A remark on maximal operators along directions in \(\mathbb R^2\)
Let \(\Omega\) be a subset of \([0,\frac\pi 4)\). The maximal function \(M_\Omega f(x)\) is defined by \[ M_\Omega f(x)=\sup_{x\in R\in \mathcal B_\Omega}|R|^{-1}\int_R |f(y)|dy, \] where \(\mathcal B_\Omega\) denotes the basis of all rectangles in \(\mathbb R^2\) with longest side forming an angle \(\alpha\) with the \(x\)-axis, for some \(\alpha\in \Omega\). Let \(\Omega_0=\{\theta_l; l=1,2,\ldots\}\subset\Omega\) with \(\theta_0=\frac\pi 4>\theta_1>\theta_2>\cdots\geq 0\), and \(\Omega_l=\{\alpha\in \Omega; \theta_l\leq\alpha<\theta_{l-1}\}\). Suppose \(\Omega=\cup_{l}\Omega_l\). The main result is: \[ \|M_\Omega\|_{L^2\to L^{2,\infty}} \leq C_1\sup_l\|M_{\Omega_l}\|_{L^2\to L^{2,\infty}} +C_2\|M_{\Omega_0}\|_{L^2\to L^{2,\infty}}. \] As an application, they give the estimate \(\|M_\Omega f\|_2\leq C(\log \# \Omega)^{\gamma+\frac 12} \|f\|_2\) for some \(\gamma\geq \frac 12\). They also give an extension of a result for lacunary sequences \(\{\theta_l\}\) by Sjögren and Sjölin. As a remark, they comment that they obtain a new result: the above estimate still holds, replacing \(L^2\to L^{2,\infty}\) by the strong \(L^2\to L^2\), and this leads to a simple proof of Katz' result: \(\|M_\Omega f\|_2 \leq C(\log \# \Omega)\|f\|_2\).
scientific article; zbMATH DE number 1918961