= Planar Kakeya maximal estimate
{title2=$\|\mathcal K_\delta f\|_2\lesssim\sqrt{\log(2/\delta)}\|f\|_2$}
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 $\sqrt{\log(2/\delta)}\lesssim_\varepsilon\delta^{-\varepsilon}$, this proves the planar version of the <Kakeya maximal conjecture>.
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