# Restricted averaging operators to cones over finite fields

## TL;DR

This paper establishes optimal L^p to L^r bounds for the restricted averaging operator over cones in finite field vector spaces, specifically in even dimensions greater or equal to 6 with certain subspace conditions.

## Contribution

It proves the sharp boundedness estimates for the operator in even dimensions d≥6 containing a d/2-dimensional subspace, extending previous results from odd dimensions.

## Key findings

- Established optimal L^p to L^r bounds for the operator in even dimensions
- Extended previous results to include even dimensions d≥6 with subspace conditions
- Provided a complete characterization of the operator's boundedness in the specified setting

## Abstract

We investigate the sharp L^p\to L^r estimates for the restricted averaging operator A_C over the cone C of the d-dimensional vector space F_q^d over the finite field F_q with q elements. The restricted averaging operator A_C for the cone C is defined by the relation that A_Cf=f\ast \sigma |_C, where \sigma denotes the normalized surface measure on the cone C, and f is a complex valued function on the space F_q^d with the normalized counting measure dx. In the previous work, the sharp boundedness of A_C was obtained in odd dimensions d\ge 3 but partial results were only given in even dimensions d\ge 4. In this paper we prove the optimal estimates in even dimensions d\ge 6 in the case when the cone C\subset F_q^d contains a d/2 dimensional subspace.