Singular Radon transforms and oscillatory integrals
type: Item
instance of: scholarly article
title: Singular Radon transforms and oscillatory integrals
zbMATH Open document ID: 0738.42011
DOI: 10.1215/S0012-7094-89-05816-X
author: D. H. Phong
author: Elias M. Stein
published in: Duke Mathematical Journal
publication date: 1992-06-25T00:00:00Z
review text: The authors are trying to provide an alternative reduction of singular Radon transforms to certain oscillatory integrals. A singular Radon transform is defined by \[ R(f)(x,t)=\int_{\mathbb{R}^{n+1}}\delta(s-t- S(x,y,t))K(x,t;x-y)f(y,s)dy ds. \] Here \(x,y\in \mathbb{R}^ n\), \(s,t\in \mathbb{R}\), \(\delta\) is the Dirac measure, \(\det{\partial^ 2S \over \partial x_ i\partial x_ j}\neq 0\) and \(K(\cdot,\cdot;x)\) is a standard singular integral kernel in \(x\). The main purpose of this paper is to show how to analyse \(R\). The authors begin by introducing a simple model of \(R\), and show a method for the study of a general \(R\). In the paper, the boundedness of \(R\) is shown by decomposing \(R\) as \(R=A+T\) so that \(A\) can be treated as a standard singular integral and the boundedness of \(T\) reduces to the weak boundedness of \(TT^*\).
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 44A12
zbMATH DE Number: 3799
zbMATH Keywords: Calderón-Zygmund singular integral
zbMATH Keywords: pseudo-differential operator
zbMATH Keywords: singular Radon transforms
zbMATH Keywords: oscillatory integrals
reviewed by: Takafumi Murai
MaRDI profile type: Publication
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full work available at URL: s0012-7094-89-05816-x
OpenAlex ID: W2041803422
description: scientific article; zbMATH DE number 3799