Sharp estimates for trilinear oscillatory integrals and an algorithm of two-dimensional resolution of singularities

We obtain sharp estimates for certain trilinear oscillatory integrals, extending Phong and Stein’s seminal result to a trilinear setting. This result partially answers a question raised by Christ, Li, Tao and Thiele, concerning sharp estimates for certain multilinear oscillatory integrals. The method in this paper relies on a self-contained algorithm of resolution of singularities in R2, which may be of independent interest.