Mapping properties for oscillatory integrals in \(d\)-dimensions (Q2371840)

The author considers the oscillatory integrals given by \[ K(f)(x)=\int_{(\mathbb R_+)^d} e^{ig(x,y)}\varphi(x,y) f(y)\,dy,\quad x\in (\mathbb R_+)^d, \] where \(g(x,y)\) is of the form \(g(x,y)=\sum_{j=1}^{d}x_j^{a_j}y_j^{b_j}\) \((a_j, b_j\geq1)\), and \(\varphi\) satisfies \(| \partial _x^\alpha\partial_y^\beta\varphi(x,y)| \leq C_{\alpha,\beta}| x-y| ^{-| \alpha| -| \beta| }\), \(\alpha,\beta\in\mathbb N_0^d\). The author states: If \(r=a_1/b_1=\cdots=a_d/b_d\) and \(p=1+1/r\), there exists \(C>0\) such that \(\| Kf\| _{L^p(\mathbb R^d)}\leq C\| f\| _{L^p(\mathbb R^d)}\). Some more general kernels are treated. These results extend the case \(d=1\) [\textit{Y.~Pan, G.~Sampson} and \textit{P.~Szeptycki}, Stud. Math. 122, No. 3, 201--224 (1997; Zbl 0876.42008)], and the case \(d=2\) [\textit{G.~Sampson} and \textit{P.~Szeptycki}, Can. J. Math. 53, No. 5, 1031--1056 (2001; Zbl 0985.42009)].