Fourier Restriction: From Linear Restriction to Multilinear Restriction

TL;DR

This dissertation explores the Fourier restriction problem, progressing from linear to multilinear restriction estimates, including proofs on curves, spheres, paraboloids, and verification of the restriction conjecture in specific cases.

Contribution

It advances the understanding of Fourier restriction by establishing new multilinear restriction estimates and providing simplified proofs for key cases.

Findings

Proved restriction estimates on curves and surfaces.

Verified the restriction conjecture on 2D paraboloid.

Presented a concise proof of multilinear restriction estimates.

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.