A Kakeya maximal function estimate in four dimensions using planebrushes

2019/02/03 by Katz, Nets Hawk, Zahl, Joshua · 3 citations
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1902.00989

Abstract

We obtain an improved Kakeya maximal function estimate and improved Kakeya Hausdorff dimension 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 3.049, and we prove that every Besicovitch set in ℝ⁴ must have Hausdorff dimension at least 3.059.

Cited by

• Restriction and Kakeya maximal estimates in ℝ⁴
• Curved Kakeya sets for generic phases in odd dimensions
• A bound for plany Kakeya sets in \mathbbF_q⁴ using the planebrush method

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