The Complete $(L^p,L^p)$ Mapping Properties of Some Oscillatory Integrals in Several Dimensions

摘要

We prove that the operators $int_{mathbb{R}_+^2} e^{ix^a cdoty^b} varphi (x,y) f(y), dy$ map $L^p(mathbb{R}^2)$ into itselffor $p in J =igl[frac{a_l+b_l}{a_l+(frac{b_l r}{2})},frac{a_l+b_l}{a_l(1-frac{r}{2})}igr]$ if $a_l,b_lge 1$ and $varphi(x,y)=|x-y|^{-r}$,$0le r <2$, the result is sharp. Generalizations to dimensions $d>2$are indicated.