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MiS Preprint

A Rough Lipschitz Function

Bernd Kirchheim and Paul F. X. Müller


A real-valued Lipschitz function on the unit interval is constructed such that $$ \sum_{k=1}^\infty \beta_f (x, 2^{-k} ) = \infty ,$$ holds for {every} $x \in [0,1]$. Here $ \beta_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.

MSC Codes:
26A16, 30D55, 26A24, 30C99
radial variation, beta-numbers

Related publications

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