$L^2$ bounds for a Kakeya type maximal operator in $\R^3$

We prove that the maximal operator obtained by taking averages at scale 1 along $N$ arbitrary directions on the sphere, is bounded in $L^2(\R^3)$ by $N^{1/4}{\log N}$. When the directions are $N^{-1/2}$ separated, we improve the bound to $N^{1/4}\sqrt{\log N}$. Apart from the logarithmic terms these bounds are optimal.

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