The Kakeya maximal conjecture in three dimensions

Theorem 1.1: for every $\varepsilon>0$ there is $C_\varepsilon$ such that $\|K_\delta f\|_{L^3(S^2)}\le C_\varepsilon\delta^{-\varepsilon}\|f\|_{L^3(\mathbb R^3)}$ for all $0<\delta<1$ and $f\in L^3(\mathbb R^3)$. This is the $n=3$ case of the maximal conjecture; it strengthens the Wang-Zahl set theorem by getting the cubic dependence on shading density. The paper also derives Nikodym maximal and local curved Kakeya estimates by cited transfers. It does not treat dimension four or higher for the maximal operator; the four-dimensional Hausdorff case is a separate entry. The same release's local smoothing result in three dimensions (family 079, entry local-smoothing-conjecture-r3) also yields this Kakeya maximal estimate, by the established Bochner-Riesz to restriction to Kakeya chain in its Section 11.1.

Theorem 1.1: for every $\varepsilon>0$ there is $C_\varepsilon$ such that $\|K_\delta f\|_{L^3(S^2)}\le C_\varepsilon\delta^{-\varepsilon}\|f\|_{L^3(\mathbb R^3)}$ for all $0<\delta<1$ and $f\in L^3(\mathbb R^3)$. This is the $n=3$ case of the maximal conjecture; it strengthens the Wang-Zahl set theorem by getting the cubic dependence on shading density. The paper also derives Nikodym maximal and local curved Kakeya estimates by cited transfers. It does not treat dimension four or higher for the maximal operator; the four-dimensional Hausdorff case is a separate entry. The same release's local smoothing result in three dimensions (family 079, entry local-smoothing-conjecture-r3) also yields this Kakeya maximal estimate, by the established Bochner-Riesz to restriction to Kakeya chain in its Section 11.1.