Singular Radon transforms and oscillatory integrals
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 \(TT^*\).
scientific article; zbMATH DE number 3799