Degenerate Fourier integral operators in the plane
type: Item
instance of: scholarly article
title: Degenerate Fourier integral operators in the plane
zbMATH Open document ID: 0806.35191
DOI: 10.1215/S0012-7094-93-07127-X
published in: Duke Mathematical Journal
publication date: 1995-02-13T00:00:00Z
review text: Soient \(X\) et \(Y\) deux ouverts bornés de \(\mathbb{R}^ 2\), \({\mathcal M}\) une hypersurface dans \(X \times Y\) et \(N^*{\mathcal M}\) son fibré normal. On pose \(\Lambda = N^* {\mathcal M} \backslash 0\) et on fait l'hypothèse que \(\Lambda \subset (T^* X \backslash 0) \times (T^* Y \backslash 0)\). On considère \({\mathcal M}_ x = \{y \in Y; (x,y) \in {\mathcal M}\}\) et \(\psi\in C^ \infty_ 0 (X \times Y)\). Si \(d \sigma_ n\) est une densité régulière sur \({\mathcal M}_ x\), dépendant régulièrement de \(x\) on définit l'opérateur \[ Rf(x) = \int_{{\mathcal M}_ x} \psi (x,y) f(y)d \sigma_ x(y) \] qui est un FIO de classe \(I^{-1/2} (X,Y, \Lambda)\). Si \(\Lambda\) est localement le graph d'une transformation canonique (donc si les projections \(\pi_ R : \Lambda \to T^* Y\) et \(\pi_ L : \Lambda \to T^* X\) sont localement des difféomorphismes), l'opérateur \(R\) a des propriétés de continuité \(L^ p \to L^ q\) et \(L^ p \to L^ p_ \alpha\) bien connues. L'auteur prouve de telles propriétés dans le cas où \(\pi_ L\), \(\pi_ R\) sont singulières.
Mathematics Subject Classification ID: 35S30
zbMATH DE Number: 558984
zbMATH Keywords: continuity properties
author: Andreas Seeger
reviewed by: Viorel Iftimie
MaRDI profile type: Publication
cites work: 𝐿^{𝑝} boundedness of Fourier integral operators
cites work: A Class of Bounded Pseudo-Differential Operators
cites work: Weak type (1,1) bounds for rough operators
cites work: Q2548398
cites work: \(H^p\) spaces of several variables
cites work: Estimates for the \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \partial \limits^ - _b $\end{document} complex and analysis on the heisenberg group
cites work: Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms
cites work: Fourier integral operators. I
cites work: Q3671563
cites work: Q3671564
cites work: Q3736112
cites work: Q3752751
cites work: Intertwining operators for semisimple groups
cites work: Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle
cites work: Local Smoothing of Fourier Integral Operators and Carleson-Sjolin Estimates
cites work: Averages Over Convex Hypersurfaces
cites work: Convolution Estimates for Some Measures on Curves
cites work: Q4847895
cites work: Hilbert integrals, singular integrals, and Radon transforms. I
cites work: Q3984203
cites work: Oscillatory integrals with polynomial phases
cites work: On analytic families of operators
cites work: Regularity properties of Fourier integral operators
cites work: Averages over hypersurfaces. II
cites work: Averages over hypersurfaces. Smoothness of generalized Radon transforms
cites work: Problems in harmonic analysis related to curvature
cites work: Q3336199
description: scientific article; zbMATH DE number 558984