{"paper":{"title":"Absolute continuity of the harmonic measure on low dimensional rectifiable sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.AP","authors_text":"Joseph Feneuil","submitted_at":"2020-06-04T20:17:49Z","abstract_excerpt":"We consider a uniformly rectifiable set $\\Gamma \\subset \\mathbb R^n$ of dimension $d<n-1$. By using degenerate elliptic operators on the complement $\\Omega = \\mathbb R^n \\setminus \\Gamma$, Guy David, Svitlana Mayboroda, and the author introduced a notion of harmonic measure on $\\Gamma$.\n  We prove in the present article that this harmonic measure on $\\Gamma$ satisfies the $A^\\infty$-property, that is the harmonic measure and the $d$-dimension Hausdorff measure on $\\Gamma$ are mutually absolutely continuous in a quantitative and scale invariant way. Thus, we give an alternate proof of a recent "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.03118","kind":"arxiv","version":4},"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/2006.03118/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"}