\(L^{2}\) bounds for a Kakeya-type maximal operator in \(\mathbb R^{3}\)
Let \(F : \mathbb{R}^d \rightarrow \mathbb{R}\) for some \(d \geq 2\), and let \(\Sigma \subset \mathbb{S}^{d-1}\) be a collection of \(N\) unit vectors. Consider the maximal operator defined by NEWLINE\[NEWLINE \mathcal{M}_{\Sigma}^{*}F(\mathrm{x}) = \max_{\mathrm{v} \in \Sigma} \left|\int_{-1/2}^{1/2} F(\mathrm{x}+ t\mathrm{v}) \, dt \right|, NEWLINE\]NEWLINE where \(\mathrm{x} = (x_1, x_2, \dotsc, x_d)\) and \(\mathrm{v} = (v_1, v_2, \dotsc, v_d)\).NEWLINENEWLINEIn this paper, two sharp \(L^2\) bounds for the maximal operator \(\mathcal{M}_{\Sigma}^{*}F\) in \(\mathbb{R}^3\) are proved.NEWLINENEWLINETheorem 1. Let \(\Sigma \subset \mathbb{S}^{d-1}\) be any collection of \(N\) unit vectors. For \(F \in L^2\left(\mathbb{R}^3\right)\), we have NEWLINE\[NEWLINE\|\mathcal{M}_{\Sigma}^{*}F \|_{L^2\left(\mathbb{R}^3\right)} \;\leq \;C \, N^{1/4}\log N \|F\|_{L^2\left(\mathbb{R}^3\right)}.NEWLINE\]NEWLINE Theorem 2. Let \(\Sigma \subset \mathbb{S}^{d-1}\) be a collection of \(N\) unit vectors such that \(\|\mathrm{v} - \mathrm{v}' \| \;\geq \;C \, N^{-1/2}\) for each \(\mathrm{v} \neq \mathrm{v}' \in \Sigma\). Then, for each \(F \in L^2\left(\mathbb{R}^3\right)\), we have NEWLINE\[NEWLINE\|\mathcal{M}_{\Sigma}^{*}F \|_{L^2\left(\mathbb{R}^3\right)} \;\leq \;C \, N^{1/4} \sqrt{\log N} \|F\|_{L^2\left(\mathbb{R}^3\right)}. NEWLINE\]NEWLINE It is also pointed out that there are no known nontrivial results for the critical case of \(L^3\left(\mathbb{R}^3\right)\)-bounds.
scientific article; zbMATH DE number 6064194