A discussion of two new proofs of Fefferman's Fourier extension theorem in the plane

TL;DR

This paper presents two novel proofs of Fefferman's Fourier extension theorem in the plane, utilizing wavelet decompositions and advanced decoupling techniques, with extensions to higher dimensions.

Contribution

It introduces two new proofs of a classical Fourier extension theorem, employing wavelet-based decoupling methods and extending these techniques to higher dimensions.

Findings

First proof combines Haar wavelets with Fefferman's decoupling.

Second proof uses smooth Alpert wavelets and a new decoupling approach.

Extensions to higher dimensions are discussed.

Abstract

The Fourier extension conjecture of E. Stein was proved in the plane in 1970 by C. Fefferman, see also Zygmund and Carleson and Sj\"olin, with simplifications given by other authors later on, in particular by L. H\"ormander and T. Tao. We discuss yet two more proofs of this classical theorem on the parabola. The first proof uses C. Fefferman's decoupling together with a decomposition into Haar wavelets. This sets the stage for the second proof in Rios and Sawyer, that uses smooth Alpert wavelets and a new decoupling method, which exploits averaging smooth Alpert projections over grids, the extraction of Dirichlet kernels, and periodic stationary phase, all of which was extended to higher dimensions in arXiv:2512.24990v7.