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UBC Theses and Dissertations
Kakeya-type sets, lacunarity, and directional maximal operators in Euclidean space Kroc, Edward
Abstract
Given a Cantor-type subset Ω of a smooth curve in ℝ(d+1), we construct random examples of Euclidean sets that contain unit line segments with directions from Ω and enjoy analytical features similar to those of traditional Kakeya sets of infinitesimal Lebesgue measure. We also develop a notion of finite order lacunarity for direction sets in ℝ(d+1), and use it to extend our construction to direction sets Ω that are sublacunary according to this definition. This generalizes to higher dimensions a pair of planar results due to Bateman and Katz [4], [3]. In particular, the existence of such sets implies that the directional maximal operator associated with the direction set Ω is unbounded on Lp(ℝ(d+1)) for all 1 ≤ p < ∞.
Item Metadata
| Title |
Kakeya-type sets, lacunarity, and directional maximal operators in Euclidean space
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| Creator | |
| Publisher |
University of British Columbia
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| Date Issued |
2015
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| Description |
Given a Cantor-type subset Ω of a smooth curve in ℝ(d+1), we construct random examples of Euclidean sets that contain unit line segments with directions from Ω and enjoy analytical features similar to those of traditional Kakeya sets of infinitesimal Lebesgue measure. We also develop a notion of finite order lacunarity for direction sets in ℝ(d+1), and use it to extend our construction to direction sets Ω that are sublacunary according to this definition. This generalizes to higher dimensions a pair of planar results due to Bateman and Katz [4], [3]. In particular, the existence of such sets implies that the directional maximal operator associated with the direction set Ω is unbounded on Lp(ℝ(d+1)) for all 1 ≤ p < ∞.
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| Genre | |
| Type | |
| Language |
eng
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| Date Available |
2015-04-07
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| Provider |
Vancouver : University of British Columbia Library
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| Rights |
Attribution-NonCommercial-NoDerivs 2.5 Canada
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| DOI |
10.14288/1.0166114
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| URI | |
| Degree | |
| Program | |
| Affiliation | |
| Degree Grantor |
University of British Columbia
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| Graduation Date |
2015-05
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| Campus | |
| Scholarly Level |
Graduate
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| Rights URI | |
| Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivs 2.5 Canada