∑math20

Real and complex analysis · Result 086

An L3 bound for the trilinear Hilbert transform

1 paper

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

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 1

First page of An <i>L</i><sup>3</sup> bound for the trilinear Hilbert transform

An L3 bound for the trilinear Hilbert transform

OpenAI · October 5, 2026

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

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@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}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.