Hilbert transforms along Lipschitz directions

Proves a uniform strong L² bound for the planar Hilbert transform along any Lipschitz unit vector field, at integration lengths bounded by an absolute multiple of its reciprocal Lipschitz constant. The estimate is uniform over inner truncations and yields an L²-bounded principal-value operator, establishing Stein's weak-type conjecture at this short scale. (Lean)

Manuscript

A uniform Hilbert transform estimate for Lipschitz directions

September 25, 2026 · source · cite as \cite{OAI:A-uniform-Hilbert-transform-estimate-for-Lipschitz-directions-September-25-2026}

We prove a uniform L² bound for the Hilbert transform along a Lipschitz unit vector field in the plane, with integration restricted to a fixed absolute multiple of the reciprocal Lipschitz seminorm. The bound is uniform in the inner truncation and holds for fields depending on both coordinates. This gives an affirmative answer to Stein's weak-(2,2) conjecture.

Result 083 · Real and complex analysis. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.