| # 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) | | |