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Time-Frequency Analysis and PDE's

Authors: Anita Tabacco

Published in: Dagstuhl Seminar Proceedings, Volume 8492, Structured Decompositions and Efficient Algorithms (2009)

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.

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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)

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@InProceedings{tabacco:DagSemProc.08492.10,
  author =	{Tabacco, Anita},
  title =	{{Time-Frequency Analysis and PDE's}},
  booktitle =	{Structured Decompositions and Efficient Algorithms},
  pages =	{1--4},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2009},
  volume =	{8492},
  editor =	{Stephan Dahlke and Ingrid Daubechies and Michal Elad and Gitta Kutyniok and Gerd Teschke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08492.10},
  URN =		{urn:nbn:de:0030-drops-18792},
  doi =		{10.4230/DagSemProc.08492.10},
  annote =	{Keywords: Fourier multipliers, modulation spaces, short-time Fourier transform}
}

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Time-Frequency Analysis and PDE's

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