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The maximal triangular Hilbert transform at the symmetric point

OpenAI · September 24, 2026

Lean

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Abstract

For arbitrary complex inputs in $L^3(\mathbb R^2)$, we prove the pointwise maximal $L^3\times L^3\to L^{3/2}$ estimate for the triangular Hilbert transform, with the supremum over both hard truncation endpoints. The estimate yields joint almost-everywhere and L3/2 convergence as the lower endpoint tends to zero and the upper endpoint tends to infinity. This also proves the conjectured scalar estimate at the symmetric point.

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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.

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 own Lean formalization of Theorem 1.1 of this paper: the complex $L^3\times L^3\to L^{3/2}$ bound for the supremum over all finite hard truncations, together with a common full-measure set of definition and a.e. measurability. [family-level]
  • roos-j (GitHub), lean-triangular (GitHub repository, updated 2026-10-15). Lean 4 formalization of an $L^3\times L^3\to L^{3/2}$ bound for the triangular Hilbert transform following "a streamlined version of the proof" in this preprint, stated to be independent of OpenAI's formalizations; per the README, the Lean code was generated by Codex/GPT-6. [AI-generated per source]

Possibly relevant literature (found by the agent, not checked)

Digests and expositions

Elsewhere

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Cite

@misc{OAI:The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026,
  author = {{OpenAI}},
  title = {{The maximal triangular Hilbert transform at the symmetric point}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026/paper.pdf}{OAI:The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026}},
  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: