Restriction operators acting on radial functions on vector spaces over finite fields

TL;DR

This paper investigates restriction estimates for radial functions on algebraic varieties over finite fields, establishing results for odd dimensions and certain even-dimensional cases with few intersection points.

Contribution

It proves the conjectured restriction estimates for radial functions on varieties in odd dimensions and certain even-dimensional cases with limited intersection points.

Findings

Restriction estimates hold for odd-dimensional vector spaces.

In even dimensions, similar estimates are valid if varieties intersect the zero sphere minimally.

Results advance understanding of restriction phenomena in finite field harmonic analysis.

Abstract

We stduy $L^p-L^r$ restriction estimates for algebraic varieties $V$ in the case when restriction operators act on radial functions in the finite field setting. We show that if the varieties $V$ lie in odd dimensional vector spaces over finite fields, then the conjectured restriction estimates are possible for all radial test functions. In addition, it is proved that if the varieties $V$ in even dimensions have few intersection points with the sphere of zero radius, the same conclusion as in odd dimensional case can be also obtained.