{"paper":{"title":"The two-phase problem for harmonic measure in VMO","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Mart\\'i Prats, Xavier Tolsa","submitted_at":"2019-04-01T12:29:39Z","abstract_excerpt":"Let $\\Omega^+\\subset\\mathbb R^{n+1}$ be an NTA domain and let $\\Omega^-= \\mathbb R^{n+1}\\setminus \\overline{\\Omega^+}$ be an NTA domain as well. Denote by $\\omega^+$ and $\\omega^-$ their respective harmonic measures. Assume that $\\Omega^+$ is a $\\delta$-Reifenberg flat domain for some $\\delta>0$ small enough. In this paper we show that $\\log\\frac{d\\omega^-}{d\\omega^+}\\in VMO(\\omega^+)$ if and only if $\\Omega^+$ is vanishing Reifenberg flat, $\\Omega^+$ and $\\Omega^-$ have joint big pieces of chord-arc subdomains, and the inner unit normal of $\\Omega^+$ has vanishing oscillation with respect to "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.00751","kind":"arxiv","version":6},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1904.00751/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}