Connections between S-operators and restriction estimates for spheres over finite fields

Abstract

In this paper, we introduce a new operator, $S$, which is closely related to the restriction problem for spheres in $F_q^d$, the d-dimensional vector space over the finite field $F_q$ with q elements. The $S$ operator is considered as a specific operator that maps functions on $F_q^d$ to functions on $F_q^{d+1}$. We explore a relationship between the boundedness of the $S$ operator and the restriction estimate for spheres in $F_q^d$. Consequently, using this relationship, we prove that the $L^2$ restriction conjectures for spheres hold in all dimensions when the test functions are restricted to homogeneous functions of degree zero.