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It is shown that $SL_2$ Besicovitch sets of measure zero exist in$\mathbb{R}^3$. The proof is constructive and uses point-line dualityanalogously to Kahane's construction of measure zero Besicovitch sets in theplane. A corollary is that the $SL_2$ Kakeya maximal inequality cannot holdwith uniform constant. A counterexample is given to show that the $SL_2$ Kakeyamaximal inequality cannot hold for $p> 3/2$; even in the model case where the$\delta$-tubes have $\delta$-separated directions and the cardinality of thetube family is $\sim \delta^{-2}$. It is then shown that, with $C_{\epsilon}\delta^{-\epsilon}$ loss, the $SL_2$ Kakeya maximal inequality does hold if $p\leq 4/3$, whenever the tubes satisfy a 2-dimensional ball condition(equivalent to the Wolff axioms in the $SL_2$ case). The proof is via an$L^{4/3}$ inequality for restricted families of projections onto planes. Forboth inequalities, the range $4/3 < p \leq 3/2$ remains an open problem. Arelated $L^{6/5}$ 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".

An $L^{4/3}$ $SL_2$ Kakeya maximal inequality