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# Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields

Published:2011-12-23
Printed: Oct 2012
• Doowon Koh,
Department of Mathematics, Chungbuk National University, Cheongju city, Chungbuk-Do 361-736, Korea
• Chun-Yen Shen,
Department of Mathematics and Statistics, McMaster University, Hamilton, L8S 4K1 Canada
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## 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.
 Keywords: extension problems, averaging operator, finite fields, Erdős-Falconer distance problems, homogeneous polynomial
 MSC Classifications: 42B05 - Fourier series and coefficients 11T24 - Other character sums and Gauss sums 52C17 - Packing and covering in $n$ dimensions [See also 05B40, 11H31]