Sharp estimates for trilinear oscillatory integrals and an algorithm of two-dimensional resolution of singularities (Q520711)

Summary: We obtain sharp estimates for certain trilinear oscillatory integrals, extending \textit{D. H. Phong} and \textit{E. M. Stein}'s seminal result to a trilinear setting [Acta Math. 179, No. 1, 105--152 (1997; Zbl 0896.35147)]. This result partially answers a question raised by \textit{M. Christ} et al. [Duke Math. J. 130, No. 2, 321--351 (2005; Zbl 1119.42005)] concerning sharp estimates for certain multilinear oscillatory integrals. The method in this paper relies on a self-contained algorithm of resolution of singularities in \(\mathbb{R}^2\), which may be of independent interest.