$\varepsilon$-Approximability of Harmonic Functions in $L^p$ Implies Uniform Rectifiability

Abstract: Suppose that $\Omega \subset \mathbb{R}^{n+1}$, $n \ge 2$, is an open set satisfying the corkscrew condition with an $n$-dimensional ADR boundary, $\partial \Omega$. In this note, we show that if harmonic functions are $\varepsilon$-approximable in $L^p$ for any $p > n/(n-1)$, then $\partial \Omega$ is uniformly rectifiable. Combining our results with those in HT gives us a new characterization of uniform rectifiability which complements the recent results in HMM, GMT and AGMT.