Extension theorems for paraboloids in the finite field setting

In this paper we study the L-p - L-r boundedness of the extension operators associated with paraboloids in F-q(d), where 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) is an element of 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 L-p to L-4 in the case when -1 is a square number in F-q. Using the sharp L-p - L-4 result, we improve upon the range of exponents r, for which the L-2 - L-r 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 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.