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

TL;DR

This paper proves that harmonic functions near uniformly rectifiable sets are $ ext{ε}$-approximable in $L^p$, linking geometric regularity with harmonic analysis properties and extending recent theoretical developments.

Contribution

It establishes the equivalence of various $ ext{ε}$-approximability properties and uniform rectifiability for codimension 1 sets, generalizing prior work.

Findings

Harmonic functions are $ ext{ε}$-approximable in $L^p$ for sets with uniform rectifiability.

Various $ ext{ε}$-approximability properties are equivalent and characterize uniform rectifiability.

Results extend recent theoretical frameworks by Hytönen, Rosén, Martell, and Mayboroda.

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 $L^p(\Omega)$ for every $p \in (1,\infty)$, where $\Omega := \mathbb{R}^{n+1} \setminus E$. 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. Hyt\"onen and A. Ros\'en and the first author, J. M. Martell and S. Mayboroda.