Fourier Restriction: From Linear Restriction to Multilinear Restriction

Abstract

This dissertation studies the Fourier restriction, which is to find the range of the constants p, q such that the L^q norm on a chosen subset of the Fourier domain is bounded above by the L^p norm in a spacial domain, up to some constant that is independent of the function. We discuss linear restriction, including Hausdorff-Young's inequality, A proof of the restriction estimate on curves, and further discussions on the restriction problem on the sphere and paraboloid via the Stein-Tomas argument. We then discuss bilinear restriction, where the estimate on 2-dimensional case is proved by the reverse square function estimate and the bilinear interaction of transverse wave packets. The result is further used to verify the restriction conjecture on the 2-dimensional paraboloid. We discuss about multi-linear restriction in the final section, focusing on a short proof of a close result of the multilinear restriction estimate from I. Bejenaru.