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

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Abstract

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.

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  1. Department of Mathematics, Chungbuk National University, Cheongju, Chungbuk, 28644, South Korea

    Doowon Koh

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This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2015R1A1A1A05001374)

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Koh, D. Conjecture and improved extension theorems for paraboloids in the finite field setting. Math. Z. 294, 51–69 (2020). https://doi.org/10.1007/s00209-019-02250-8

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  • DOI: https://doi.org/10.1007/s00209-019-02250-8

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