# American Institute of Mathematical Sciences

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

## Asymptotic estimates for unimodular Fourier multipliers on modulation spaces

 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

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.
Citation: Jiecheng Chen, Dashan Fan, Lijing Sun. Asymptotic estimates for unimodular Fourier multipliers on modulation spaces. Discrete & 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.  Google Scholar [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.  Google Scholar [3] J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction," Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [7] H. G. Feichtinger, Modulation spaces: Looking back and ahead, Sampl Theory Signal Image Process, 5 (2006), 109-140.  Google Scholar [8] K. Gröchening, "Foundations of Time-Frequency Analysis," Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001.  Google Scholar [9] L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math, 104 (1960), 93-139. doi: 10.1007/BF02547187.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] J. Sjöstrand, An algebra of pseudodifferential operators, Math. Res. Lett, 1 (1994), 185-192.  Google Scholar [13] E. M. Stein, "Beijing Lectures In Harmonic Analysis," Annals of Mathematics Studies, 112, Princeton University Press, Princeton, NJ, 1982. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] B. Wang, C. Hao and C. Huo, "Introduction on Nonlinear Developing Equations," Unpublished Lecture Notes, Beijing University, 2009. Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [3] J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction," Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [7] H. G. Feichtinger, Modulation spaces: Looking back and ahead, Sampl Theory Signal Image Process, 5 (2006), 109-140.  Google Scholar [8] K. Gröchening, "Foundations of Time-Frequency Analysis," Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001.  Google Scholar [9] L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math, 104 (1960), 93-139. doi: 10.1007/BF02547187.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] J. Sjöstrand, An algebra of pseudodifferential operators, Math. Res. Lett, 1 (1994), 185-192.  Google Scholar [13] E. M. Stein, "Beijing Lectures In Harmonic Analysis," Annals of Mathematics Studies, 112, Princeton University Press, Princeton, NJ, 1982. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] B. Wang, C. Hao and C. Huo, "Introduction on Nonlinear Developing Equations," Unpublished Lecture Notes, Beijing University, 2009. Google Scholar [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.  Google Scholar
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