A Real Analytic Approach to Estimating Oscillatory Integrals

We develop an asymptotic expansion for oscillatory integrals with real analytic phases. We assume the phases satisfy a nondegeneracy condition originally considered by Varchenko, which is related to the Newton polyhedron. Analogous estimates for smooth and $C^k$ phases are also proved. With algebraic techniques such as resolution of singularities, Varchenko was the first to obtain sharp estimates for oscillatory integrals with nondegenerate analytic phases, assuming the Newton distance of the phase is greater than 1. This condition has also been frequently used in modern literature; for example, Greenblatt and later Kamimoto-Nose obtained more general results by also using resolution of singularities. Using only real analytic methods that are very much in the spirit of van der Corput, we develop a full asymptotic expansion for analytic phases satisfying Varchenko\u27s condition, and an asymptotic expansion with finitely many terms for $C^k$ phases under the additional assumption that the Newton polyhedron intersects each coordinate axis. We demonstrate how the exponents in the asymptotic expansion of these integrals can be obtained completely geometrically via the Newton polyhedron. Important techniques include: dyadic decomposition; proving and then using a lower bound similar to that of Lojaciewicz for analytic functions, together with the method of stationary phase to integrate by parts; linear programming to get sharpest estimates (matching Varchenko\u27s); and finally, repeated differentiation of the integral with respect to the oscillatory parameter in order to obtain higher order terms of the expansion