Harmonic analysis related to homogeneous varieties in three dimensional vector space over finite fields

In this paper we study extension problems, averaging problems, and generalized Erdos-Falconer distance problems associated with arbitrary homogeneous varieties in three dimensional vector space over finite fields. In the case when homogeneous varieties in three dimension do not contain any plane passing through the origin, we obtain the general best possible results on aforementioned three problems. In particular, our results on extension problems recover and generalize the work due to Mockenhaupt and Tao who completed the particular conical extension problems in three dimension. Investigating the Fourier decay on homogeneous varieties, we give the complete mapping properties of averaging operators over homogeneous varieties in three dimension. In addition, studying the generalized Erd\H os-Falconer distance problems related to homogeneous varieties in three dimensions, we improve the cardinality condition on sets where the size of distance sets is nontrivial. Finally, we address a question of our problems for homogeneous varieties in higher odd dimensions.