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

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

2011-05-05

arxiv.org/abs/1105.1115v1

Mathematics

Classical Analysis and ODEs

13 pages

Scientific paper

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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