Endpoint restriction estimates for the paraboloid over finite fields

TL;DR

This paper establishes improved endpoint restriction estimates for the paraboloid over finite fields in three and higher dimensions, removing previous logarithmic factors and advancing the understanding of harmonic analysis in finite field settings.

Contribution

It introduces a bilinear approach that extends estimates from characteristic functions to general functions without logarithmic losses, improving prior results.

Findings

Removed logarithmic factors from restriction estimates in 3D and higher dimensions.

Extended estimates from characteristic functions to all functions.

Achieved sharper bounds in finite field harmonic analysis.

Abstract

We prove certain endpoint restriction estimates for the paraboloid over finite fields in three and higher dimensions. Working in the bilinear setting, we are able to pass from estimates for characteristic functions to estimates for general functions while avoiding the extra logarithmic power of the field size which is introduced by the dyadic pigeonhole approach. This allows us to remove logarithmic factors from the estimates obtained by Mockenhaupt and Tao in three dimensions and those obtained by Iosevich and Koh in higher dimensions.