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Annular variation of the triangular Hilbert transform at the symmetric point
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Abstract
We prove the annular r-variation estimate for the triangular Hilbert transform from complex $L^3\times L^3$ to L3/2 for every r > 2. The partitions may depend on the output point and range over all positive scales. The estimate yields the two-endpoint maximal bound and joint almost-everywhere and norm principal values, and resolves the symmetric scalar triangular Hilbert transform problem.
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work in progress
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.
Follow-up works
- Zhuowen Guo, Rongzhong Xiao, Shuhao Zhang, Pointwise convergence of bilinear ergodic averages for commuting transformations (arXiv:2610.26388, 19 Oct 2026). Proves a.e. convergence of $\frac1N\sum F(S^nx)G(Q^nx)$ for commuting $S,Q$ and $1/p+1/q\le1$, citing this paper as [23] and taking its matrix energy, Gaussian heat calculations and several lemmas (e.g. Lemmas 3.4–3.6, 4.1, 4.3, 4.4) from it; it declares that ChatGPT (GPT-6) was used.
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 the symmetric maximal $L^3\times L^3\to L^{3/2}$ estimate of the companion paper; it does not cover the $r$-variation bound or the disjoint-annulus count estimate of this paper. [family-level]
- Floris van Doorn, Polona Durcik, Joris Roos, Lenka Slavíková, Christoph Thiele (GitHub roos-j), Pointwise convergence of ergodic averages with respect to two commuting transformations (Lean 4 repository, PALOMAR-2026-10-18-000026, Oct 2026). Lean 4 formalization of a.e. convergence for two commuting measure-preserving transformations with $1/p+1/q\le1$; the README says a GPT-6 model adapted this preprint and produced the blueprint, Claude generated the formalization, and only
README.mdandTheorems.leanare human-authored. [AI-generated per source]
Possibly relevant literature (found by the agent, not checked)
- Yen Do, Camil Muscalu, Christoph Thiele, Variational estimates for paraproducts (arXiv:1009.5187, 2010). Proves $r$-variation estimates for bilinear (paraproduct) operators extending Lépingle's inequality, which may give context for the bilinear $r$-variation bound, $r>2$, in Theorem 1.1.
- Michael Christ, Polona Durcik, Vjekoslav Kovač, Joris Roos, Pointwise convergence of certain continuous-time double ergodic averages (arXiv:2011.06370; Ergodic Theory Dynam. Systems 42 (2022)). Derives a.e. convergence of double ergodic averages from a curved triangular Hilbert transform model, related to the a.e. principal-value consequences of the variation estimate.
Digests and expositions
- 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
- Palomar: Pointwise a.e. convergence of ergodic averages with respect to two commuting transformations in Lean (Lean formalization (adapts))
- Emergent Mind: family 082
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026,
author = {{OpenAI}},
title = {{Annular variation of the triangular Hilbert transform at the symmetric point}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-October-5-2026/annular-variation.pdf}{OAI:Annular-variation-of-the-triangular-Hilbert-transform-at-the-symmetric-point-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:
- An L³ bound for the dyadic triangular Hilbert form · October 5, 2026
- The maximal triangular Hilbert transform at the symmetric point · September 24, 2026