A remark on a maximal function over a Cantor set of directions
Let \(\Sigma\) be a set of unit vectors in \(\mathbb{R}^2\), and \[ {\mathfrak M}_\Sigma f(x) = \sup_{v \in \Sigma, s > 0} {1 \over 2s} \int_{|t |< s} \bigl |f(x - tv) \bigr |dt \] and \[ {\mathfrak M}_\Sigma^0 f(x) = \sup_{v \in \Sigma} {1 \over 2} \int_{|t |< 1} \bigl |f(x - tv) \bigr |dt. \] In the case \(\Sigma\) is lacunary or uniformly distributed, \({\mathfrak M}_\Sigma\) is treated by Strömberg, Córdoba-R. Fefferman, Nagel-Stein-Wainger, and Córdoba and Strömberg. As for \({\mathfrak M}_\Sigma^0 f(x)\), Wainger showed in 1979 that it is \(L^2\) bounded with bound \(C(1 + \log N)\), if \(\Sigma\) has \(N\) elements. However, when \(\Sigma\) is the set \({\mathfrak C}\) of unit vectors whose slopes belong to the Cantor's ternary set, there is a conjecture that it is still \(L^2\) bounded, like as lacunary case for \({\mathfrak M}_\Sigma\). The authoress gives a step to this direction. Let \({\mathcal A}_0 = \{0,1\}\) and \({\mathcal A}_N = {1 \over 3} {\mathcal A}_{N - 1} \cup [1 - {1 \over 3} {\mathcal A}_{N - 1}]\), \(N = 1,2, \ldots\), and \({\mathfrak A}_k\) be the set of unit vectors whose slopes belong to \(A_k\). She gives a better result than Wainger's: \(|{\mathfrak M}^0_{{\mathfrak A}_n} f |_{L^2} \leq C \sqrt {N + 1} |f |_{L^2}\). As for the case \({\mathfrak C}\), it is bounded from some subsets of \(L^2\) to \(L^2\). These subsets are composed of positive functions whose Fourier transforms have a very weak decay or supported in a horizontal strip.
scientific article; zbMATH DE number 792942