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

2005/12/15 by Richard Oberlin, Oberlin, Richard · 1 citation
Computer Science · Mathematics · #42B25 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.math/0512377

openalex publication_date 2005/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A (d,k) set is a subset of R^d containing a translate of every k-dimensional plane. Bourgain showed that for k ≥ k_cr(d), where k_cr(d) solves 2^k_cr-1+k_cr = 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 k_cr could be obtained if improved mixed-norm estimates for the x-ray transform were known.

Citations

• A recursive bound for a Kakeya-type maximal operator

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• On the packing dimension of unions and extensions of k-planes

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