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- Title
A Kakeya maximal function estimate in four dimensions using planebrushes.
- Authors
Katz, Nets Hawk; Zahl, Joshua
- Abstract
We obtain an improved Kakeya maximal function estimate in R4 using a new geometric argument called the planebrush. A planebrush is a higher dimensional analogue of Wolff's hairbrush, which gives effective control on the size of Besicovitch sets when the lines through a typical point concentrate into a plane. When Besicovitch sets do not have this property, the existing trilinear estimates of Guth-Zahl can be used to bound the size of a Besicovitch set. In particular, we establish a maximal function estimate in R4 at dimension 3.059. As a consequence, every Besicovitch set in R4 must have Hausdorff dimension at least 3.059.
- Subjects
FRACTAL dimensions; ESTIMATES; MAXIMAL functions; HAIRBRUSHES
- Publication
Revista Mathematica Iberoamericana, 2021, Vol 37, Issue 1, p317
- ISSN
0213-2230
- Publication type
Article
- DOI
10.4171/rmi/1219