Maximal operators over arbitrary sets of directions
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.
scientific article; zbMATH DE number 1425195