The fixed-width Lipschitz Kakeya maximal-function conjecture

About 19 years old · traced to 2007

Let $v$ be a Lipschitz map from $\mathbb R^2$ to the unit circle. For rectangles $R$ satisfying $|\operatorname V R|\geq\delta|R|$ and $\mathsf w\leq\operatorname W R\leq2\mathsf w$ , define

\operatorname M_{\delta,v,\mathsf w}f(x)=\sup\frac{\mathbf 1_R(x)}{|R|}\int_R|f(y)|\,dy.

Fixed-width Lipschitz Kakeya conjecture. For some $1<p<2$ and some finite $N$ , for all $0<\delta<1$ , all Lipschitz vector fields $v$ , and $0<\mathsf w<\frac1{100\|v\|_{\operatorname{Lip}}}$ , the operator $\operatorname M_{\delta,v,\mathsf w}$ maps $L^p(\mathbb R^2)$ to $L^{p,\infty}(\mathbb R^2)$ with norm at most $\lesssim\delta^{-N}$ . This fixed-width form is stronger than what is needed for the paper's conditional Hilbert-transform results and is not verified in the source.

References

Primary source

Michael Lacey and Xiaochun Li, “On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields”, arXiv:0704.0808 (2008).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A detailed proof blueprint was submitted on September 3, 2026, but it does not prove the conjecture, so the general case remains open.

The conjecture asks for a polynomial weak-type bound for fixed-width rectangles following an arbitrary Lipschitz direction field, for some $1<p<2$ and finite $N$ . Lacey and Li treated this estimate as a conjectural hypothesis in their 2008 conditional work on Hilbert transforms.

Known results

For one-variable fields $v(x,y)=(1,u(x))$ , Lacey–Li proved an $L^p$ bound for every $p>1$ , with loss at most $\delta^{-1}$ .
In 2011, fixed-width one-variable estimates gave an $L^2$ bound with logarithmic loss and interpolated bounds for $1<p<2$ .
For a related variable-width operator, Lacey–Li proved a weak- $L^2$ bound with loss at most $\delta^{-1/2}$ ; this does not imply the stated conjecture.

Community submission (unverified), September 3, 2026

A submitted article expands a fourteen-stage roadmap involving discretization, local freezing, angular decompositions, incidence estimates, multiplicity bounds, and recursive $\delta$ -loss bookkeeping. It presents proof obligations and verification checkpoints, but no verified proof of the conjecture.

Current status (as of September 2026): The one-variable and related variable-width cases are known, while the general fixed-width conjecture remains open and the September submission is unverified.

Sources

ProofTHE FIXED-WIDTH LIPSCHITZ KAKeya MAXIMAL-FUNCTION CONJECTURE A More Analytical 14-Stage Research Article Definitions • Geometric Reduction • Incidence Theory • Weak-Type Analysis • Proof Assembly Research manuscript — exploratory and proof-oriented; not a claim that the conjecture is solved.See full solutionHide full solution

This article develops a substantially more analytical version of the fourteen-stage roadmap for the fixed-width Lipschitz Kakeya maximal-function conjecture. The objective is to expose the precise mathematical interfaces between the geometry of admissible rectangles and the desired weak-L^p estimate for some 1<p<2. The argument is organized around normalization, dyadic discretization, local freezing of the Lipschitz field, good-set and angular decompositions, multiplicity estimates, longitudinal multiscale analysis, recursive polynomial δ-losses, and the final distributional inequality. Each stage is expanded into concrete substeps, candidate lemmas, bookkeeping requirements, and verification checkpoints.

CODE ANALYSIS.pdf371,756 bytesOpen
Fixed_Width_Lipschitz_Kakeya_Analytical_14_Stage_Article.pdf294,520 bytesOpen
Fixed_Width_Lipschitz_Kakeya_Second_Analytical_Report.pdf220,840 bytesOpen