Maximal operators over arbitrary sets of directions
type: Item
instance of: scholarly article
title: Maximal operators over arbitrary sets of directions
zbMATH Open document ID: 0942.42009
DOI: 10.1215/S0012-7094-99-09702-8
author: Nets Hawk Katz
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: Let \(\Omega\) be a collection of \(N\) directions in \(S^1 \subset R^2\), and let \(M_\Omega\) be the (Kakeya-) Nikodým maximal operator \[ M_\Omega f(x) = \sup_{r > 0, \omega \in \Omega} \frac{1}{2r} \int_{-r}^r |f(x + t\omega)|\;dt. \] It is known that the \(L^2\) operator norm of these maximal operators are not uniformly bounded in \(N\). For instance, when \(\Omega\) is uniformly distributed on the unit cirlcle, then a result of Strömberg shows that the operator norm is comparable to \(\log N\). The author extends this to the case when \(\Omega\) is arbitrary. (The best bound that was previously known was of Barrionuevo, who showed that this operator was bounded by \(O(N^{2/\sqrt{\log N}}).\)) The author also shows the slightly stronger estimate, that the restricted \(L^2\) norm of this operator is bounded by \(\sqrt{\log N}\). The argument is based on \(TT^*\) methods and the theory of Carleson measures in the two-parameter setting. An interesting feature in the argument is that estimates on \(\mathbb R^2\) are ``lifted'' to estimates on the space of all dyadic line segments in \(\mathbb R^2\), which is a more tractable space for the analysis.
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 1425195
zbMATH Keywords: Kakeya maximal operator
zbMATH Keywords: Nikodým maximal operator
zbMATH Keywords: Besicovitch sets
reviewed by: Terence Tao
MaRDI profile type: Publication
cites work: A note on the Kakeya maximal operator
cites work: A continuous version of duality of \(H^ 1\) with BMO on the bidisc
cites work: On the equivalence between the boundedness of certain classes of maximal and multiplier operators in Fourier analysis
cites work: Maximal functions associated to rectangles with uniformly distributed directions
full work available at URL: s0012-7094-99-09702-8
OpenAlex ID: W2002188470
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description: scientific article; zbMATH DE number 1425195