$L^p$ type mapping estimates for oscillatory integrals in higher dimensions
Volume 172 / 2006
Studia Mathematica 172 (2006), 101-123
MSC: 42B20, 46B70, 47G10.
DOI: 10.4064/sm172-2-1
Abstract
We show in two dimensions that if $Kf=\int_{{\mathbb R}_+^2} k(x,y) f(y) \,dy$, $k(x,y)={e^{i x^a\cdot y^b}}/{|x-y|^{\eta}},$ $p={4}/{(2+\eta)}$, $a\ge b\ge \bar{1}=(1,1)$, $v_p(y)=y^{({p}/{p^{\prime}})(\bar{1}-b/a)}$, then $\|Kf\|_p\le C\|f\|_{p,v_p}$ if $\eta+\alpha_1+\alpha_2<2,$ $\alpha_j=1-{b_j}/{a_j}$, $j=1,2$. Our methods apply in all dimensions and also for more general kernels.
