Stein's conjecture on the Hilbert transform along Lipschitz vector fields

For a unit vector field $v:\mathbb R^2\to S^1$ consider the truncated directional Hilbert transform $H_{v,a}f(x)=\mathrm{p.v.}\int_{|t|<a}f(x-tv(x))\,dt/t$ , which integrates along the line through $x$ in direction $v(x)$ . Stein conjectured that Lipschitz regularity suffices: there is an absolute $\epsilon_0>0$ such that, with outer length $a=\epsilon_0/\|v\|_{\mathrm{Lip}}$ , $H_{v,a}$ is of weak type $(2,2)$ with a constant independent of $v$ . Lacey and Li trace it to Zygmund's question on differentiation along Lipschitz directions; Holder fields of any exponent below one do not suffice. Is $H_{v,a}$ uniformly weak- $(2,2)$ bounded for all Lipschitz unit fields at this scale?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Harmonic analysis; directional singular integrals
Posed by: Elias M. Stein
Year posed: —
Years open: —
Solved: 2026-09-25
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 50 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: there are absolute $a\in(0,1/2)$ and $C$ such that every Lipschitz unit field $v$ on $\mathbb R^2$ with $\mathrm{Lip}(v)\le1$ satisfies $\sup_{0<\epsilon<a}\|H^{\epsilon}_{v,a}f\|_2\le C\|f\|_2$ ; the principal value is $L^2$ bounded and weak-(2,2). By scaling this holds at outer length $a/\mathrm{Lip}(v)$ . Fields may depend on both coordinates. It does not give $L^p$ bounds for $p\neq2$ , does not address the maximal (Zygmund) averaging operator, and does not give estimates at arbitrary outer lengths for varying fields.

What the AI did

The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Single-manuscript family dated September 25, 2026.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Section 1.1 were read against Stein's conjecture as formulated by Lacey and Li; the theorem gives a strong $L^2$ bound, uniform in the inner truncation, for every 1-Lipschitz unit field at an absolute outer length, and scaling transfers it to outer length $a/\mathrm{Lip}(v)$ , which is the conjecture's scale. The challenge LipschitzHilbert is not in the formalization catalogue (lean/formalization.yaml); it is linked from lean/docs/083.md, its solution module exists at the pinned commit, and its statement was read here: absolute $a\in(0,1/2)$ and $C$ with uniform truncation, principal-value, weak-(2,2) and bounded $L^2$ extension conclusions for every 1-Lipschitz unit field, plus a version for Lipschitz constant $K$ with $R\le1/(10^6K)$ . This states the headline. Not rebuilt here. The paper itself stresses it does not treat arbitrary outer lengths.

Stein's conjecture on the Hilbert transform along Lipschitz vector fields

What was actually shown

What the AI did

Verification

Sources

Discussion