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

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.

19 pages. More detail added to Subsection 2.3 on well localized operators, and Subsubsection 2.3.2 added with a new theorem on extension to strongly localized operators. All old results unchanged