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 ℝ3. We first establish an abstract L4 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 ℓ2L6 decoupling for arbitrary partitions of the parabola, with an Nε 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].
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BibTeX
@misc{cushman2026diameterfree,
title = {{Diameter-free reverse inequalities and superorthogonality}},
author = {Adam Cushman and Ciprian Demeter and Shukun Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13105},
eprint = {2609.13105},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13105},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Diameter-free reverse inequalities and superorthogonality
AU - Adam Cushman
AU - Ciprian Demeter
AU - Shukun Wu
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13105
UR - https://arxiv.org/abs/2609.13105
AB - 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 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 ℓ^2L^6 decoupling for arbitrary partitions of the parabola, with an N^ε 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].
ER -