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On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps
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 |
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. |
show all references
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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