Local single-scale modular Fourier square function extension conjecture (paraboloid in R^3)

Prove the local single-scale modular Fourier square function extension inequality for the paraboloid in R^3, namely: for all ε > 0, q > 3, s ∈ N, all modulation sequences u = {u_I}_{I ∈ G_s[U]} ∈ V, and all bounded functions f ∈ L^∞(U), show that ||S_Fourier^{s,u} f||_{L^q(R^3)} ≤ C_{ε,q} 2^{ε s} ||f||_{L^∞(U)}, where S_Fourier^{s,u} is the modulated Fourier square function built from the pushforward to the paraboloid and dyadic wavelet projections over a 2^{-s}-separated grid.

Background

In Section 2, the authors define a modulated Fourier square function and formulate a conjectural local single-scale inequality (FECUS) that, if true for all q > 3, would imply the dual form of the Kakeya maximal operator conjecture. They subsequently establish equivalences between this conjecture and certain trilinear square function inequalities, but do not prove FECUS itself.

The conjecture is stated in precise quantitative form with parameters s (scale), modulations u over a separated grid G_s[U], and bounded functions f on U. It serves as a bridge between Fourier extension phenomena on the paraboloid and Kakeya-type estimates, and remains an explicit target for future work.