Bounds for Kakeya-type maximal operators associated with $k$-planes

A $(d,k)$ set is a subset of $\rea^d$ containing a translate of every $k$-dimensional plane. Bourgain showed that for $k \geq \kcrit(d)$, where $\kcrit(d)$ solves $2^{\kcrit-1}+\kcrit = d$, every $(d,k)$ set has positive Lebesgue measure. We give a short proof of this result which allows for an improved $L^p$ estimate of the corresponding maximal operator, and which demonstrates that a lower value of $\kcrit$ could be obtained if improved mixed-norm estimates for the $x$-ray transform were known.

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Published 1 January 2007