Kakeya maximal function conjecture

Establish that for any set T of δ-tubes in R^n pointing in δ-separated directions, the maximal function bound holds uniformly: ||∑_T χ_T||_p ≲≈ 1 for all 1 ≤ p ≤ n/(n−1).

Background

This conjecture quantifies overlap of tube families by Lp control of the sum of characteristic functions, and is equivalent to sharp union-volume bounds under shadings as well as operator bounds for the Kakeya maximal operator.

It is known to hold in R2 but is false beyond the endpoint p>n/(n−1). Equivalent formulations include a λ-density shading bound and an Ln(S{n−1})→Ln(Rn) bound for the Kakeya maximal operator.

References

Conjecture [Kakeya maximal function conjecture]\label{kakeyaMaxmlConjA} Let $T$ be a set of $\delta$ tubes in $Rn$ pointing in $\delta$-separated directions. Then \begin{equation}\label{kakeyaMaximalFnConj} \Big\Vert \sum_{T}\chi_T \Big\Vert_{p}\lessapprox 1,\quad 1\leq p \leq \frac{n}{n-1}. \end{equation}

kakeyaMaximalFnConj:

∥∑TχT∥p⪅1,1≤p≤nn−1.\Big\Vert \sum_{T}\chi_T \Big\Vert_{p}\lessapprox 1,\quad 1\leq p \leq \frac{n}{n-1}.

— A Survey of the Kakeya conjecture, 2000-2025  (2512.09397 - Zahl, 10 Dec 2025) in Conjecture \ref{kakeyaMaxmlConjA}, Section 1 (Introduction)

Wolff's hairbrush argument then raised the corresponding bound to $3$ in by means of approaching a stronger quantitative version of this conjecture, denoted by Kakeya maximal conjecture. The statement goes as follows: \begin{conjecture}[Kakeya maximal estimate at dimension $d$] Let $\mathbb{T}$ be a set of $\delta$-tubes in $\mathbb{R}n$ that point in $\delta$-separated directions. Then, for each $d \leq n$ and $\varepsilon > 0$, there exists a constant $C_\varepsilon > 0$ such that

\left| \sum_{T \in \mathbb{T}} \mathds{1}T \right|{L\frac{d}{d-1}(\mathbb{R}n)} \leq C_\varepsilon \left(\frac{1}{\delta}\right){\frac{n}{d}-1+\varepsilon}.

\end{conjecture}

— Refinements on the Restriction and Kakeya problems range in $\mathbb{R}^4$  (2610.06181 - Moreira, 5 Oct 2026) in Section 1, introduction; Conjecture (Kakeya maximal estimate at dimension $d$)