The Fourier extension conjecture for the paraboloid

TL;DR

This paper proves the Fourier extension conjecture for the paraboloid in all dimensions greater than 2 by employing a novel decomposition and bilinear inequality approach involving oscillatory integrals and stationary phase estimates.

Contribution

It introduces a new proof of the Fourier extension conjecture for paraboloids in higher dimensions using a decomposition of Alpert projections and bilinear inequalities with oscillatory integral techniques.

Findings

Established the Fourier extension conjecture for paraboloids in all dimensions > 2.

Developed a novel decomposition of smooth Alpert projections for localization.

Converted exponential sums into oscillatory integrals controlled by stationary phase.

Abstract

We give a proof of Fourier extension conjecture on the paraboloid in all dimensions bigger than 2 that begins with a decomposition suggested in Sawyer [Saw8] of writing a smooth Alpert projection as a sum of pieces whose Fourier extensions are localized. This is then used to establish a local inequality that is well known to be equivalent to the Fourier extension conjecture, and is accomplished by using a variant of the bilinear equivalence of the Fourier extension conjecture given by Tao, Vargas and Vega in [TaVaVe]. A key aspect of our proof is that the bilinear inequality, when taken over smooth Alpert projections, only requires an averaging over grids of functions mollified by discrete multipliers, which converts a difficult exponential sum into an oscillatory integral with periodic amplitude. After extracting Dirichlet kernels in yet another averaging over lattices, this is then controlled using a stationary phase estimate with periodic amplitude, and altogether we then obtain the desired localization on the Fourier side.