Asymptotic estimates for unimodular Fourier multipliers on modulation spaces

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February 2012, 32(2): 467-485. doi: 10.3934/dcds.2012.32.467

Asymptotic estimates for unimodular Fourier multipliers on modulation spaces

Jiecheng Chen ^1, , Dashan Fan ^2, and Lijing Sun ^2,

1.	Department of Mathematics, Zhejiang Normal University, 321004 Jinhua, China
2.	Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, United States, United States

Received August 2010 Revised June 2011 Published September 2011

Abstract
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Recently, it has been shown that the unimodular Fourier multipliers $e^{it|\Delta |^{\frac{\alpha }{2}}}$ are bounded on all modulation spaces. In this paper, using the almost orthogonality of projections and some techniques on oscillating integrals, we obtain asymptotic estimates for the unimodular Fourier multipliers $e^{it|\Delta |^{\frac{\alpha }{2}}}$ on the modulation spaces. As applications, we give the grow-up rates of the solutions for the Cauchy problems for the free Schrödinger equation, the wave equation and the Airy equation with the initial data in a modulation space. We also obtain a quantitative form about the solution to the Cauchy problem of the nonlinear dispersive equations.

Keywords: Airy equation., Unimodular multipliers, Schrödinger equation, modulation spaces, wave equation.

Mathematics Subject Classification: Primary: 42B15, 42B35; Secondary: 42C1.

Citation: Jiecheng Chen, Dashan Fan, Lijing Sun. Asymptotic estimates for unimodular Fourier multipliers on modulation spaces. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 467-485. doi: 10.3934/dcds.2012.32.467

References:

[1]	A. Bényi, K. Gröchenig, K. A. Okoudjou and L. G. Rogers, Unimodular Fourier multipliers for modulation spaces, J. Func. Anal., 246 (2007), 366-384. doi: 10.1016/j.jfa.2006.12.019.
[2]	A. Bényi and K. A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. Lond. Math. Soc., 41 (2009), 549-558. doi: 10.1112/blms/bdp027.
[3]	J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction," Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976.
[4]	E. Cordero and F. Nicola, Some new Strichartz estimates for the Schrödinger equation, J. Differential Equations, 245 (2008), 1945-1974. doi: 10.1016/j.jde.2008.07.009.
[5]	Y. Domar, On the spectral synthesis problem for $(n-1)$-dimensional subset of $ \mathbbR ^n,$ $n\geq 2,$ Ark Math, 9 (1971), 23-37. doi: 10.1007/BF02383635.
[6]	H. G. Feichtinger, Modulation spaces on locally compact abelian groups, Technical Report, University of Vienna, 1983, and in "Wavelets and Their Applications" (eds. M. Krishna, R. Radha and S. Thangavelu), 99-140, Allied Publishers, New Delhi, 2003.
[7]	H. G. Feichtinger, Modulation spaces: Looking back and ahead, Sampl Theory Signal Image Process, 5 (2006), 109-140.
[8]	K. Gröchening, "Foundations of Time-Frequency Analysis," Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001.
[9]	L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math, 104 (1960), 93-139. doi: 10.1007/BF02547187.
[10]	W. Littman, Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc., 69 (1963), 766-770. doi: 10.1090/S0002-9904-1963-11025-3.
[11]	A. Miyachi, F. Nicola, S. Rivetti, A. Taracco and N. Tomita, Estimates for unimodular Fourier multipliers on modulation spaces, Proc. Amer. Math. Soc., 137 (2009), 3869-3883. doi: 10.1090/S0002-9939-09-09968-7.
[12]	J. Sjöstrand, An algebra of pseudodifferential operators, Math. Res. Lett, 1 (1994), 185-192.
[13]	E. M. Stein, "Beijing Lectures In Harmonic Analysis," Annals of Mathematics Studies, 112, Princeton University Press, Princeton, NJ, 1982.
[14]	H. Triebel, "Theory of Function Spaces," Mathematik und ihre anwendugen in Physik und Technik [Mathematics and its Applications in Physics and Technology], 38, Akademische Verlagsgesellchaft Geest & Portig K.-G., Leipzig, 1983. doi: 10.1007/978-3-0346-0416-1.
[15]	J. Toft, Continuity properties for modulation spaces with applications to pseudo-differential calculus. II, Ann Global Anal Geom, 26 (2004), 73-106. doi: 10.1023/B:AGAG.0000023261.94488.f4.
[16]	B. Wang and H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data, J. Differential Equations, 232 (2007), 36-73. doi: 10.1016/j.jde.2006.09.004.
[17]	B. Wang, L. Han and C. Huang, Global well-posedness and scattering for the derivative nonlinear Schrödinger equation with small rough data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2253-2281.
[18]	B. Wang, C. Hao and C. Huo, "Introduction on Nonlinear Developing Equations," Unpublished Lecture Notes, Beijing University, 2009.
[19]	B. Wang, L. Zhao and B. Guo, Isometric decomposition operators function spaces $E_{p,q}^{\lambda}$ and applications to nonlinear evolution equations, J. Func. Anal., 233 (2006), 1-39. doi: 10.1016/j.jfa.2005.06.018.

