Conjecture and improved extension theorems for paraboloids in the finite field setting

  • Doowon KohEmail author


We study the extension estimates for paraboloids in d-dimensional vector spaces over finite fields \(\mathbb F_q\) with q elements. We use the connection between \(L^2\) based restriction estimates and \(L^p\rightarrow L^r\) extension estimates for paraboloids. As a consequence, we improve the \(L^2\rightarrow L^r\) extension results obtained by Lewko and Lewko (Proc Am Math Soc 140:2013–2028, 2012) in even dimensions \(d\ge 6\) and odd dimensions \(d=4\ell +3\) for \(\ell \in \mathbb N.\) Our results extend the consequences for 3-D paraboloids due to Lewko (Adv Math 270(1):457–479, 2015) to higher dimensions. We also clarifies conjectures on finite field extension problems for paraboloids.


Restriction theorem Extension theorem Paraboloid Finite field 

Mathematics Subject Classification




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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsChungbuk National UniversityCheongjuSouth Korea

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