Bounds for Kakeya-type maximal operators associated with $k$-planes
نویسندگان
چکیده
منابع مشابه
BOUNDS FOR KAKEYA-TYPE MAXIMAL OPERATORS ASSOCIATED WITH k-PLANES
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 imp...
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We completely characterize the boundedness of planar directional maximal operators on L. More precisely, if Ω is a set of directions, we show that MΩ, the maximal operator associated to line segments in the directions Ω, is unbounded on L, for all p < ∞, precisely when Ω admits Kakeya-type sets. In fact, we show that if Ω does not admit Kakeya sets, then Ω is a generalized lacunary set, and hen...
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We prove that the bound on the L norms of the Kakeya type maximal functions studied by Cordoba [2], and by Bourgain [1] are sharp for p > 2. The proof is based on a construction originally due to Schoenberg [5], for which we provide an alternative derivation. We also show that r log(1/r) is the exact Minkowski dimension of the class of Kakeya sets in R, and prove that the exact Hausdorff dimens...
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ژورنال
عنوان ژورنال: Mathematical Research Letters
سال: 2007
ISSN: 1073-2780,1945-001X
DOI: 10.4310/mrl.2007.v14.n1.a7