it all} homogeneous functions $mu$ whose Hessian matrix is
non-degenerate at some point.
Keywords
- Fourier multipliers
- modulation spaces
- short-time Fourier transform
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