# Extension theorems for Hamming varieties over finite fields

## TL;DR

This paper establishes Stein-Tomas extension estimates for Hamming varieties over finite fields, despite poor Fourier decay, advancing understanding of harmonic analysis in finite field settings.

## Contribution

It proves the Stein-Tomas $L^2\to L^r$ extension estimate for Hamming varieties over finite fields, a novel result given their Fourier decay properties.

## Key findings

- Stein-Tomas extension estimate holds for Hamming varieties
- Fourier decay bound is not optimal but extension estimate still valid
- Advances finite field harmonic analysis understanding

## Abstract

We study the finite field extension estimates for Hamming varieties $H_j, j\in \mathbb F_q^*,$ defined by $H_j=\{x\in \mathbb F_q^d: \prod_{k=1}^d x_k=j\},$ where $\mathbb F_q^d$ denotes the $d$-dimensional vector space over a finite field $\mathbb F_q$ with $q$ elements. We show that although the maximal Fourier decay bound on $H_j$ away from the origin is not good, the Stein-Tomas $L^2\to L^r$ extension estimate for $H_j$ holds.