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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 $$ \sum_{k=1}^\infty \beta_f (x, 2^{-k} ) = \infty ,$$ 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.
