Examples and counterexamples to a conjecture in the theory of differentiation of integrals
type: Item
instance of: scholarly article
title: Examples and counterexamples to a conjecture in the theory of differentiation of integrals
zbMATH Open document ID: 0593.42007
DOI: 10.2307/1971350
author: F. Soria
published in: Annals of Mathematics. Second Series
publication date: 1986-01-01T00:00:00Z
review text: Let \(\phi_ 1,\phi_ 2,...,\phi_ n\) be n positive real functions of k variables, increasing separately on each variable and assuming arbitrarily small values. Consider the basis \({\mathcal B}\) of all intervals in \({\mathbb{R}}^ n\), whose side length in the ith direction is determined by the function \(\phi_ i\). It was conjectured that the differentiability properties of such a basis would only depend upon the number of parameters, degrees of freedom in some sense, involved. In particular, it should behave not worse than the basis of all intervals in \({\mathbb{R}}^ k\); that is, it should differentiate the Orlicz class \(L(\log^+L)^{k-1}\), independently of the choice of functions \(\phi_ 1,\phi_ 2,...,\phi_ n.\) In this paper we show that such a wild conjecture is false in general. In fact, we give a counterexample in the simplest possible case, when \(n=3\) and \(k=2\), since the cases \(n\leq 2\), for any k, and \(n\geq 3\), for \(k=1\) or \(k\geq n\), follow by general principles such as Vitali's covering lemma or the classical Jessen-Marcinkiewicz-Zygmund theorem. We also explore with some positive results, the possibility of some aspects of the conjecture remaining true. In all these examples we are forced to assume extra conditions on the \(\phi_ i's\).
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 28A20
zbMATH DE Number: 3953496
zbMATH Keywords: differentiability properties
zbMATH Keywords: Vitali's covering lemma
zbMATH Keywords: Jessen- Marcinkiewicz-Zygmund theorem
zbMATH Keywords: examples
Wikidata QID: Q122947146
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971350
OpenAlex ID: W2328162041
description: scientific article; zbMATH DE number 3953496