In this talk, I will introduce the Hormander-type oscillatory integral operator. A classical result of Hormander and Stein gives the (L^\infty, L^p) estimate when p=2(n+1)/(n-1). By adding some additional condition, we prove a (L^\infty, L^q) estimate for some q<2(n+1)/(n-1). The new ingredient is a sublevel set estimate for real analytic functions.
