{"paper":{"title":"The Riesz transform and quantitative rectifiability for general Radon measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Daniel Girela-Sarri\\'on, Xavier Tolsa","submitted_at":"2016-01-29T12:26:48Z","abstract_excerpt":"In this paper we show that if $\\mu$ is a Borel measure in $\\mathbb R^{n+1}$ with growth of order $n$, so that the $n$-dimensional Riesz transform $R_\\mu$ is bounded in $L^2(\\mu)$, and $B\\subset\\mathbb R^{n+1}$ is a ball with $\\mu(B)\\approx r(B)^n$ such that: (a) there is some $n$-plane $L$ passing through the center of $B$ such that for some $\\delta>0$ small enough, it holds $\\int_B \\frac{dist(x,L)}{r(B)}\\,d\\mu(x)\\leq \\delta\\,\\mu(B),$ (b) for some constant $\\epsilon>0$ small enough, $\\int_B |R_\\mu1(x) - m_{\\mu,B}(R_\\mu1)|^2\\,d\\mu(x) \\leq \\epsilon \\,\\mu(B)$, where $m_{\\mu,B}(R_\\mu1)$ stands for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.08079","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":""},"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"}