Kakeya Sets and Directional Maximal Operators in the Plane

Michael Bateman

Kakeya Sets and Directional Maximal Operators in the Plane

经典分析与常微分方程 2007-05-23 v1 组合数学概率论

作者: Michael Bateman

摘要

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.

关键词

maximal operators

引用

@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}
}

备注

20 pages

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