Mapping properties of the $S$-operator

TL;DR

This paper investigates the boundedness of the $S$-operator in finite field restriction problems, providing necessary and sufficient conditions and analyzing special cases like radial functions.

Contribution

It establishes a complete characterization of the $S$-operator's boundedness and explores sharp results for radial test functions.

Findings

Derived necessary and sufficient conditions for $S$-operator boundedness

Identified sharp bounds for radial functions

Enhanced understanding of restriction phenomena over finite fields

Abstract

In this paper, we study the $\ell^p\to \ell^r$ estimates for the $S$-operator arising in restriction problems for spheres over finite fields. We establish a necessary and sufficient condition for the boundedness of the $S$-operator. Furthermore, we investigate this problem under certain restrictions on test functions. In particular, we address the sharp results when test functions are restricted to radial functions.