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

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.