Extension theorems for Hamming varieties over finite fields

Cheong, Daewoong; Koh, Doowon; Pham, Thang

doi:10.1090/proc/15738

Extension theorems for Hamming varieties over finite fields
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by Daewoong Cheong, Doowon Koh and Thang Pham PDF

Proc. Amer. Math. Soc. 150 (2022), 161-170 Request permission

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.

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