A stationary set method for estimating oscillatory integrals

TL;DR

This paper introduces a stationary set method for estimating oscillatory integrals, leading to sharp convergence exponents in Tarry's problems and improved Fourier extension estimates for monomial surfaces.

Contribution

The paper presents a novel stationary set method that advances the estimation of oscillatory integrals and applies it to solve Tarry's problems and enhance Fourier extension estimates.

Findings

Sharp convergence exponents for Tarry's problems in dimension two

Improved Fourier extension estimates for monomial surfaces

Introduction of a new stationary set method

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.