Averages and maximal averages over Product j-varieties in finite fields

TL;DR

This paper establishes sharp bounds for averaging and maximal averaging operators over Product j-varieties in finite fields, advancing understanding of their harmonic analysis properties.

Contribution

It provides the first sharp $L^p o L^r$ bounds for averaging operators and optimal $L^p$ estimates for maximal operators over Product j-varieties in finite fields.

Findings

Proved sharp $L^p o L^r$ bounds for averaging operators.

Established optimal $L^p$ estimates for maximal averaging operators.

Enhanced understanding of harmonic analysis on finite field varieties.

Abstract

We study both averaging and maximal averaging problems for Product $j$-varieties defined by $\Pi_j=\{x\in \mathbb F_q^d: \prod_{k=1}^d x_k=j\}$ for $j\in \mathbb F_q^*,$ where $\mathbb F_q^d$ denotes a $d$-dimensional vector space over the finite field $\mathbb F_q$ with $q$ elements. We prove the sharp $L^p\to L^r$ boundedness of averaging operators associated to Product $j$-varieties. We also obtain the optimal $L^p$ estimate for a maximal averaging operator related to a family of Product $j$-varieties $\{\Pi_j\}_{j\in \mathbb F_q^*}.$