Extension theorems for paraboloids in the finite field setting (Q5962181)

Let \(\mathbb F_q\) be a finite field of \(q\) elements. The paper under review deals with the \(L^p - L^r\) boundedness of the extension operators associated with paraboloids in \(\mathbb F_q^d\). In even dimensions \(d \geq 4\), the authors 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, they obtain the sharp range of exponents \(p\) for which the extension operator is bounded, independently of \(q\), from \(L^p\) to \(L^4\) in the case when \(- 1\) is a square number in \(\mathbb F_q\). Using the sharp \(L^p - L^4\) result, they improve upon the range of exponents \(r\), for which the \(L^2 - L^r\) estimate holds, obtained by \textit{G. Mockenhaupt} and \textit{T. Tao} [Duke Math. J. 121, No. 1, 35--74 (2004; Zbl 1072.42007)] in even dimensions \(d \geq 4\). In addition, assuming that \(-1\) is not a square number in \(\mathbb F_q\), the authors extend their work done in three dimension to specific odd dimensions \(d \geq 7\). The discrete Fourier analytic machinery and Gauss sum estimates play an important role in the proof.