On a stopping process for oscillatory integrals
The authors obtain new \(L^ 2\) estimates about oscillatory integral operators of the form \[ T_ \lambda f(x)= \int^ \infty_{-\infty} e^{i\lambda S(x,y)}\chi(x,y) f(y) dy, \] where \(f\in L^ 2(\mathbb{R})\) and \(\chi\in C^ \infty_ 0\). One would like to determine sharp bounds for the operator norm \(\| T_ \lambda\|\) as \(\lambda\to\infty\). Results on \(T_ \lambda\) can be applied to prove smoothing estimates for generalized Radon transforms of the form \[ Rg(x_ 1, x_ 2)= \int f(y_ 1, x_ 2- S(x_ 1, y_ 1)) \chi(x_ 1, y_ 1) dy_ 1 \] and related operators in \(\mathbb{R}^ n\). It is assumed that the phase functions satisfy the two-sided finite type condition \(\partial_ x \partial^{n-1}_ y S(x,y)\neq 0\), \(\partial_ y \partial^{k-1}_ x S(x,y)\neq 0\), where \(k\leq n\). It was previously known that \(\| T_ \lambda\|\leq C_ \varepsilon(\lambda^{- a(k,n)+\varepsilon})\), where \(a(k,n)= (k+ n-4)[2(kn- n- k)]^{-1}\) (in particular \(a(n,n)= 1/n\)) and \(\varepsilon> 0\). Here \(C_ \varepsilon\) blows up rather badly as \(\varepsilon\to 0\) [see the reviewers article in Duke Math. J. 71, No. 3, 685-745 (1993)]. The main result in the present paper states that for \(k= n\) one has the estimate \[ \| T_ \lambda\|\leq C\lambda^{-1/n}[\log(2+ |\lambda|)]^{1/n}. \] This is a significant improvement and the proof relies on a very elegant stopping time argument. The authors have announced [Math. Res. Letters 1, 27-34 (1994)] that the logarithmic factor can be dropped if the phase functions are real analytic. The results are applied to obtain estimates for Radon transforms associated to families of curves given by \(y_ i= x_ i+ S_ i(x_ 1,y_ 1)\), \(i= 2,\dots,n\).
scientific article; zbMATH DE number 579777
scientific article