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            <title xml:lang="so">An $L^{3/2}$ $SL_2$ Kakeya maximal inequality</title>
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                <title xml:lang="so">An $L^{3/2}$ $SL_2$ Kakeya maximal inequality</title>
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                    <forename type="first">John</forename>
                    <surname>Green</surname>
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                    <forename type="first">Terence</forename>
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              <p>We prove a special case of the Kakeya maximal function conjecture in $\mathbb{R}^3$, with $C_{\epsilon}\delta^{-\epsilon}$ loss, when the centre lines of the tubes are  $SL_2$ lines and the tubes satisfy a 2-dimensional ball condition (equivalent to the Wolff axioms in this case). We show that the exponent $p=3/2$ is sharp and that some loss (such as $C_{\epsilon} \delta^{-\epsilon}$) is necessary, even in the $SL_2$ case where the $\delta$-tubes have $\delta$-separated directions and the cardinality of the tube family is maximal ($\sim \delta^{-2}$).  The $SL_2$ Kakeya maximal inequality is deduced from an $L^{3/2}$ inequality for restricted families of projections onto planes. A related $L^{3/2 -\epsilon}$ inequality is also derived for restricted projections onto lines, and an application is given to generic intersections of sets in $\mathbb{R}^3$ with ``light rays'' and ``light planes''.</p>
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