An abstract $L^2$ Fourier restriction theorem

TL;DR

This paper generalizes an $L^2$ Fourier restriction theorem to locally compact abelian groups, leading to new restriction estimates for varieties over local fields and rings of integers, expanding the scope of Fourier analysis techniques.

Contribution

It introduces an abstract framework for $L^2$ Fourier restriction theorems on locally compact abelian groups, enabling new estimates for varieties over local fields and modular rings.

Findings

Established new restriction estimates for varieties in local fields.

Extended Fourier restriction techniques to algebraic structures over rings of integers.

Provided a unified approach to Fourier restriction in abstract harmonic analysis.

Abstract

An $L^2$ Fourier restriction argument of Bak and Seeger is abstracted to the setting of locally compact abelian groups. This is used to prove new restriction estimates for varieties lying in modules over local fields or rings of integers $\mathbb{Z}/N\mathbb{Z}$.