An $L^{4\mathrm{/}3}$ $S{L}_2$ Kakeya maximal inequality

It is shown that $S{L}_2$ Besicovitch sets of measure zero exist in ${\mathbb{R}}^3$. The proof is constructive and uses point-line duality analogously to Kahane’s construction of measure zero Besicovitch sets in the plane. A corollary is that the $S{L}_2$ Kakeya maximal inequality cannot hold with uniform constant. A counterexample is given to show that the $S{L}_2$ Kakeya maximal inequality cannot hold for $p>3\mathrm{/}2$; even in the model case where the $\delta$-tubes have $\delta$-separated directions and the cardinality of the tube family is $\sim {\delta }^{-2}$. It is then shown that, with $C_{\epsilon }{\delta }^{-\epsilon }$ loss, the $S{L}_2$ Kakeya maximal inequality does hold if $p\le 4\mathrm{/}3$, whenever the tubes satisfy a 2-dimensional ball condition (equivalent to the Wolff axioms in the $S{L}_2$ case). The proof is via an $L^{4\mathrm{/}3}$ inequality for restricted families of projections onto planes. For both inequalities, the range $4\mathrm{/}3<p\le 3\mathrm{/}2$ remains an open problem. A related $L^{6\mathrm{/}5}$ inequality is derived for restricted projections onto lines. Finally, an application is given to generic intersections of sets in ${\mathbb{R}}^3$ with “light rays” and “light planes”.