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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Estimates for unimodular Fourier multipliers on modulation spaces


Authors: Akihiko Miyachi, Fabio Nicola, Silvia Rivetti, Anita Tabacco and Naohito Tomita
Journal: Proc. Amer. Math. Soc. 137 (2009), 3869-3883
MSC (2000): Primary 42B15, 42B35, 42C15
Published electronically: June 22, 2009
MathSciNet review: 2529896
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Abstract: We study the action on modulation spaces of Fourier multipliers with symbols $ e^{i\mu(\xi)}$, for real-valued functions $ \mu$ having unbounded second derivatives. In a simplified form our result reads as follows: if $ \mu$ satisfies the usual symbol estimates of order $ \alpha\geq2$, or if $ \mu$ is a positively homogeneous function of degree $ \alpha$, then the corresponding Fourier multiplier is bounded as an operator between the weighted modulation spaces $ M^{p,q}_s$ and $ M^{p,q}$, for all $ 1\leq p,q\leq\infty$ and $ s\geq (\alpha-2)n\vert{1/p}-1/2\vert$. Here $ s$ represents the loss of derivatives. The above threshold is shown to be sharp for any homogeneous function $ \mu$ whose Hessian matrix is non-degenerate at some point.


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Additional Information

Akihiko Miyachi
Affiliation: Department of Mathematics, Tokyo Woman’s Christian University, Zempukuji, Suginami-ku, Tokyo 167-8585, Japan
Email: miyachi@lab.twcu.ac.jp

Fabio Nicola
Affiliation: Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Email: fabio.nicola@polito.it

Silvia Rivetti
Affiliation: Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Email: silvia.rivetti@polito.it

Anita Tabacco
Affiliation: Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Email: anita.tabacco@polito.it

Naohito Tomita
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Email: tomita@math.sci.osaka-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09968-7
PII: S 0002-9939(09)09968-7
Keywords: Fourier multipliers, modulation spaces, short-time Fourier transform, Schr\"odinger operators
Received by editor(s): October 30, 2008
Received by editor(s) in revised form: March 11, 2009
Published electronically: June 22, 2009
Additional Notes: The second, third, and fourth authors were partially supported by the Progetto MIUR Cofinanziato 2007 “Analisi Armonica”
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.