Kakeya maximal conjecture ($\Vert {\mathcal{K}}_{\delta }f{\Vert }_n{\lesssim }_{n,\epsilon }{\delta }^{-\epsilon }\Vert f{\Vert }_n$)

The maximal estimate used in the classical Kakeya set problem asks, for every $\epsilon >0$, for a bound from $L^n({\mathbb{R}}^n)$ to $L^n(S^{n-1})$ with loss at most ${\delta }^{-\epsilon }$. Constants may depend on dimension and $\epsilon$, but not tube thickness or the function. Applied to an indicator function of a neighborhood of a Kakeya set, it gives the Kakeya Minkowski dimension conjecture.

For a separated angular net, two translated unit rectangles of width $\delta$ intersect in area at most a constant times ${\delta }^2/(\delta +\alpha )$, where $\alpha$ is their unoriented angular distance. Each row of their intersection matrix therefore has sum at most $C\delta \log (2/\delta )$. Applying the Schur test to the adjoint averaging operator gives the displayed bound. A wider-tube comparison extends it from the net to all directions. Since $\sqrt{\log (2/\delta )}{\lesssim }_{\epsilon }{\delta }^{-\epsilon }$, this proves the planar version of the Kakeya maximal conjecture.