Sharp L^p-L^r estimates for k-plane transforms in finite fields

TL;DR

This paper establishes optimal L^p-L^r estimates for k-plane transforms over finite fields, utilizing geometric combinatorics to recover critical endpoints and extend results to multilinear transforms.

Contribution

It provides the first comprehensive proof of optimal estimates for all k-plane transforms in finite fields, including endpoint and multilinear cases.

Findings

Recovered critical endpoint estimates using geometric combinatorics

Established optimal L^p-L^r bounds for all k-plane transforms

Derived multilinear transform estimates via Holder's inequality

Abstract

We study mapping properties of finite field k-plane transforms. Using geometric combinatorics, we do an elaborate analysis to recover the critical endpoint estimate. As a consequence, we obtain optimal L^p-L^r estimates for all k-plane transforms in the finite field setting. In addition, applying Holder's inequality to our results, we obtain an estimate for multilinear k-plane transforms.