Stein’s Hilbert-transform bound along Lipschitz directions
For a vector field $v : \mathbb{R}^2 \to S^1$, a constant $K \ge 1$ and a Schwartz function $f$ on $\mathbb{R}^2$, let $$H_v f(x) = \lim_{\varepsilon \downarrow 0} \int_{\varepsilon < |t| < 1/K} f(x - t v(x))\, \frac{dt}{t},$$ the Hilbert transform of $f$ along the line through $x$ in the direction $v(x)$, truncated at length $1/K$. Lacey and Li recall that the estimate below for $C^2$ fields would imply Carleson's theorem on the almost everywhere convergence of Fourier series. Bateman and Thiele proved $L^p$ bounds, for a range of exponents $p$, for the untruncated transform along measurable non-vanishing fields that depend on only one of the two coordinates.
Problem. Do absolute constants $K \ge 1$ and $C < \infty$ exist such that $$\lambda^2 \bigl|\{x \in \mathbb{R}^2 : |H_v f(x)| > \lambda\}\bigr| \le C \|f\|_{L^2(\mathbb{R}^2)}^2$$ for every $v : \mathbb{R}^2 \to S^1$ with $|v(x) - v(y)| \le |x - y|$ for all $x, y$, every Schwartz function $f$ and every $\lambda > 0$? The constants must be uniform over all such fields, including fields depending on both coordinates.
Source: Stein (1986, ICM address); in this form, Lacey and Li (2010), Conjecture 1.5, with the Lipschitz constant normalised to 1 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- CA — Classical Analysis and ODEs
- Secondary
- FA
- Keywords
- AIM Open Applied Problems, directional Hilbert transform, Lipschitz vector fields, singular integrals
References
@inproceedings{stein1987,
author = {Stein, Elias M.},
title = {Problems in harmonic analysis related to curvature and oscillatory integrals},
booktitle = {Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986)},
volume = {1},
publisher = {Amer. Math. Soc.},
address = {Providence, RI},
year = {1987},
pages = {196--221},
}
@article{laceyli2010,
author = {Lacey, Michael and Li, Xiaochun},
title = {On a conjecture of {E}. {M}. {S}tein on the {H}ilbert transform on vector fields},
journal = {Mem. Amer. Math. Soc.},
volume = {205},
number = {965},
year = {2010},
doi = {10.1090/S0065-9266-10-00572-7},
eprint = {0704.0808},
archivePrefix = {arXiv},
}
@article{bateman2013,
author = {Bateman, Michael and Thiele, Christoph},
title = {{$L^p$} estimates for the {H}ilbert transforms along a one-variable vector field},
journal = {Anal. PDE},
volume = {6},
number = {7},
year = {2013},
pages = {1577--1600},
doi = {10.2140/apde.2013.6.1577},
eprint = {1109.6396},
archivePrefix = {arXiv},
}
@misc{aim-stein-lipschitz-direction-hilbert-bound,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Stein’s Hilbert-transform bound along Lipschitz directions}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/450-stein-lipschitz-direction-hilbert-bound.md},
note = {Entry 450 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
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.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00722, "Stein’s Hilbert-transform bound along Lipschitz directions", clopen.solutions.