Browse › Real and complex analysis › 082 Annular variation and dyadic absolute bounds for the triangular Hilbert transform
An L³ bound for the dyadic triangular Hilbert form
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Abstract
We prove a uniform $L^3\times L^3\times L^3$ estimate for the dyadic triangular Hilbert form with unrestricted real inputs. The sum of the absolute local contributions over any finite set of scales is bounded by forty times the product of the input norms. In particular, the bound allows coefficients of modulus at most one to vary independently among admissible interval triples.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Annular variation and dyadic absolute bounds for the triangular Hilbert transform: Lean scope (release Lean docs, Oct 2026). The release's Lean formalization covers only the continuous symmetric maximal $L^3\times L^3\to L^{3/2}$ estimate of the companion paper; it does not cover the dyadic bound with constant 40 proved in this paper. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Fred Yu-Hsiang Lin, On the family of singular Brascamp-Lieb inequalities with dimension datum $(1,2,2,1)$ (arXiv:2212.11066, 2022). Its abstract describes a short proof of a variant of an estimate for the dyadic triangular Hilbert transform, which may be compared with the paper's main dyadic estimate and the earlier Kovač–Thiele–Zorin-Kranich dyadic results.
Digests and expositions
- Open Mathematics Courses, The dyadic triangular Hilbert form (three-lesson course, undated). This Claude-authored exposition develops the paper's absolute bound with constant 40 through matrix energies, the trace inequality and telescoping over four admissible children; the site states that independent review is incomplete. [AI-generated per source]
- AI Proved What?, Triangle, tamed? The triangular Hilbert transform at the symmetric point (Episode 082, October 2026). This AI-assisted video and transcript distinguish the three manuscripts' claims and outline their matrix-energy, Gaussian heat-flow and dyadic telescoping arguments. [AI-generated per source] [family-level]
Elsewhere
- Emergent Mind: family 082
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:An-L3-bound-for-the-dyadic-triangular-Hilbert-form-October-5-2026,
author = {{OpenAI}},
title = {{An $L^3$ bound for the dyadic triangular Hilbert form}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/An-L3-bound-for-the-dyadic-triangular-Hilbert-form-October-5-2026/dyadic-triangular-hilbert.pdf}{OAI:An-L3-bound-for-the-dyadic-triangular-Hilbert-form-October-5-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/082.md. The Lean statements have not been checked by people against the paper's claims.
Family 082: Annular variation and dyadic absolute bounds for the triangular Hilbert transform
Proves maximal and annular r-variation bounds, for every r > 2, from complex $L^3(\mathbb R^2)\times L^3(\mathbb R^2)$ to $L^{3/2}(\mathbb R^2)$. The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric $L^3\times L^3\times L^3$ point.
Other papers in this family:
- Annular variation of the triangular Hilbert transform at the symmetric point · October 5, 2026
- The maximal triangular Hilbert transform at the symmetric point · September 24, 2026