×

Criterion on \(L^ p\)-boundedness for a class of oscillatory singular integrals with rough kernels. (English) Zbl 0786.42007

The authors extend a theorem of Ricci and Stein concerning \(L^ p\) boundedness of singular oscillatory integral operators of the form \(Tf(x)=\text{p.v.}\int e^{iP(x,y)} K(x- y)f(y)dy\), where \(P\) is a real- valued polynomial in \(\mathbb{R}^ n\times\mathbb{R}^ n\). They assume that the Calderón-Zygmund kernel \(K\) is given by \(K(x)= \Omega(x/| x|)| x|^{-n}\), where \(\Omega\in L^ q(S^{n-1})\) for some \(q>1\) and the mean value of \(\Omega\) over the sphere vanishes. The conclusion is that \(T\) is bounded on \(L^ p(\mathbb{R}^ n)\) if \(1<p<\infty\). The authors also prove a stronger version where \(K(x-y)\) is replaced by \(K(x-y)b(| x-y|)\) and \(b\) is a function of bounded variation.
Reviewer: A.Seeger (Madison)

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)