The complete \((L^p,L^p)\) mapping properties of some oscillatory integrals in several dimensions (Q2753433)

The main result is : Assume \(\frac{a_1}{b_1}=\frac{a_2}{b_2}\) and \(a_j, b_j\geq 1\) for \(j=1,2\). Let \(0\leq r<2\) and \(\varphi (x,y)=|x-y|^{-r+i\tau}\) \((-\infty<\tau<\infty)\). Then, for the oscillatory integral operator \(K\) defined by NEWLINE\[NEWLINEKf(x)=\int_{(\mathbb R_+)^2}e^{i(x_1^{a_1}y_1^{b_1} +x_2^{a_2}y_2^{b_2})}\varphi(x,y)f(y) dy,NEWLINE\]NEWLINE there exists \(C>0\) such that NEWLINE\[NEWLINE\|Kf\|_p\leq C\|f\|_p,NEWLINE\]NEWLINE if and only if \(\frac{a_1+b_1}{a_1+b_1 r/2}\leq p \leq \frac{a_1+b_1}{a_1(1-r/2))}\). A sufficiency result is obtained also in the case NEWLINE\[NEWLINE|\nabla^j\varphi(x,y)|\leq C_j|x-y|^{-j-r}, \quad j=0,1,2,3.NEWLINE\]NEWLINE The authors treat also singular cases like as \(\varphi(x,y)=|x-y|^{-2}\). These are extensions of the one dimensional results [\textit{Y. Pan, G. Sampson} and \textit{P. Szeptycki}, Stud. Math. 122, No. 3, 201-224 (1997; Zbl 0876.42008)].