Global Calculus of Fourier Integral Operators, Weighted Estimates, and Applications to Global Analysis of Hyperbolic Equations
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.
KeywordsFourier integral operators hyperbolic equations global estimates
Unable to display preview. Download preview PDF.
- P. Boggiato, E. Buzano and L. Rodino, Global Hypoellipticity and Spectral Theory, Akademie Verlag, Berlin, 1996.Google Scholar
- R. R. Coifman and Y. Meyer, Au-delà des opérateurs pseudo-différentiels, Astérisque 57 (1978).Google Scholar
- H. O. Cordes, The Technique of Pseudodifferential Operators, Cambridge University Press 1995.Google Scholar
- I. Kamotski and M. Ruzhansky, Regularity properties, representation of solutions and spectral asymptotics of systems with multiplicities, Preprint, arXiv:math.AP/0402203.Google Scholar
- H. Kumano-go, Pseudo-Differential Operators, MIT Press, 1981.Google Scholar
- M. Ruzhansky and M. Sugimoto, Global L2-boundedness theorems for a class of Fourier integral operators, Comm. Partial Differential Equations, to appear.Google Scholar
- M. Ruzhansky and M. Sugimoto, A smoothing property of Schrödinger equations in the critical case, Math. Ann., to appear.Google Scholar
- 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.Google Scholar
- M. Ruzhansky and M. Sugimoto, Weighted L2 estimates for a class of Fourier integral operators, Preprint.Google Scholar
- A. Seeger, C. D. Sogger and E. M. Stein, Regularity properties of Fourier integral operators, Ann. Math. 134 (1991), 231–251.Google Scholar
- E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, 1993.Google Scholar