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An L3 bound for the trilinear Hilbert transform
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Abstract
We prove that the trilinear Hilbert transform with fixed slopes 1, 2, 3 maps $L^3(\mathbb R)\times L^3(\mathbb R)\times L^3(\mathbb R)$ to $L^1(\mathbb R)$, resolving this case of the trilinear Hilbert transform conjecture.
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Comments and discussion
- OSVerify (F. Nie, H. Ye, S. Liu, Z. Wang, S. Ermon, J. Zou, Stanford), An L³ bound for the trilinear Hilbert transform: issues (OSVerify case page, October 2026). A referee-style report on the October 5 version listing one major issue (a missing endpoint hypothesis in Lemma 8.6) and two minor ones (a predictor identity in Lemma 5.3 and the first-failing-role convention in Section 9.1), each with a proposed repair; the page says the findings come from "OSVerify's internal review" (the OSVerify site says its reports are verified by human experts) and "do not necessarily establish that the paper's main theorem is false".
Formalization
- Joris Roos, lean-trilinearht: Formalization of boundedness of the trilinear Hilbert transform (GitHub, Apache-2.0, October 2026). An independent Lean 4 formalization (about 194k lines) of the L³×L³×L³→L¹ bound, with human-written statement (
TrilinearHT/Theorems.lean) and operator definition (TrilinearHT/Defs.lean) and a proof machine-generated by coding agents from a machine-generated blueprint based on this preprint. [AI-generated per source] - Joris Roos, Formalization of L^3 boundedness of the trilinear Hilbert transform (PALOMAR-2026-10-20-000001, version 1, 2026-10-20). The Palomar registry record verifying
TrilinearHT.trilinear_ht_boundat commit0fac9f4ed4aa3391931897a681ce547981fb5f46(Lean v4.35.0-rc3), listing this preprint as the source and a separate machine-generated "revised shorter argument" as an adaptation; its review warnings note that Comparator checks only the boundedness theorem, not the operator definition. [AI-generated per source]
Possibly relevant literature (found by the agent, not checked)
- Terence Tao, Open question: boundedness of the trilinear Hilbert transform (What's new, 2007-05-10). States the Hölder-range boundedness conjecture for the same operator with slopes 1, 2, 3, discussing its quadratic modulation symmetry and the logarithmic obstruction in summing time-frequency forests; this bears on the paper's claimed scale-uniform L³ estimate.
- Ciprian Demeter, Divergence of combinatorial averages and the unboundedness of the trilinear Hilbert transform (arXiv:0712.2494, December 2007). Proves that the trilinear Hilbert transform is unbounded in a range of L^p spaces with p close to 1; this bears on the conjectured exponent range that the paper cites from Hu and Lie.
Digests and expositions
- Latent Seminar, The trilinear Hilbert transform at the symmetric exponent (2026-10-06). Outlines the manuscript's quadratic charts, projection compression, curvature localization, and sparse counting, stating that it is not an independent verification of the complete proof.
- Orply / Perplexity, A Fixed-Slope Trilinear Hilbert Transform Maps Three L³ Inputs to L¹ (2026-10-09). Explains the quadratic phase obstruction and the claimed absorption argument, distinguishing the bounded extension from pointwise principal-value convergence for arbitrary L³ inputs. [AI-generated per source]
Elsewhere
- Palomar: Formalization of L^3 boundedness of the trilinear Hilbert transform (Lean formalization (formalizes))
- VibeMathed: Problem record
- Emergent Mind: family 086
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:An-L3-bound-for-the-trilinear-Hilbert-transform-October-5-2026,
author = {{OpenAI}},
title = {{An $L^3$ bound for the trilinear Hilbert transform}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/An-L3-bound-for-the-trilinear-Hilbert-transform-October-5-2026/paper.pdf}{OAI:An-L3-bound-for-the-trilinear-Hilbert-transform-October-5-2026}},
year = {2026}
}Family 086: An L3 bound for the trilinear Hilbert transform
Proves that the principal-value trilinear Hilbert transform with shifts $x-t$, $x-2t$, $x-3t$ is bounded from $L^3(\mathbb R)^3$ to $L^1(\mathbb R)$. This resolves the L3 exponent case of the standard conjecture for slopes 1, 2, 3.
This is the only paper in this family.