Uniform rectifiability and $\varepsilon$-approximability of harmonic functions in $L^p$

Abstract

Suppose that $E \subset \mathbb{R}^{n+1}$ is a uniformly rectifiable set of codimension $1$. We show that every harmonic function is $\varepsilon$-approximable in $L^p(Ω)$ for every $p \in (1,\infty)$, where $Ω:= \mathbb{R}^{n+1} \setminus E$. Together with results of many authors this shows that pointwise, $L^\infty$ and $L^p$ type $\varepsilon$-approximability properties of harmonic functions are all equivalent and they characterize uniform rectifiability for codimension $1$ Ahlfors-David regular sets. Our results and techniques are generalizations of recent works of T. Hytönen and A. Rosén and the first author, J. M. Martell and S. Mayboroda.

34 pages. v2: accepted version; introduction updated, details added and some proofs re-written. To appear in Annales de l'Institut Fourier