Gabor representations of evolution operators

Cordero, Elena; Nicola, Fabio; Rodino, Luigi

doi:10.1090/S0002-9947-2015-06302-8

Gabor representations of evolution operators
HTML articles powered by AMS MathViewer

by Elena Cordero, Fabio Nicola and Luigi Rodino PDF

Trans. Amer. Math. Soc. 367 (2015), 7639-7663 Request permission

Abstract:

We perform a time-frequency analysis of Fourier multipliers and, more generally, pseudodifferential operators with symbols of Gevrey, analytic and ultra-analytic regularity. As an application we show that Gabor frames, which provide optimally sparse decompositions for Schrödinger-type propagators, reveal to be an even more efficient tool for representing solutions to a wide class of evolution operators with constant coefficients, including weakly hyperbolic and parabolic-type operators. Besides the class of operators, the main novelty of the paper is the proof of super-exponential (as opposed to super-polynomial) off-diagonal decay for the Gabor matrix representation.

References

Kenji Asada and Daisuke Fujiwara, On some oscillatory integral transformations in $L^{2}(\textbf {R}^{n})$, Japan. J. Math. (N.S.) 4 (1978), no. 2, 299–361. MR 528863, DOI 10.4099/math1924.4.299
Árpád Bényi and Kasso A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. Lond. Math. Soc. 41 (2009), no. 3, 549–558. MR 2506839, DOI 10.1112/blms/bdp027
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, DOI 10.1016/j.jfa.2006.12.019
Emmanuel J. Candès and Laurent Demanet, The curvelet representation of wave propagators is optimally sparse, Comm. Pure Appl. Math. 58 (2005), no. 11, 1472–1528. MR 2165380, DOI 10.1002/cpa.20078
Lamberto Cattabriga, Alcuni problemi per equazioni differenziali lineri con coefficienti costanti, Quaderni dell’Unione Matematica Italiana, 24, Pitagora, Bologna, 1983.
Elena Cordero, Gelfand-Shilov window classes for weighted modulation spaces, Integral Transforms Spec. Funct. 18 (2007), no. 11-12, 829–837. MR 2374584, DOI 10.1080/10652460701510709
H. Feichtinger, F. Luef, and E. Cordero, Banach Gelfand triples for Gabor analysis, Pseudo-differential operators, Lecture Notes in Math., vol. 1949, Springer, Berlin, 2008, pp. 1–33. MR 2477142, DOI 10.1007/978-3-540-68268-4_{1}
Elena Cordero, Karlheinz Gröchenig, and Fabio Nicola, Approximation of Fourier integral operators by Gabor multipliers, J. Fourier Anal. Appl. 18 (2012), no. 4, 661–684. MR 2984364, DOI 10.1007/s00041-011-9214-1
Elena Cordero and Fabio Nicola, Remarks on Fourier multipliers and applications to the wave equation, J. Math. Anal. Appl. 353 (2009), no. 2, 583–591. MR 2508960, DOI 10.1016/j.jmaa.2008.12.027
Elena Cordero and Fabio Nicola, Boundedness of Schrödinger type propagators on modulation spaces, J. Fourier Anal. Appl. 16 (2010), no. 3, 311–339. MR 2643585, DOI 10.1007/s00041-009-9111-z
Elena Cordero, Fabio Nicola, and Luigi Rodino, Sparsity of Gabor representation of Schrödinger propagators, Appl. Comput. Harmon. Anal. 26 (2009), no. 3, 357–370. MR 2502369, DOI 10.1016/j.acha.2008.08.003
Elena Cordero, Fabio Nicola, and Luigi Rodino, Boundedness of Fourier integral operators on ${\scr F}L^p$ spaces, Trans. Amer. Math. Soc. 361 (2009), no. 11, 6049–6071. MR 2529924, DOI 10.1090/S0002-9947-09-04848-X
Elena Cordero, Fabio Nicola, and Luigi Rodino, Time-frequency analysis of Fourier integral operators, Commun. Pure Appl. Anal. 9 (2010), no. 1, 1–21. MR 2556742, DOI 10.3934/cpaa.2010.9.1
Elena Cordero, Fabio Nicola, and Luigi Rodino, On the global boundedness of Fourier integral operators, Ann. Global Anal. Geom. 38 (2010), no. 4, 373–398. MR 2733369, DOI 10.1007/s10455-010-9219-z
