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
DOI:
https://doi.org/10.1090/S0002-9939-09-09968-7
Published electronically:
June 22, 2009
MathSciNet review:
2529896
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We study the action on modulation spaces of Fourier multipliers with symbols , for real-valued functions
having unbounded second derivatives. In a simplified form our result reads as follows: if
satisfies the usual symbol estimates of order
, or if
is a positively homogeneous function of degree
, then the corresponding Fourier multiplier is bounded as an operator between the weighted modulation spaces
and
, for all
and
. Here
represents the loss of derivatives. The above threshold is shown to be sharp for any homogeneous function
whose Hessian matrix is non-degenerate at some point.
- 1. Árpád Bényi, Karlheinz Gröchenig, Kasso A. Okoudjou, and Luke G. Rogers, Unimodular Fourier multipliers for modulation spaces, J. Funct. Anal. 246 (2007), no. 2, 366–384. MR 2321047, https://doi.org/10.1016/j.jfa.2006.12.019
- 2. A. Bényi and K.A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces. Preprint, April 2007. Available at arXiv:0704.0833v1.
- 3. Francesco Concetti and Joachim Toft, Trace ideals for Fourier integral operators with non-smooth symbols, Pseudo-differential operators: partial differential equations and time-frequency analysis, Fields Inst. Commun., vol. 52, Amer. Math. Soc., Providence, RI, 2007, pp. 255–264. MR 2385329
- 4. F. Concetti, G. Garello and J. Toft, Trace ideals for Fourier integral operators with non-smooth symbols II. Preprint, 2007. Available at arXiv:0710.3834.
- 5. Elena Cordero and Fabio Nicola, Some new Strichartz estimates for the Schrödinger equation, J. Differential Equations 245 (2008), no. 7, 1945–1974. MR 2433493, https://doi.org/10.1016/j.jde.2008.07.009
- 6. E. Cordero and F. Nicola, Boundedness of Fourier integral operators on modulation spaces. Preprint, 2008. Available at arXiv:0807.2380.
- 7.
E. Cordero, F. Nicola and L. Rodino,
Time-frequency analysis of Fourier integral operators.
Commun. Pure Appl. Anal., to appear. Available at arXiv:0710.3652v1. - 8.
E. Cordero, F. Nicola and L. Rodino,
Boundedness of Fourier integral operators onspaces, Trans. Amer. Math. Soc., to appear. Available at ArXiv:0801.1444.
- 9. Yngve Domar, On the spectral synthesis problem for (𝑛-1)-dimensional subsets of 𝑅ⁿ,𝑛≥2, Ark. Mat. 9 (1971), 23–37. MR 324319, https://doi.org/10.1007/BF02383635
- 10.
H.G. Feichtinger,
Modulation spaces on locally compact abelian groups.
Technical Report, University of Vienna, 1983, and also in
Wavelets and Their Applications, M. Krishna, R. Radha, S. Thangavelu, editors,
99-140, Allied Publishers, New Delhi, 2003. - 11. Hans G. Feichtinger, Modulation spaces: looking back and ahead, Sampl. Theory Signal Image Process. 5 (2006), no. 2, 109–140. MR 2233968
- 12. Hans G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. II, Monatsh. Math. 108 (1989), no. 2-3, 129–148. MR 1026614, https://doi.org/10.1007/BF01308667
- 13. Hans G. Feichtinger and Ghassem Narimani, Fourier multipliers of classical modulation spaces, Appl. Comput. Harmon. Anal. 21 (2006), no. 3, 349–359. MR 2274842, https://doi.org/10.1016/j.acha.2006.04.010
- 14. Karlheinz Gröchenig, Foundations of time-frequency analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1843717
- 15. Karlheinz Gröchenig and Christopher Heil, Modulation spaces and pseudodifferential operators, Integral Equations Operator Theory 34 (1999), no. 4, 439–457. MR 1702232, https://doi.org/10.1007/BF01272884
- 16. Lars Hörmander, Estimates for translation invariant operators in 𝐿^{𝑝} spaces, Acta Math. 104 (1960), 93–140. MR 121655, https://doi.org/10.1007/BF02547187
- 17.
L. Hörmander,
The Analysis of Linear Partial Differential Operators, Vol. IV. Springer-Verlag, Berlin, 1985. - 18. Liang-pan Li, Some remarks on a multiplier theorem of Hörmander, Math. Appl. (Wuhan) 15 (2002), no. suppl., 152–154 (English, with English and Chinese summaries). MR 1960194
- 19. Walter Littman, Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc. 69 (1963), 766–770. MR 155146, https://doi.org/10.1090/S0002-9904-1963-11025-3
- 20. Takahiro Mizuhara, On Fourier multipliers of homogeneous Besov spaces, Math. Nachr. 133 (1987), 155–161. MR 912425, https://doi.org/10.1002/mana.19871330110
- 21. S. Rivetti, Fourier multipliers on modulation spaces. Ph.D. Thesis, Politecnico di Torino, in preparation.
- 22. Johannes Sjöstrand, An algebra of pseudodifferential operators, Math. Res. Lett. 1 (1994), no. 2, 185–192. MR 1266757, https://doi.org/10.4310/MRL.1994.v1.n2.a6
- 23. Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- 24. Mitsuru Sugimoto and Naohito Tomita, The dilation property of modulation spaces and their inclusion relation with Besov spaces, J. Funct. Anal. 248 (2007), no. 1, 79–106. MR 2329683, https://doi.org/10.1016/j.jfa.2007.03.015
- 25. Joachim Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus. II, Ann. Global Anal. Geom. 26 (2004), no. 1, 73–106. MR 2054576, https://doi.org/10.1023/B:AGAG.0000023261.94488.f4
- 26. H. Triebel, Modulation spaces on the Euclidean 𝑛-space, Z. Anal. Anwendungen 2 (1983), no. 5, 443–457 (English, with German and Russian summaries). MR 725159, https://doi.org/10.4171/ZAA/79
- 27. Baoxiang Wang and Chunyan Huang, Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations, J. Differential Equations 239 (2007), no. 1, 213–250. MR 2341554, https://doi.org/10.1016/j.jde.2007.04.009
- 28. Wang Baoxiang, Zhao Lifeng, and Guo Boling, Isometric decomposition operators, function spaces 𝐸^{𝜆}_{𝑝,𝑞} and applications to nonlinear evolution equations, J. Funct. Anal. 233 (2006), no. 1, 1–39. MR 2204673, https://doi.org/10.1016/j.jfa.2005.06.018
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 42B15, 42B35, 42C15
Retrieve articles in all journals with MSC (2000): 42B15, 42B35, 42C15
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:
https://doi.org/10.1090/S0002-9939-09-09968-7
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.