{"paper":{"title":"Uniform rectifiability from Carleson measure estimates and $\\varepsilon$-approximability of bounded harmonic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"John Garnett, Mihalis Mourgoglou, Xavier Tolsa","submitted_at":"2016-11-01T15:12:12Z","abstract_excerpt":"Let $\\Omega\\subset\\mathbb R^{n+1}$, $n\\geq1$, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that $\\partial\\Omega$ is uniformly $n$-rectifiable if every bounded harmonic function on $\\Omega$ is $\\varepsilon$-approximable or if every bounded harmonic function on $\\Omega$ satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when $\\Omega=\\mathbb R^{n+1}\\setminus E$ and $E$ is Ahlfors-David regular. Our results solve a conjecture posed by Hofmann, Martell, and Mayboroda in a recent work where they proved the conv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.00264","kind":"arxiv","version":3},"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"}