Elena Cordero, Stevan Pilipović, Luigi Rodino, and Nenad Teofanov, Localization operators and exponential weights for modulation spaces, Mediterr. J. Math. 2 (2005), no. 4, 381–394. MR 2191800, DOI 10.1007/s00009-005-0052-8
Elena Cordero and Davide Zucco, The Cauchy problem for the vibrating plate equation in modulation spaces, J. Pseudo-Differ. Oper. Appl. 2 (2011), no. 3, 343–354. MR 2831662, DOI 10.1007/s11868-011-0032-7
Antonio Córdoba and Charles Fefferman, Wave packets and Fourier integral operators, Comm. Partial Differential Equations 3 (1978), no. 11, 979–1005. MR 507783, DOI 10.1080/03605307808820083
Ingrid Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), no. 5, 961–1005. MR 1066587, DOI 10.1109/18.57199
Hans G. Feichtinger, Modulation spaces on locally compact abelian groups, Technical Report, University Vienna, 1983, and also in Wavelets and Their Applications, M. Krishna, R. Radha, S. Thangavelu, editors, Allied Publishers, 2003, pp. 99–140.
Dan-Andrei Geba and Daniel Tataru, A phase space transform adapted to the wave equation, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1065–1101. MR 2353138, DOI 10.1080/03605300600854308
I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 3: Theory of differential equations, Academic Press, New York-London, 1967. Translated from the Russian by Meinhard E. Mayer. MR 0217416
Karlheinz Gröchenig, Foundations of time-frequency analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1843717, DOI 10.1007/978-1-4612-0003-1
Karlheinz Gröchenig, Time-frequency analysis of Sjöstrand’s class, Rev. Mat. Iberoam. 22 (2006), no. 2, 703–724. MR 2294795, DOI 10.4171/RMI/471
Karlheinz Gröchenig and Yurii Lyubarskii, Gabor (super)frames with Hermite functions, Math. Ann. 345 (2009), no. 2, 267–286. MR 2529475, DOI 10.1007/s00208-009-0350-8
Karlheinz Gröchenig and Ziemowit Rzeszotnik, Banach algebras of pseudodifferential operators and their almost diagonalization, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 7, 2279–2314 (English, with English and French summaries). MR 2498351, DOI 10.5802/aif.2414
Karlheinz Gröchenig and Georg Zimmermann, Spaces of test functions via the STFT, J. Funct. Spaces Appl. 2 (2004), no. 1, 25–53. MR 2027858, DOI 10.1155/2004/498627
Kanghui Guo and Demetrio Labate, Sparse shearlet representation of Fourier integral operators, Electron. Res. Announc. Math. Sci. 14 (2007), 7–19. MR 2314296
Lars Hörmander, The analysis of linear partial differential operators. II, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 257, Springer-Verlag, Berlin, 1983. Differential operators with constant coefficients. MR 705278, DOI 10.1007/978-3-642-96750-4
A. J. E. M. Janssen, Duality and biorthogonality for Weyl-Heisenberg frames, J. Fourier Anal. Appl. 1 (1995), no. 4, 403–436. MR 1350700, DOI 10.1007/s00041-001-4017-4
Keiichi Kato, Masaharu Kobayashi, and Shingo Ito, Representation of Schrödinger operator of a free particle via short-time Fourier transform and its applications, Tohoku Math. J. (2) 64 (2012), no. 2, 223–231. MR 2948820, DOI 10.2748/tmj/1341249372
Keiichi Kato, Masaharu Kobayashi, and Shingo Ito, Remark on wave front sets of solutions to Schrödinger equation of a free particle and a harmonic oscillator, SUT J. Math. 47 (2011), no. 2, 175–183. MR 2953118
Keiichi Kato, Masaharu Kobayashi, and Shingo Ito, Remarks on Wiener amalgam space type estimates for Schrödinger equation, Harmonic analysis and nonlinear partial differential equations, RIMS Kôkyûroku Bessatsu, B33, Res. Inst. Math. Sci. (RIMS), Kyoto, 2012, pp. 41–48. MR 3050804
B. S. Mitjagin, Nuclearity and other properties of spaces of type $S$, Amer. Math. Soc. Transl. 93(2) (1970), 45–59.
