A stationary set method for estimating oscillatory integrals

Abstract

We propose a new method of estimating oscillatory integrals, which we call a stationary set method. We use it to obtain the sharp convergence exponents of Tarry's problems in dimension two for every degree $k\ge 2$. As a consequence, we obtain sharp Fourier extension estimates for a family of monomial surfaces.

Added an argument showing sharpness of the main result in an average sense; shared to us by Wright