Result 086, Real and complex analysis

An L3 bound for the trilinear Hilbert transform

Proves that the principal-value trilinear Hilbert transform with shifts x−tx-t, x−2tx-2t, x−3tx-3t is bounded from L3(R)3L^3(\mathbb R)^3 to L1(R)L^1(\mathbb R). This resolves the L3 exponent case of the standard conjecture for slopes 1, 2, 3.

Proof

The bigger picture

Why it matters

Combining three functions through a singular integral can amplify their interaction in ways that ordinary size estimates miss. This manuscript claims a bound showing that, for one specific configuration, their combined output remains controlled.

What changes?

The unreviewed manuscript reports that the trilinear Hilbert transform with fixed slopes 1, 2, 3 maps three L3 functions on the real line into L1. This transform combines values at x-t, x-2t and x-3t using the singular weight 1/t, interpreted by symmetric truncation around zero. The output's integral of absolute value is bounded by a constant times the product of the inputs' L3 norms. Each norm measures cube-integrability. The constant is independent of the inputs; other slopes and exponents are not covered.

What does that help mathematicians do?

The estimate rules out arbitrarily large total output when all three input norms remain bounded, despite the singular weight. It also gives stability: if each input is approximated in L3, the corresponding outputs converge in L1. Researchers could therefore pass from estimates or constructions using well-behaved inputs to general L3 data for the bounded operator. This is a concrete consequence of the claimed bound, not a resolution of the full conjecture.

Are there practical applications?

The immediate value is foundational: this would establish a precise control principle for an operator combining multiplication, translation and singular cancellation. It supplies a bound that further analysis of this fixed-slope transform could use. The supplied abstract identifies no practical application, and the result alone does not establish a computational method or performance improvement.

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.

Manuscript

An L3 bound for the trilinear Hilbert transform

October 5, 2026 93 pages

We prove that the trilinear Hilbert transform with fixed slopes 1, 2, 3 maps L3(R)×L3(R)×L3(R)L^3(\mathbb R)\times L^3(\mathbb R)\times L^3(\mathbb R) to L1(R)L^1(\mathbb R), resolving this case of the trilinear Hilbert transform conjecture.

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Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 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.