Note on Besicovitch's theorem on the possible values of upper and lower derivatives
type: Item
instance of: scholarly article
title: Note on Besicovitch's theorem on the possible values of upper and lower derivatives
zbMATH Open document ID: 1273.28002
DOI: 10.1134/S0001434613010306
author: Yanyan Li
published in: Mathematical Notes
publication date: 2013-04-26T00:00:00Z
review text: The author proves the following result. For all natural numbers \(k\), \(n_1,\dotsc,n_k\) and bases \(B_i\) in \({\mathbb R}^{n_i}\) that are Busemann-Feller, regular and complete, the basis \(B_1\times \dotsb \times B_k\) possesses the Besicovitch property. Let us recall the definitions of the concepts in the statement. A (differentiation) basis \(B\) on \(\mathbb R^n\) is a mapping defined on \(\mathbb R^n\) such that, for each \(x\in \mathbb R^n\), \(B(x)\) is a family of measurable sets of positive finite measure containing \(x\) such that there exists a sequence \(\{R_k\}\subset B(x)\) with the diameter of \(R_k\) tending to zero as \(k\) approaches infinity. For example \(\mathbb I(\mathbb R^n)\), defined such that for each \(x\in \mathbb R^n\), the family \(\mathbb I(\mathbb R^n)(x)\) consists of all \(n\)-dimensional intervals containing \(x\), is a basis. For a basis \(B\), \(\overline{B}\) shall denote the family \(\cup_{x\in \mathbb R^n} B(x)\). The basis \(B\) is said to be Busemann-Feller if \(R\in B(x)\) for all \(R\in \overline{B}\) and \(x\in R\). Let \(I_1,\dotsc,I_n\) be intervals in \(\mathbb R\) and \(I=I_1\times\dotsb \times I_n\). A basis \(B \subset \mathbb I(\mathbb R^n)\) is said to be regular if \[ \sup_{I\in \overline{B}} \frac{\max_{1\leq i\leq n} |I_i|}{\min_{1\leq i\leq n} |I_i|}<\infty. \] We say that a Busemann-Feller basis is complete if there are numbers \(0<c_1<c_2<1\) such that, for any \(I\in \overline{B}\), there exists a partition \(\{I_1,\dotsc,I_k\}\subset \overline{B}\) of the interval \(I\) with the property \[ c_1\leq \frac{|I_i|}{|I|}\leq c_2\quad i=1,\dotsc,k. \] For \(f\in L_{{\roman{loc}}}({\mathbb R}^n)\), we set \[ \overline{D}_B\Big(\int f,x\Big)=\overline{\lim} \frac{1}{|R|}\int_R f \] and \[ \underline{D}_B\Big(\int f,x\Big)=\underline{\lim}\frac{1}{|R|}\int_R f, \] where the limes inferior and superior are taken with respect to \(R\in B(x)\) with diameter of \(R\) converging to zero. A basis is said to possess th…
reviewed by: Thomas Zürcher
Mathematics Subject Classification ID: 28A15
zbMATH DE Number: 6158185
zbMATH Keywords: Besicovitch's theorem on the values of upper and lower derivatives
zbMATH Keywords: Busemann-Feller basis
zbMATH Keywords: regular differentiation basis
MaRDI profile type: Publication
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full work available at URL: s0001434613010306
OpenAlex ID: W2083298672
description: scientific article; zbMATH DE number 6158185