Extension theorems for paraboloids in the finite field setting

Abstract

In this paper we study the Lp − Lr boundedness of the extension operators associated with paraboloids in \({{\mathbb F}_{q}^{d}}\) , where \({\mathbb{F}_{q}}\) is a finite field of q elements. In even dimensions d ≥ 4, we estimate the number of additive quadruples in the subset E of the paraboloids, that is the number of quadruples \({(x,y,z,w) \in E^4}\) with x + y = z+w. As a result, in higher even dimensions, we obtain the sharp range of exponents p for which the extension operator is bounded, independently of q, from Lp to L4 in the case when −1 is a square number in \({\mathbb{F}_{q}}\) . Using the sharp Lp −L4 result, we improve upon the range of exponents r, for which the L2 − Lr estimate holds, obtained by Mockenhaupt and Tao (Duke Math 121:35–74, 2004) in even dimensions d ≥ 4. In addition, assuming that −1 is not a square number in \({\mathbb{F}_{q}}\), we extend their work done in three dimension to specific odd dimensions d ≥ 7. The discrete Fourier analytic machinery and Gauss sum estimates make an important role in the proof.