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References:

[1]	A. Bényi, K. Gröchenig, K. A. Okoudjou and L. G. Rogers, Unimodular Fourier multipliers for modulation spaces, J. Func. Anal., 246 (2007), 366-384. doi: 10.1016/j.jfa.2006.12.019.
[2]	A. Bényi and K. A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. Lond. Math. Soc., 41 (2009), 549-558. doi: 10.1112/blms/bdp027.
[3]	J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction," Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976.
[4]	E. Cordero and F. Nicola, Some new Strichartz estimates for the Schrödinger equation, J. Differential Equations, 245 (2008), 1945-1974. doi: 10.1016/j.jde.2008.07.009.
[5]	Y. Domar, On the spectral synthesis problem for $(n-1)$-dimensional subset of $ \mathbbR ^n,$ $n\geq 2,$ Ark Math, 9 (1971), 23-37. doi: 10.1007/BF02383635.
[6]	H. G. Feichtinger, Modulation spaces on locally compact abelian groups, Technical Report, University of Vienna, 1983, and in "Wavelets and Their Applications" (eds. M. Krishna, R. Radha and S. Thangavelu), 99-140, Allied Publishers, New Delhi, 2003.
[7]	H. G. Feichtinger, Modulation spaces: Looking back and ahead, Sampl Theory Signal Image Process, 5 (2006), 109-140.
[8]	K. Gröchening, "Foundations of Time-Frequency Analysis," Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001.
[9]	L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math, 104 (1960), 93-139. doi: 10.1007/BF02547187.
[10]	W. Littman, Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc., 69 (1963), 766-770. doi: 10.1090/S0002-9904-1963-11025-3.
[11]	A. Miyachi, F. Nicola, S. Rivetti, A. Taracco and N. Tomita, Estimates for unimodular Fourier multipliers on modulation spaces, Proc. Amer. Math. Soc., 137 (2009), 3869-3883. doi: 10.1090/S0002-9939-09-09968-7.
[12]	J. Sjöstrand, An algebra of pseudodifferential operators, Math. Res. Lett, 1 (1994), 185-192.
[13]	E. M. Stein, "Beijing Lectures In Harmonic Analysis," Annals of Mathematics Studies, 112, Princeton University Press, Princeton, NJ, 1982.
[14]	H. Triebel, "Theory of Function Spaces," Mathematik und ihre anwendugen in Physik und Technik [Mathematics and its Applications in Physics and Technology], 38, Akademische Verlagsgesellchaft Geest & Portig K.-G., Leipzig, 1983. doi: 10.1007/978-3-0346-0416-1.
[15]	J. Toft, Continuity properties for modulation spaces with applications to pseudo-differential calculus. II, Ann Global Anal Geom, 26 (2004), 73-106. doi: 10.1023/B:AGAG.0000023261.94488.f4.
[16]	B. Wang and H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data, J. Differential Equations, 232 (2007), 36-73. doi: 10.1016/j.jde.2006.09.004.
[17]	B. Wang, L. Han and C. Huang, Global well-posedness and scattering for the derivative nonlinear Schrödinger equation with small rough data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2253-2281.
[18]	B. Wang, C. Hao and C. Huo, "Introduction on Nonlinear Developing Equations," Unpublished Lecture Notes, Beijing University, 2009.
[19]	B. Wang, L. Zhao and B. Guo, Isometric decomposition operators function spaces $E_{p,q}^{\lambda}$ and applications to nonlinear evolution equations, J. Func. Anal., 233 (2006), 1-39. doi: 10.1016/j.jfa.2005.06.018.

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