Balanced L3 boundedness of the trilinear Hilbert transform
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Definitions and mathematical statement
Statement recorded in the supplied source.
For Schwartz functions f1,f2,f3 on R, let $T(f1,f2,f3)(x)=pv$ integral f1(x+t)f2(x+2t)f3(x+3t) dt/t. Is $||T(f1,f2,f3)||_L1$ $<=$ C $product_j$ $||fj||_L3$ for one finite absolute C?Short English statement
For Schwartz functions f1,f2,f3 on R, let $T(f1,f2,f3)(x)=pv$ integral f1(x+t)f2(x+2t)f3(x+3t) dt/t. Is $||T(f1,f2,f3)||_L1$ $<=$ C $product_j$ $||fj||_L3$ for one finite absolute C?Sources
- S1 Original question source or mathematical context. https://terrytao.wordpress.com/2007/05/10/open-question-boundedness-of-the-trilinear-hilbert-transform/.
- S2 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-L3-bound-for-the-trilinear-Hilbert-transform-October-5-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2959
A trilinear Hilbert-transform estimate: The primary statement says: The one-dimensional trilinear Hilbert transform with slopes one, two and three maps three L3 Schwartz inputs to L1. The target retains these quantified objects and hypotheses, so its positive assertion is covered.
With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.
Balanced L3 boundedness of the trilinear Hilbert transform
OpenAI's claim
A trilinear Hilbert-transform estimate: The manuscript claims an $L^1$ output bound for three $L^3(\mathbb R)$ inputs to the Hilbert transform with slopes $1,2,3$. This is a fixed-slope symmetric-exponent case.Mathematical statement of the claimed result
Mathematical claim 1.
For Schwartz $f_1,f_2,f_3:\mathbb R\to\mathbb C$ define \[T(f_1,f_2,f_3)(x)=\operatorname{p.v.}\int_{\mathbb R}f_1(x-t)f_2(x-2t)f_3(x-3t)\,\frac{dt}{t}.\] The claim is $\|T(f_1,f_2,f_3)\|_1\le C\prod_{j=1}^3\|f_j\|_3$. It yields a unique bounded trilinear extension $(L^3)^3\to L^1$, for these three distinct shifts and balanced exponents.Outline of the argument
A trilinear Hilbert-transform estimate: Local forms are compressed using projection and quadratic localization tools. A base form has few active scales on each path, and curvature localization splits it into side terms and a core. Square-energy bounds control the sides, counting bounds control the core, and the final parameter choices absorb the remaining errors.Formal verification
A trilinear Hilbert-transform estimate: The pinned catalogue lists no Lean formalization for this result.This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.
Status caveat
OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.Sources
- OpenAI, An L${}^{3}$ bound for the trilinear Hilbert transform, 2026-10-05; citation key
OAI:An-L3-bound-for-the-trilinear-Hilbert-transform-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - OpenAI catalogue entry.