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cfa2fd2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | # Hilbert transforms along Lipschitz directions
The following describes the scope of the Lean formalization related to the following accompanying paper(s):
- [A uniform Hilbert transform estimate for Lipschitz directions](../../preprints/A-uniform-Hilbert-transform-estimate-for-Lipschitz-directions-September-25-2026/main.pdf)
## Scope
The formalization proves uniform $L^2$ bounds for short Hilbert transforms along Lipschitz unit vector fields in the plane, including fields depending on both coordinates. For Lipschitz constant $K>0$, every hard truncation with $0<\varepsilon\le R\le1/(10^6K)$ has one universal $L^2$ bound on Schwartz functions, independent of the inner cutoff.
At a fixed short scale for $1$-Lipschitz fields, the formalization also gives principal-value and weak-$(2,2)$ estimates and bounded extensions to all of $L^2$. These results establish the stated short-scale form of Stein's question.
## Comparator links
| Result | Comparator statement |
| --- | --- |
| Uniform short Hilbert-transform bounds for Lipschitz directions | [LipschitzHilbert.lean](../ComparatorChallenges/LipschitzHilbert.lean) |
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