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

Published online by Cambridge University Press:  20 November 2018

Doowon Koh  and
Chun-Yen Shen
Doowon Koh
Affiliation:
Department of Mathematics, Chungbuk National University, Cheongju city, Chungbuk-Do 361-736, Korea email: koh131@chungbuk.ac.kr
Chun-Yen Shen
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1 email: shenc@umail.iu.edu
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Abstract

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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

42B0511T2452C17extension problemsaveraging operatorfinite fieldsErdős–Falconer distance problemshomogeneous polynomials
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 5 , 01 October 2012 , pp. 1036 - 1057
Copyright
Copyright © Canadian Mathematical Society 2012

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