{"paper":{"title":"A new family of singular integral operators whose $L^2$-boundedness implies rectifiability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Petr Chunaev","submitted_at":"2016-01-27T10:32:43Z","abstract_excerpt":"Let $E \\subset \\mathbb{C}$ be a Borel set such that $0<\\mathcal{H}^1(E)<\\infty$. David and L\\'eger proved that the Cauchy kernel $1/z$ (and even its coordinate parts $\\textrm{Re}\\, z/|z|^2$ and $\\textrm{Im}\\, z/|z|^2$, $z\\in \\mathbb{C}\\setminus\\{0\\}$) has the following property $(*)$: the $L^2(\\mathcal{H}^1\\lfloor E)$-boundedness of the corresponding singular integral operator implies the rectifiability of $E$. Recently Chousionis, Mateu, Prat and Tolsa extended this result to any kernel of the form $(\\textrm{Re}\\, z)^{2n-1}/|z|^{2n}$, $n\\in \\mathbb{N}$. In this paper, we prove that the proper"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.07319","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"}