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

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

💡 Research Summary

The paper investigates the relationship between two central conjectures in harmonic analysis in three dimensions: the paraboloid Fourier extension conjecture and the Kakeya strong maximal operator conjecture. Both conjectures have been reformulated in recent work as trilinear testing conditions—one involving a smooth Alpert testing condition for the extension operator, the other a modulated testing condition for the Kakeya maximal function. The author’s main goal is to translate the modulated condition into a restricted smooth Alpert testing condition, thereby establishing a bridge between the two frameworks.

The exposition begins by recalling the Fourier extension operator (E) associated with the paraboloid (P^2\subset\mathbb R^3) and the smooth Alpert pseudoprojection (Q_{s,U}) built from doubly smooth Alpert wavelets (h_{\eta_I;\kappa}) on a dyadic grid (G). The smooth Alpert trilinear inequality (Definition 1) asserts that for any three (\nu)-disjoint squares (U_1,U_2,U_3) and any scale (s),

\