The Fourier extension conjecture for the paraboloid

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.