## Estimates for unimodular Fourier multipliers on modulation spaces

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- by Akihiko Miyachi, Fabio Nicola, Silvia Rivetti, Anita Tabacco and Naohito Tomita
- Proc. Amer. Math. Soc.
**137**(2009), 3869-3883 - DOI: https://doi.org/10.1090/S0002-9939-09-09968-7
- Published electronically: June 22, 2009
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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 $\alpha \geq 2$, 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 $s\geq (\alpha -2)n|{1/p}-1/2|$. 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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## Bibliographic Information

**Akihiko Miyachi**- Affiliation: Department of Mathematics, Tokyo Woman’s Christian University, Zempukuji, Suginami-ku, Tokyo 167-8585, Japan
- MR Author ID: 193440
- 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
- MR Author ID: 739282
- Email: tomita@math.sci.osaka-u.ac.jp
- 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
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**137**(2009), 3869-3883 - MSC (2000): Primary 42B15, 42B35, 42C15
- DOI: https://doi.org/10.1090/S0002-9939-09-09968-7
- MathSciNet review: 2529896