Kakeya sets and directional maximal operators in the plane
Bateman, Michael
Duke Math. J., Tome 146 (2009) no. 1, p. 55-77 / Harvested from Project Euclid
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 $p \lt \infty$ 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 $p>1$
Publié le : 2009-03-15
Classification:  42B25,  60K35
@article{1235657188,
     author = {Bateman, Michael},
     title = {Kakeya sets and directional maximal operators in the plane},
     journal = {Duke Math. J.},
     volume = {146},
     number = {1},
     year = {2009},
     pages = { 55-77},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1235657188}
}
Bateman, Michael. Kakeya sets and directional maximal operators in the plane. Duke Math. J., Tome 146 (2009) no. 1, pp.  55-77. http://gdmltest.u-ga.fr/item/1235657188/