Averaging operators over homogeneous varieties over finite fields

TL;DR

This paper investigates the behavior of averaging operators over homogeneous varieties in finite fields, providing optimal results for two-dimensional cases and partial insights for higher dimensions, based on algebraic and geometric analysis.

Contribution

It offers new optimal results for averaging over two-dimensional homogeneous varieties and explores properties of higher-dimensional varieties in finite fields.

Findings

Optimal averaging results for 2D varieties

Varieties are not contained in hyperplanes

Higher-dimensional results are partial

Abstract

In this paper we study the mapping properties of the averaging operator over a variety given by a system of homogeneous equations over a finite field. We obtain optimal results on the averaging problems over two dimensional varieties whose elements are common solutions of diagonal homogeneous equations. The proof is based on a careful study of algebraic and geometric properties of such varieties. In particular, we show that they are not contained in any hyperplane and are complete intersections. We also address partial results on averaging problems over arbitrary dimensional homogeneous varieties which are smooth away from the origin.