OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
Proves a uniform strong L^(2) 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 L^(2)-bounded principal-value operator, establishing Stein's weak-type conjecture at this short scale.
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MANUSCRIPT 01
A uniform Hilbert transform estimate for Lipschitz directions
We prove a uniform L^(2) 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…
Full source abstract
We prove a uniform L^(2) 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.
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