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 $L^{\infty }\to L^p$ Fourier extension estimates for a family of monomial surfaces.