Bounds for Kakeya-type maximal operators associated with k-planes
Abstract
A (d,k) set is a subset of Rd containing a translate of every k-dimensional plane. Bourgain showed that for k ≥ kcr(d), where kcr(d) solves 2kcr-1+kcr = d, every (d,k) set has positive Lebesgue measure. We give a short proof of this result which allows for an improved Lp estimate of the corresponding maximal operator, and which demonstrates that a lower value of kcr could be obtained if improved mixed-norm estimates for the x-ray transform were known.
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