For a planar Lipschitz unit vector field with Lipschitz constant K>0, short Hilbert transforms with outer scale at most 1/(10⁶K) have a universal L² bound independent of the inner hard truncation, including fields depending on both coordinates.
For a planar Lipschitz unit vector field with Lipschitz constant K>0, short Hilbert transforms with outer scale at most 1/(10⁶K) have a universal L² bound independent of the inner hard truncation, including fields depending on both coordinates.
For a planar Lipschitz unit vector field with Lipschitz constant K>0, short Hilbert transforms with outer scale at most 1/(10⁶K) have a universal L² bound independent of the inner hard truncation, including fields depending on both coordinates.
If correct, gives an affirmative strong-L² form of Stein’s short-scale directional Hilbert-transform question.
Are short directional Hilbert transforms uniformly L² bounded for arbitrary planar Lipschitz direction fields?
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OpenAI reports uniform Hilbert-transform bounds for Lipschitz directions
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