{"paper":{"title":"Rectifiable measures, square functions involving densities, and the Cauchy transform","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Xavier Tolsa","submitted_at":"2014-08-29T10:59:30Z","abstract_excerpt":"This paper is devoted to the proof of two related results. The first one asserts that if $\\mu$ is a Radon measure in $\\mathbb R^d$ satisfying $$\\limsup_{r\\to 0} \\frac{\\mu(B(x,r))}{r}>0\\quad \\text{ and }\\quad \\int_0^1\\left|\\frac{\\mu(B(x,r))}{r} - \\frac{\\mu(B(x,2r))}{2r}\\right|^2\\,\\frac{dr}r< \\infty$$ for $\\mu$-a.e. $x\\in\\mathbb R^d$, then $\\mu$ is rectifiable. Since the converse implication is already known to hold, this yields the following characterization of rectifiable sets: a set $E\\subset\\mathbb R^d$ with finite $1$-dimensional Hausdorff measure $H^1$ is rectifiable if and only $$\\int_0^1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.6979","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"}