Averaging operators over nondegenerate quadratic surfaces in finite fields

We study mapping properties of the averaging operator related to the variety $V=x\in {\mathbb{F}}_q^d:Q(x)=0,$ where $Q(x)$ is a nondegenerate quadratic polynomial over a finite field ${\mathbb{F}}_q$ with $q$ elements. This paper is devoted to eliminating the logarithmic bound appearing in the paper of Koh and Shen. As a consequence, we settle down the averaging problems over the quadratic surfaces $V$ in the case when the dimensions $d\ge 4$ are even and $V$ contains a $d\mathrm{/}2$-dimensional subspace.