Kakeya Sets and Directional Maximal Operators in the Plane
Classical Analysis and ODEs
2007-05-23 v1 Combinatorics
Probability
Abstract
We completely characterize the boundedness of planar directional maximal operators on L^p. More precisely, if Omega is a set of directions, we show that M_Omega, the maximal operator associated to line segments in the directions Omega, is unbounded on L^p, for all p < infinity, precisely when Omega admits Kakeya-type sets. In fact, we show that if Omega does not admit Kakeya sets, then Omega is a generalized lacunary set, and hence M_Omega is bounded on L^p, for p>1.
Keywords
Cite
@article{arxiv.math/0703559,
title = {Kakeya Sets and Directional Maximal Operators in the Plane},
author = {Michael Bateman},
journal= {arXiv preprint arXiv:math/0703559},
year = {2007}
}
Comments
20 pages