Connections between $\mathcal{S}$-operators and restriction estimates for spheres over finite fields

In this paper, we introduce a new operator, $\mathcal{S}$, which is closely related to the restriction problem for spheres in ${\mathbb{F}}_q^d$, the $d$-dimensional vector space over the finite field ${\mathbb{F}}_q$ with $q$ elements. The $\mathcal{S}$ operator is considered as a specific operator that maps functions on ${\mathbb{F}}_q^d$ to functions on ${\mathbb{F}}_q^{d+1}$. We explore a relationship between the boundedness of the $\mathcal{S}$ operator and the restriction estimate for spheres in ${\mathbb{F}}_q^d$. 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.