Abstract
The aim of this paper is to present certain global regularity properties of hyperbolic equations. In particular, it will be determined in what way the global decay of Cauchy data implies the global decay of solutions. For this purpose, global weighted estimates in Sobolev spaces for Fourier integral operators will be reviewed. We will also present elements of the global calculus under minimal decay assumptions on phases and amplitudes.
This work was completed with the aid of UK-Japan Joint Project Grant by The Royal Society and Japan Society for the Promotion of Science.
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References
K. Asada and D. Fujiwara, On some oscillatory integral transformations in L2(ℝn), Japan. J. Math. (N.S.) 4 (1978), 299–361.
P. Boggiato, E. Buzano and L. Rodino, Global Hypoellipticity and Spectral Theory, Akademie Verlag, Berlin, 1996.
A. Boulkhemair, Estimations L2 precisees pour des integrales oscillantes, Comm. Partial Differential Equations 22 (1997), 165–184.
A. P. Calderón and R. Vaillancourt, On the boundedness of pseudo-differential operators, J. Math. Soc. Japan 23 (1971), 374–378.
R. R. Coifman and Y. Meyer, Au-delà des opérateurs pseudo-différentiels, Astérisque 57 (1978).
H. O. Cordes, On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators, J. Funct. Anal. 18 (1975), 115–131.
H. O. Cordes, The Technique of Pseudodifferential Operators, Cambridge University Press 1995.
S. Coriasco, Fourier integral operators in SG classes I: composition theorems and action on SG Sobolev spaces, Rend. Sem. Mat. Univ. Pol. Torino 57 (1999), 249–302.
I. Kamotski and M. Ruzhansky, Regularity properties, representation of solutions and spectral asymptotics of systems with multiplicities, Preprint, arXiv:math.AP/0402203.
H. Kumano-go, A calculus of Fourier integral operators on ℝn and the fundamental solution for an operator of hyperbolic type, Comm. Partial Differential Equations 1 (1976), 1–44.
H. Kumano-go, Pseudo-Differential Operators, MIT Press, 1981.
D. S. Kurtz and R. L. Wheeden, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc. 255 (1979), 343–362.
M. Ruzhansky, Singularities of affine fibrations in the regularity theory of Fourier integral operators, Russian Math. Surveys 55, 99–170 (2000).
M. Ruzhansky and M. Sugimoto, Global L2 estimates for a class of Fourier integral operators with symbols in Besov spaces, Russian Math. Surveys 58 (2003), 201–202.
M. Ruzhansky and M. Sugimoto, Global L2-boundedness theorems for a class of Fourier integral operators, Comm. Partial Differential Equations, to appear.
M. Ruzhansky and M. Sugimoto, A smoothing property of Schrödinger equations in the critical case, Math. Ann., to appear.
M. Ruzhansky and M. Sugimoto, A new proof of global smoothing estimates for dispersive equations, in Advances in Pseudo-Differential Operators, Editors: R. Ashino, P. Boggiatto and M. W. Wong, Birkhäuser, 2004, 65–75.
M. Ruzhansky and M. Sugimoto, Weighted L2 estimates for a class of Fourier integral operators, Preprint.
A. Seeger, C. D. Sogger and E. M. Stein, Regularity properties of Fourier integral operators, Ann. Math. 134 (1991), 231–251.
E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, 1993.
M. Sugimoto, L2-boundedness of pseudo-differential operators satisfying Besov estimates I, J. Math. Soc. Japan 40 (1988), 105–122.
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Ruzhansky, M., Sugimoto, M. (2006). Global Calculus of Fourier Integral Operators, Weighted Estimates, and Applications to Global Analysis of Hyperbolic Equations. In: Boggiatto, P., Rodino, L., Toft, J., Wong, M.W. (eds) Pseudo-Differential Operators and Related Topics. Operator Theory: Advances and Applications, vol 164. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7514-0_5
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