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

Abstract

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.