Bordering functions for differential bases
type: Item
instance of: scholarly article
title: Bordering functions for differential bases
zbMATH Open document ID: 0836.42014
DOI: 10.1007/BF01209086
published in: Mathematical Notes
publication date: 1995-09-12T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 795653
zbMATH Keywords: halo function
zbMATH Keywords: differential basis
zbMATH Keywords: maximal function
zbMATH Keywords: bordering function
author: A. A. Soljanik
reviewed by: Petr Gurka
MaRDI profile type: Publication
cites work: Q3895987
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description: scientific article; zbMATH DE number 795653