Publications
2002
Issue 115

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server end of 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

115/2002

A Rough Lipschitz Function

Bernd Kirchheim and Paul F. X. Müller

Abstract

A real-valued Lipschitz function on the unit interval is constructed such that $\sum_{k = 1}^{\infty} β_{f} (x, 2^{- k}) = \infty,$ holds for {every} $x \in [0, 1]$ . Here $β_{f} (x, 2^{- k})$ measures the distance of $f$ to the best approximating linear functions at scale $2^{- k}$ around $x$ .

This problem is linked to the ongoing efforts to provide geometric understanding for J. Bourgain's results that there exist points $x \in [0, 1],$ at which bounded harmonic functions have finite radial variation.

Contact the author per mail Download full preprint 133 KB

Received:: 22.12.02

Published:: 22.12.02

MSC Codes:: 26A16, 30D55, 26A24, 30C99

Keywords:: radial variation, beta-numbers

Related publications

inJournal

2008 Repository Open Access

Paul F. X. Müller and Bernd Kirchheim

A rough differentiable function

In: Proceedings of the American Mathematical Society, 136 (2008) 11, pp. 3875-3881

BibTex DOI: 10.1090/S0002-9939-08-09322-2