A comparison of trilinear testing conditions for the paraboloid Fourier extension and Kakeya conjectures in three dimensions

TL;DR

This paper compares two testing conditions related to the paraboloid Fourier extension and Kakeya conjectures in three dimensions, revealing connections between them through a conversion process.

Contribution

It introduces a method to convert the modulated testing condition for the Kakeya conjecture into a restricted smooth Alpert testing condition for the paraboloid Fourier extension conjecture.

Findings

Establishes a link between testing conditions for two major conjectures.

Provides a new perspective on the relationship between Fourier extension and Kakeya problems.

Suggests potential avenues for transferring techniques between these conjectures.

Abstract

We compare the smooth Alpert testing condition for the paraboloid Fourier extension conjecture in <cite>RiSa3</cite> to the modulated testing condition for the Kakeya conjecture in <cite>RiSa2</cite>. To this end, the modulated testing condition is converted to a certain restricted smooth Alpert testing condition for the paraboloid Fourier extension conjecture.