Extension theorems for Hamming varieties over finite fields

Abstract

We study the finite field extension estimates for Hamming varieties $H_j,j\in {\mathbb{F}}_q^{\ast },$ 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.