  In this paper we study $L^p-L^r$ estimates of both extension operators and
averaging operators associated with the algebraic variety $S=\{x\in {\mathbb
F}_q^d: Q(x)=0\}$ where $Q(x)$ is a nondegenerate quadratic form over the
finite field ${\mathbb F}_q.$ In the case when $d\geq 3$ is odd and the surface
$S$ contains a $(d-1)/2$-dimensional subspace, we obtain the exponent $r$ where
the $L^2-L^r$ extension estimate is sharp. In particular, we give the complete
solution to the extension problems related to specific surfaces $S$ in three
dimension. In even dimensions $d\geq 2$, we also investigates the sharp
$L^2-L^r$ extension estimate. Such results are of the generalized version and
extension to higher dimensions for the conical extension problems which
Mochenhaupt and Tao studied in three dimensions. The boundedness of averaging
operators over the surface $S$ is also studied. In odd dimensions $d\geq 3$ we
completely solve the problems for $L^p-L^r$ estimates of averaging operators
related to the surface $S.$ On the other hand, in the case when $d\geq 2$ is
even and $S$ contains a $d/2$-dimensional subspace, using our optimal $L^2-L^r$
results for extension theorems we, except for endpoints, have the sharp
$L^p-L^r$ estimates of the averaging operator over the surface $S$ in even
dimensions.
