A stationary set method for estimating oscillatory integrals

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.

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