Result 082, Real and complex analysis

Annular variation and dyadic absolute bounds for the triangular Hilbert transform

Proves maximal and annular r-variation bounds, for every r > 2, from complex L3(R2)×L3(R2)L^3(\mathbb R^2)\times L^3(\mathbb R^2) to L3/2(R2)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 L3×L3×L3L^3\times L^3\times L^3 point.

Lean formalization New or sharp bound

The bigger picture

Why it matters

The triangular Hilbert transform combines two functions on the plane through interactions across scales. The unreviewed manuscripts report bounds that keep these interactions controlled even when the smallest and largest included scales change.

What changes?

The main manuscript claims bounds from two complex-valued L3 functions on the plane to L3/2, where Lp means the pth power is integrable. It controls annular r-variation, measuring accumulated changes across scale bands, for every r greater than 2. Scale partitions may depend on the output point and range over all positive scales. The maximal bound controls both hard cutoff endpoints, yielding joint almost-everywhere and L3/2-norm convergence as the lower cutoff approaches zero and the upper cutoff tends to infinity.

What does that help mathematicians do?

The dyadic manuscript reports a complementary bound for a discrete-scale form: for unrestricted real inputs, the sum of absolute local contributions over any finite set of scales is at most forty times the product of the three L3 norms. Coefficients of modulus at most one can vary independently among admissible interval triples. Researchers could therefore control these weighted sums without relying on cancellation between local contributions, with no deterioration as more scales are included.

Are there practical applications?

The immediate value is foundational in analysis. Pairing the continuous transform's output with a third L3 function gives the claimed symmetric three-input estimate, not a solution for all exponent choices. The convergence statements also make the limiting operator well-defined almost everywhere and in norm as both cutoffs are removed. The supplied sources describe no practical deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

Annular variation of the triangular Hilbert transform at the symmetric point

October 5, 2026 37 pages

We prove the annular r-variation estimate for the triangular Hilbert transform from complex L3×L3L^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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

An L³ bound for the dyadic triangular Hilbert form

October 5, 2026 11 pages

We prove a uniform L3×L3×L3L^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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

The maximal triangular Hilbert transform at the symmetric point

September 24, 2026 27 pages Main result formalized in Lean

For arbitrary complex inputs in L3(R2)L^3(\mathbb R^2), we prove the pointwise maximal L3×L3→L3/2L^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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/082.md.

Annular variation and dyadic absolute bounds for the triangular Hilbert transform

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized result proves the symmetric maximal estimate for the triangular Hilbert transform: for arbitrary complex F,G∈L3(R2)F,G\in L^3(\mathbb R^2), the L3/2L^{3/2} norm of the supremum over all finite hard-truncation intervals is at most C∥F∥3∥G∥3C\|F\|_3\|G\|_3 for one absolute constant CC. It also establishes a common full-measure set on which the truncated integrals are defined and almost-everywhere measurability of the maximal output.

Comparator links

Result Comparator statement
Maximal triangular Hilbert transform bound TriangularHilbert.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.