Models of degenerate Fourier integral operators and Radon transforms
type: Item
instance of: scholarly article
title: Models of degenerate Fourier integral operators and Radon transforms
zbMATH Open document ID: 0833.43004
DOI: 10.2307/2118622
author: D. H. Phong
author: Elias M. Stein
published in: Annals of Mathematics. Second Series
publication date: 1996-03-17T00:00:00Z
review text: Let \(S(x,y)\) be a homogeneous polynomial of degree \(n\), and consider the oscillatory integral \(T_\lambda\) defined by \[ T_\lambda(f) (x) = \int^\infty_{-\infty} e^{i\lambda S(x,y)} \Psi(x,y) f(y) dy \] where \(f\) (resp. \(\Psi\)) denotes a smooth compactly supported function on \(\mathbb{R}\) (resp. \(\mathbb{R}^2\)). The operator \(T_\lambda\) is closely related to a class of Radon transforms which we describe as follows: The phase \(S(x,y)\) gives rise to a family \((t,x) \to \gamma_{(t,x)}\) of curves in the plane via \[ \gamma_{(t,x)} = \{(s,y); s = t + S(x,y)\} \] with curves \(\gamma_{(t,x)}\) and suitable densities \(d \mu_{(t,x)}\) defined on them; the associated Radon transform \(R\) is given by \(R(f) (t,x) = \int_{\gamma_{(t,x)}} fd\mu_{(t,x)}.\) In this paper the authors give necessary and sufficient conditions for optimal \(L^2\) estimates of the operator \(T_\lambda\). Next, by viewing \(R\) as a pseudo-differential operator whose symbol is the operator-valued function \(\lambda \to T_\lambda\) they deduce the \(L^p - L^q\) boundedness and Sobolev regularity of the transform \(R\).
Mathematics Subject Classification ID: 43A32
Mathematics Subject Classification ID: 35S30
zbMATH DE Number: 750886
zbMATH Keywords: oscillatory integral
zbMATH Keywords: Radon transforms
zbMATH Keywords: pseudo-differential operator
zbMATH Keywords: Sobolev regularity
reviewed by: Khalifa Trimèche
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2118622
OpenAlex ID: W2313790782
description: scientific article; zbMATH DE number 750886