Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T07:10:35.777Z Has data issue: false hasContentIssue false
  • English
  • Français

An Open Problem in Complex Analytic Geometry Arising inHarmonic Analysis

Published online by Cambridge University Press:  28 January 2013

M. Ruzhansky
Get access

Abstract

In this paper, an open problem in the multidimensional complex analysis is presented thatarises in the harmonic analysis related to the investigation of the regularity propertiesof Fourier integral operators and in the regularity theory for hyperbolic partialdifferential equations. The problem is discussed in a self-contained elementary way andsome results towards its resolution are presented. A conjecture concerning the structureof appearing affine fibrations is formulated.

Keywords

complex analytic geometryfibrationsLagranigian manifoldsFourier integral operators
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 1: Harmonic analysis , 2013 , pp. 230 - 236
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J. J. Duistermaat, Fourier integral operators. Birkhäuser, Boston, 1996.
S. Łojasiewicz. Introduction to complex analytic geometry. Birkhäuser, Basel, 1991.
Nicola, F.. Boundedness of Fourier integral operators on Fourier Lebesgue spaces and affine fibrations. Studia Math., 198 (2010), 207–219. CrossRefGoogle Scholar
Remmert, R.. Holomorphe und meromorphe Abbildungen komplexer Räume. Math. Ann. 133 (1957), 328–370. CrossRefGoogle Scholar
Ruzhansky, M.. Analytic Fourier integral operators, Monge–Ampere equation and holomorphic factorization. Arch. Math., 72 (1999), 68–76. CrossRefGoogle Scholar
Ruzhansky, M.. On singularities of affine fibrations of certain types. Russian Math. Surveys, 55 (2000), 353–354. CrossRefGoogle Scholar
Ruzhansky, M.. Singularities of affine fibrations in the regularity theory of Fourier integral operators. Russian Math. Surveys, 55 (2000), 93–161. CrossRefGoogle Scholar
M. Ruzhansky. Regularity theory of Fourier integral operators with complex phases and singularities of affine fibrations. CWI Tracts , vol. 131, 2001.
Ruzhansky, M.. On the failure of the factorization condition for non-degenerate Fourier integral operators. Proc. Amer. Math. Soc., 130 (2002), 1371–1376. CrossRefGoogle Scholar
Seeger, A., Sogge, C. D., Stein, E. .M.. Regularity properties of Fourier integral operators. Ann. of Math. 134 (1991), 231–251. CrossRefGoogle Scholar