{"paper":{"title":"$\\varepsilon$-Approximability of Harmonic Functions in $L^p$ Implies Uniform Rectifiability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Olli Tapiola, Simon Bortz","submitted_at":"2018-01-18T13:02:10Z","abstract_excerpt":"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] (Hofmann-Tapiola) gives us a new characterization of uniform rectifiability which complements the recent results in [HMM] (Hofmann-Martell-Mayboroda), [GMT] (Garnett-Mourgoglou-Tolsa) and [AGMT] (Azzam-Garnett-Mourgoglou-Tolsa)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.05996","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}