Planar Kakeya maximal estimate ($\Vert {\mathcal{K}}_{\delta }f{\Vert }_2\lesssim \sqrt{\log (2/\delta )}\Vert f{\Vert }_2$)

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.