Diameter-free reverse inequalities and superorthogonality

Abstract

We prove three results as part of the program of diameter-free estimates initiated in [CDW26]. The first two are reverse square function estimates for the light cone in $\mathbb{R}^3$. We first establish an abstract $L^4$ inequality of independent interest, under an ordered superorthogonality hypothesis: for every four distinct indices, only the two nonalternating pairings are required to vanish. The loss is $C(1+\log N)^2$, where $N$ is the number of functions. An alternating-determinant argument verifies this hypothesis for separated cone sectors. This gives a diameter-free estimate for arbitrary disjoint angular intervals, at the thickness determined by their smallest width, with no upper restriction on the radial parameter. For the canonical equal-width partition on a fixed radial annulus, we obtain the loss $C(1+\log N)^{1/4}$, which is sharp up to constants. This refines the estimate by Guth-Wang-Zhang, via a different approach. The improvement uses additional orthogonality between diagonal and off-diagonal differences, together with bounded overlap of dyadic difference shells. Our third result is the diameter-free $\ell^2L^6$ decoupling for arbitrary partitions of the parabola, with an $N^\varepsilon$ loss independent of the interval widths. The argument is an adaptation of the method from \cite{Cushman-Demeter-Wu} and implies their three-fold additive-energy estimate for the parabola. The three proofs use variants of interlacing and special orthogonality in place of wave packet analysis, multilinearity and parabolic/Lorentz rescaling. Together, these results provide further evidence for the scope of the paradigm introduced in [CDW26].