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

Richard Oberlin

Type: Article

Publication Date: 2007-01-01

Citations: 6

DOI: https://doi.org/10.4310/mrl.2007.v14.n1.a7

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.

Locations

Mathematical Research Letters - View - PDF
arXiv (Cornell University) - View

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Works Cited by This (14)

Action	Title	Year	Authors
+ PDF Chat	Norm estimates for a Kakeya‐type maximal operator	2005	Themis Mitsis
+ PDF Chat	$L^{p}$ estimates for the $X$-ray transform	1983	S.W. Drury
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+	An incidence bound for $k$-planes in $F^n$ and a planar variant of the Kakeya maximal function	2006	John Bueti
+ PDF Chat	An improved bound for Kakeya type maximal functions	1995	Thomas Wolff
+ PDF Chat	New bounds for Kakeya problems	2002	N. Katz Terence Tao
+	Lower dimensional integrability of L2 functions	1975	Kennan T. Smith Donald C. Solmon
+	Besicovitch type maximal operators and applications to fourier analysis	1991	Jean Bourgain
+ PDF Chat	Corrigenda: $(n,2)$-sets have full Hausdorff dimension (Rev. Mat. Iberoamericana 20 (2004), no. 2, 381-393)	2005	Themis Mitsis