The square function estimates for cones in arbitrary dimensions and elliptic quadrics

Abstract

We demonstrate that the square function estimate for the light cone in $\R^d$, $d\ge3$, at the exponent $p=2(d-1)/(d-2)$, with loss $[\log(2R)]^{5/p}$ can be deduced from local-smoothing and auxiliary-energy statements in the October 25, 2026 OpenAI manuscript. Exact affine sections preserve these logarithmic bounds and give corresponding estimates for spheres, ellipsoids, elliptic paraboloids, and compact patches of two-sheet elliptic hyperboloids in $\R^{d-1}$.