Kakeya maximal operator L^p bounds in higher dimensions

Establish L^p boundedness (identify the full range of p) for the Kakeya maximal operator on ℝ^n for n ≥ 3.

Background

The Kakeya maximal operator is central in modern harmonic analysis and is deeply connected to geometric measure phenomena. Its Lp bounds in higher dimensions are intertwined with major open problems such as Bochner–Riesz summability and Fourier restriction.

The paper notes recent advances on dimensional consequences in n=3 by Wang and Zahl, but the operator’s complete Lp theory in higher dimensions is still unresolved and impacts several other core conjectures.

References

Three of the most important open questions in euclidean harmonic analysis are: (i) Lp bounds for the Kakeya maximal operator (the corresponding dimensional consequence of these when n=3 has recently been established in spectacular work of Wang and Zahl), (ii) Lp bounds for the Bochner--Riesz operators S\delta, and (iii) Lp -Lq bounds for the Fourier restriction and extension operators for the sphere. These are known to be intimately related, and all of them are resolved when n=2, but all are open in all higher dimensions.

— Littlewood, Paley and Almost-Orthogonality: a theory well ahead of its time (2511.22605 - Carbery, 27 Nov 2025) in Section 7.3 (Angular decompositions — reverse inequalities and curvature)