Extension Theorems for Paraboloids in the Finite Field Setting

TL;DR

This paper advances the understanding of extension operators for paraboloids over finite fields by improving bounds in higher even dimensions and exploring new estimates in odd dimensions using Fourier analysis and Gauss sums.

Contribution

It provides sharper $L^p-L^4$ bounds in higher even dimensions and extends the analysis to odd dimensions where -1 is not a square, using discrete Fourier analysis and Gauss sum estimates.

Findings

Improved $L^p-L^4$ bounds in higher even dimensions.

Established $L^p-L^r$ bounds in odd dimensions when -1 is not a square.

Estimated the number of additive quadruples in paraboloids.

Abstract

In this paper we study the $L^p-L^r$ boundedness of the extension operators associated with paraboloids in vector spaces over finite fields.In higher even dimensions, 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 improve upon the standard Tomas-Stein exponents which Mockenhaupt and Tao obtained for the boundedness of extension operators for paraboloids by estimating the decay of the Fourier transform of measures on paraboloids. In particular, we obtain the sharp $L^p-L^4$ bound up to endpoints in higher even dimensions. Moreover, we also study the $L^2-L^r$ estimates.In the case when -1 is not a square number in the underlying finite field, we also study the $L^p-L^r$ bound in higher odd dimensions.The discrete Fourier analytic machinery and Gauss sum estimates make an important role in the proof.