Harmonie analysis related to homogeneous varieties in three dimensional vector spaces over finite fields
Journal
Canadian Journal of Mathematics
Journal Volume
64
Journal Issue
5
Pages
1036-1057
Date Issued
2012
Author(s)
Koh, D.
Abstract
Abstract 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.
Type
journal article