Akihiko Miyachi, Fabio Nicola, Silvia Rivetti, Anita Tabacco, and Naohito Tomita, Estimates for unimodular Fourier multipliers on modulation spaces, Proc. Amer. Math. Soc. 137 (2009), no. 11, 3869–3883. MR 2529896, DOI 10.1090/S0002-9939-09-09968-7
Fabio Nicola and Luigi Rodino, Global pseudo-differential calculus on Euclidean spaces, Pseudo-Differential Operators. Theory and Applications, vol. 4, Birkhäuser Verlag, Basel, 2010. MR 2668420, DOI 10.1007/978-3-7643-8512-5
Jianliang Qian and Lexing Ying, Fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beams for the wave equation, Multiscale Model. Simul. 8 (2010), no. 5, 1803–1837. MR 2728710, DOI 10.1137/100787313
Jianliang Qian and Lexing Ying, Fast Gaussian wavepacket transforms and Gaussian beams for the Schrödinger equation, J. Comput. Phys. 229 (2010), no. 20, 7848–7873. MR 2674307, DOI 10.1016/j.jcp.2010.06.043
Jeffrey Rauch, Partial differential equations, Graduate Texts in Mathematics, vol. 128, Springer-Verlag, New York, 1991. MR 1223093, DOI 10.1007/978-1-4612-0953-9
Richard Rochberg and Kazuya Tachizawa, Pseudodifferential operators, Gabor frames, and local trigonometric bases, Gabor analysis and algorithms, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 1998, pp. 171–192. MR 1601103
Michael Ruzhansky, Mitsuru Sugimoto, and Baoxiang Wang, Modulation spaces and nonlinear evolution equations, Evolution equations of hyperbolic and Schrödinger type, Progr. Math., vol. 301, Birkhäuser/Springer Basel AG, Basel, 2012, pp. 267–283. MR 3014810, DOI 10.1007/978-3-0348-0454-7_{1}4
Laurent Schwartz, Mathematics for the physical sciences, Dover, 2008.
Hart F. Smith, A parametrix construction for wave equations with $C^{1,1}$ coefficients, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 3, 797–835 (English, with English and French summaries). MR 1644105, DOI 10.5802/aif.1640
Thomas Strohmer, Pseudodifferential operators and Banach algebras in mobile communications, Appl. Comput. Harmon. Anal. 20 (2006), no. 2, 237–249. MR 2207837, DOI 10.1016/j.acha.2005.06.003
Daniel Tataru, Phase space transforms and microlocal analysis, Phase space analysis of partial differential equations. Vol. II, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2004, pp. 505–524. MR 2208883
Nenad Teofanov, Ultradistributions and time-frequency analysis, Pseudo-differential operators and related topics, Oper. Theory Adv. Appl., vol. 164, Birkhäuser, Basel, 2006, pp. 173–192. MR 2233590, DOI 10.1007/3-7643-7514-0_{1}3
Joachim Toft, The Bargmann transform on modulation and Gelfand-Shilov spaces, with applications to Toeplitz and pseudo-differential operators, J. Pseudo-Differ. Oper. Appl. 3 (2012), no. 2, 145–227. MR 2925181, DOI 10.1007/s11868-011-0044-3
François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131
David F. Walnut, Lattice size estimates for Gabor decompositions, Monatsh. Math. 115 (1993), no. 3, 245–256. MR 1233956, DOI 10.1007/BF01300672
Baoxiang Wang. Sharp global well-posedness for non-elliptic derivative Schrödinger equations with small rough data. arXiv:1012.0370.
Wang Baoxiang, Zhao Lifeng, and Guo Boling, Isometric decomposition operators, function spaces $E^\lambda _{p,q}$ and applications to nonlinear evolution equations, J. Funct. Anal. 233 (2006), no. 1, 1–39. MR 2204673, DOI 10.1016/j.jfa.2005.06.018
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, DOI 10.1016/j.jde.2007.04.009
Baoxiang Wang and Henryk Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data, J. Differential Equations 232 (2007), no. 1, 36–73. MR 2281189, DOI 10.1016/j.jde.2006.09.004
Baoxiang Wang, Zhaohui Huo, Chengchun Hao, and Zihua Guo, Harmonic analysis method for nonlinear evolution equations. I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. MR 2848761, DOI 10.1142/9789814360746