In this paper we study Lp−Lr estimates of both the extension operator and the averaging operator associated with the algebraic variety S = {x ∈ : Q(x) = 0}, where Q(x) is a non-degenerate quadratic form over the finite field with q elements. We show that the Fourier decay estimate on S is good enough to establish the sharp averaging estimates in odd dimensions. In addition, the Fourier decay estimate enables us to simply extend the sharp L2−L4 conical extension result in , due to Mockenhaupt and Tao, to the L2−L2(d+1)/(d−1) estimate in all odd dimensions d ≥ 3. We also establish a sharp estimate of the mapping properties of the average operators in the case when the variety S in even dimensions d ≥ 4 contains a d/2-dimensional subspace.

In this paper we study the extension problem, the averaging problem, and the generalized Erdős–Falconer distance problem associated with arbitrary homogeneous varieties in three dimensional vector spaces over finite fields. In the case when the varieties do not contain any plane passing through the origin, we obtain the best possible results on the aforementioned three problems. In particular, our result on the extension problem modestly generalizes the result by Mockenhaupt and Tao who studied the particular conical extension problem. In addition, investigating the Fourier decay on homogeneous varieties enables us to give complete mapping properties of averaging operators. Moreover, we improve the size condition on a set such that the cardinality of its distance set is nontrivial.