{"paper":{"title":"The Riesz transform, rectifiability, and removability for Lipschitz harmonic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Alexander Volberg, Fedor Nazarov, Xavier Tolsa","submitted_at":"2012-12-21T13:43:30Z","abstract_excerpt":"We show that, given a set $E\\subset \\mathbb R^{n+1}$ with finite $n$-Hausdorff measure $H^n$, if the $n$-dimensional Riesz transform $$R_{H^n|E} f(x) = \\int_{E} \\frac{x-y}{|x-y|^{n+1}} f(y) dH^n(y)$$ is bounded in $L^2(H^n|E)$, then $E$ is $n$-rectifiable. From this result we deduce that a compact set $E\\subset\\mathbb R^{n+1}$ with $H^n(E)<\\infty$ is removable for Lipschitz harmonic functions if and only if it is purely $n$-unrectifiable, thus proving the analog of Vitushkin's conjecture in higher dimensions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.5431","kind":"arxiv","version":2},"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"}