{"paper":{"title":"Calder\\'on-Zygmund kernels and rectifiability in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Joan Mateu, Laura Prat, Vasilis Chousionis, Xavier Tolsa","submitted_at":"2011-10-06T15:43:16Z","abstract_excerpt":"Let $E \\subset \\C$ be a Borel set with finite length, that is, $0<\\mathcal{H}^1 (E)<\\infty$. By a theorem of David and L\\'eger, the $L^2 (\\mathcal{H}^1 \\lfloor E)$-boundedness of the singular integral associated to the Cauchy kernel (or even to one of its coordinate parts $x / |z|^2,y / |z|^2,z=(x,y) \\in \\C$) implies that $E$ is rectifiable. We extend this result to any kernel of the form $x^{2n-1} /|z|^{2n}, z=(x,y) \\in \\C,n \\in \\mathbb{N}$. We thus provide the first non-trivial examples of operators not directly related with the Cauchy transform whose $L^2$-boundedness implies rectifiability"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.1302","kind":"arxiv","version":3},"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"}