Revista Matemática Iberoamericana


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Volume 28, Issue 1, 2012, pp. 157–178
DOI: 10.4171/RMI/672

Published online: 2012-01-20

Sharp extension theorems and Falconer distance problems for algebraic curves in two dimensional vector spaces over finite fields

Doowon Koh[1] and Chun-Yen Shen[2]

(1) Chungbuk National University, Cheongju, South Korea
(2) National Central University, Jhongli City, Taoyuan County, Taiwan

In this paper we study extension theorems associated with general varieties in two dimensional vector spaces over finite fields. Applying Bezout’s theorem, we obtain the sufficient and necessary conditions on general curves where sharp $L^p$-$L^r$ extension estimates hold. Our main result can be considered as a nice generalization of works by Mockenhaupt and Tao in [17] and Iosevich and Koh in [10]. As an application of our sharp extension estimates, we also study the Falconer distance problems in two dimensions.

Keywords: Finite fields, extension/restriction theorems, Falconer distances, Bezout’s theorem

Koh Doowon, Shen Chun-Yen: Sharp extension theorems and Falconer distance problems for algebraic curves in two dimensional vector spaces over finite fields. Rev. Mat. Iberoamericana 28 (2012), 157-178. doi: 10.4171/RMI/672