Restriction estimates for the flat disks over finite fields

TL;DR

This paper advances the understanding of restriction estimates for flat disks over finite fields, providing sharp $L^2 o L^r$ bounds for all dimensions $n \\geq 6$, surpassing previous results limited to specific dimensions.

Contribution

It extends restriction estimates for flat disks over finite fields to all dimensions $n \\geq 6$, including sharp bounds and analysis of the Fourier transform of the surface measure.

Findings

Established sharp $L^2 o L^r$ restriction estimates for all dimensions $n \\geq 6$.

Derived explicit Fourier transform formulas for the surface measure on the flat disk.

Improved restriction estimates beyond $L^2$ using recent paraboloid results.

Abstract

In this paper we study the restriction estimate for the flat disk over finite fields. Mockenhaupt and Tao initially studied this problem but their results were addressed only for dimensions $n=4,6$. We improve and extend their results to all dimensions $n\geq 6$. More precisely, we obtain the sharp $L^2 \to L^r$ estimates, which cannot be proven by applying the usual Stein-Tomas argument over a finite field even with the optimal Fourier decay estimate on the flat disk. One of main ingredients is to discover and analyze an explicit form of the Fourier transform of the surface measure on the flat disk. In addition, based on the recent results on the restriction estimates for the paraboloids, we address improved restriction estimates for the flat disk beyond the $L^2$ restriction estimates.