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

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.

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Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Stein's conjecture on the Hilbert transform along Lipschitz vector fields · Stein's Lipschitz vector field conjecture

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

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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.