Bordering functions for differential bases
Let \({\mathcal B} = \{\cup {\mathcal B} (x); x \in \mathbb{R}^n\}\) be a differential basis in \(\mathbb{R}^n\). For \(f \in L^1 (\mathbb{R}^n)\) we define the maximal function with respect to the basis \({\mathcal B}\) by \[ M_{\mathcal B} f(x) = \sup_{B \in {\mathcal B} (x)} {1 \over |B |} \int_B |f |. \] The bordering function for the basis \({\mathcal B}\) is defined by \[ \varphi_{\mathcal B} (t) = \sup \left\{ {1 \over |E |} \left |\left\{ M_{{\mathcal B} \chi_E} > {1 \over t} \right\} \right |; \;|E |> 0,\;E \text{ bounded } \right\} \] where \(|E |\) denotes the Lebesgue measure of the set \(E\) and \(\chi_E\) denotes the characteristic function of the set \(E\). The author finds asymptotic behaviors of the functions \(\varphi_{\mathcal B} (t) - 1\), \(t \to 1^+\), for the bases of cubes, rectangles, and centered cubes. In the one-dimensional case he proves that \(\varphi_{\mathcal B} (t) = 2t - 1\) provided \({\mathcal B}\) is the basis of intervals. Remarks of the reviewer: Instead of the term ``bordering'' function it is usually used the term ``halo'' function [see, e.g., \textit{M. de Guzmán}, ``Differentiation of integrals in \(\mathbb{R}^n\)'' (1975; Zbl 0327.26010)].
scientific article; zbMATH DE number 795653
scientific article