Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the failure of the factorization condition for non-degenerate Fourier integral operators

Author: Michael Ruzhansky
Journal: Proc. Amer. Math. Soc. 130 (2002), 1371-1376
MSC (1991): Primary 35A20, 35S30, 58G15, 32D20
Published electronically: October 12, 2001
MathSciNet review: 1879959
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we give examples of polynomial phase functions for which the factorization condition of Seeger, Sogge and Stein (Ann. Math. 134 (1991)) fails. The corresponding Fourier integral operators turn out to be still continuous in $L^p$. We also give examples of the failure of the factorization condition for translation invariant operators. In this setting the frequency space must be at least 5-dimensional, which shows that the examples are optimal. We briefly discuss the stationary phase method for the corresponding operators.

References [Enhancements On Off] (What's this?)

  • 1. M. Beals, $L^p$ boundedness of Fourier integrals, Mem. Amer. Math. Soc., 264 (1982). MR 84m:42026
  • 2. J.J. Duistermaat, Fourier integral operators, Birkhäuser, Boston, 1996. MR 96m:58245
  • 3. M. Ruzhansky, Analytic Fourier integral operators, Monge-Ampère equation and holomorphic factorization, Arch. Mat., 72 (1999), 68-76. MR 99k:58178
  • 4. M. Ruzhansky, On the sharpness of Seeger-Sogge-Stein orders, Hokkaido Math. J., 28 (1999), 357-362. MR 2000e:35247
  • 5. M. Ruzhansky, Singularities of affine fibrations in the regularity theory of Fourier integral operators, Russian Math. Surveys, 55 (2000), 93-161. CMP 2000:11
  • 6. M. Ruzhansky, Regularity theory of Fourier integral operators with complex phases and singularities of affine fibrations, CWI Tract, 131, Amsterdam, 2001.
  • 7. A. Seeger, C.D. Sogge and E.M. Stein, Regularity properties of Fourier integral operators, Ann.of Math., 134 (1991), 231-251. MR 92g:35252
  • 8. C.D. Sogge, Fourier integrals in classical analysis, Cambridge Univ. Press, Cambridge, 1993. MR 94c:35178
  • 9. E.M. Stein, Harmonic analysis, Princeton Univ. Press, Princeton, 1993. MR 95c:42002

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 35A20, 35S30, 58G15, 32D20

Retrieve articles in all journals with MSC (1991): 35A20, 35S30, 58G15, 32D20

Additional Information

Michael Ruzhansky
Affiliation: Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2BZ, United Kingdom

Received by editor(s): June 22, 1999
Received by editor(s) in revised form: October 30, 2000
Published electronically: October 12, 2001
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society