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Thiele's Problem 13: the symmetric L3 bound for the triangular Hilbert transform

For functions f,g,hf,g,h on the plane, the triangular Hilbert transform is the trilinear form Λ3(f,g,h)=p.v.∫R3f(x,y) g(y,z) h(z,x) dx dy dzx+y+z\Lambda_3(f,g,h)=\mathrm{p.v.}\int_{\mathbb R^3}f(x,y)\,g(y,z)\,h(z,x)\,\frac{dx\,dy\,dz}{x+y+z}. Each input sees one pair of the three variables and the pairs form a cycle, so it is a basic entangled singular integral; its two-variable analogue is equivalent to the Hilbert transform. Before this work only growing bounds were known: truncations to 1<∣x+y+z∣<N1<|x+y+z|<N obey C(log⁡N)1/2∥f∥3∥g∥3∥h∥3C(\log N)^{1/2}\|f\|_3\|g\|_3\|h\|_3 (Durcik, Kovac, Thiele), and a dyadic model was bounded only under structural assumptions on one input (Kovac, Thiele, Zorin-Kranich). Thiele's Problem 13: is there a universal constant CC with ∣Λ3(f,g,h)∣≤C∥f∥L3∥g∥L3∥h∥L3|\Lambda_3(f,g,h)|\le C\|f\|_{L^3}\|g\|_{L^3}\|h\|_{L^3}?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Harmonic analysis: entangled multilinear singular integrals
Posed by
Christoph Thiele, Problem 13 in Section 9 of Grafakos, Oliveira e Silva, Pramanik, Seeger and Stovall, Some problems in harmonic analysis (arXiv:1701.06637, 2017)
Year posed
2017
Years open
9y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
46 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Principal manuscript: for complex F,G∈L3(R2)F,G\in L^3(\mathbb R^2), ∥sup⁡0<ε<R<∞∣Bε,R(F,G)∣∥L3/2≤C∥F∥3∥G∥3\|\sup_{0<\varepsilon<R<\infty}|B_{\varepsilon,R}(F,G)|\|_{L^{3/2}}\le C\|F\|_3\|G\|_3, where Bε,R(F,G)(x,y)=∫ε<∣t∣<RF(x+t,y)G(x,y+t) dt/tB_{\varepsilon,R}(F,G)(x,y)=\int_{\varepsilon<|t|<R}F(x+t,y)G(x,y+t)\,dt/t, with joint principal values a.e. and in L3/2L^{3/2}; pairing with a third L3L^3 function gives the scalar bound of Problem 13. Companions (October 5): rr-variation bounds for every r>2r>2, and ∑k∑I∣LI∣≤40∏v∥Fv∥3\sum_k\sum_{\mathbf I}|L_{\mathbf I}|\le40\prod_v\|F_v\|_3 for Thiele's dyadic model with no structural assumption. Only the symmetric exponent point is treated; other triples with 1/p+1/q+1/r=11/p+1/q+1/r=1 and the four-linear simplex form are not.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. Three manuscripts form the family: the principal maximal-estimate paper (September 24) and two October 5 companions, one strengthening it to variation bounds and one treating the dyadic model.

Verification

No independent mathematician has checked this yet. Checked here: the abstracts, introductions and main theorems of the TeX sources of all three manuscripts, and Thiele's Problem 13 read in arXiv:1701.06637v1, Section 9; the proofs were not refereed. Lean-checked on the Comparator challenge TriangularHilbert, listed in the release's formalization catalogue (OAI.TriangularHilbert.main_estimate, OAI/Analysis/TriangularHilbert/Main.lean). Its statement: one constant CC such that for all complex F,GF,G in L3(R2)L^3(\mathbb R^2), almost every point has every annular truncation integrable, the maximal function over all finite truncations is a.e. measurable, and its L3/2L^{3/2} norm is at most C∥F∥3∥G∥3C\|F\|_3\|G\|_3. That is Theorem 1.1 of the principal manuscript, the bilinear maximal estimate. The step to the scalar trilinear form of Problem 13 (duality with a third L3L^3 function and a change of variables), the principal values, the variation bound and the dyadic bound are not formalized. Permitted axioms: propext, Quot.sound, Classical.choice. Not rebuilt here. Listed as Unreviewed rather than Lean-checked because its formal statement covers only the bilinear maximal bound.

Sources

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