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

Abstract

We prove that the maximal operator obtained by taking averages at scale 1 along $N$ arbitrary directions on the sphere, is bounded in $L^2({\mathbb{R}}^3)$ 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.

13 pages

Cited by in corpus (3)

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