{"paper":{"title":"The $s$-Riesz transform of an $s$-dimensional measure in $\\R^2$ is unbounded for $1<s<2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.MG"],"primary_cat":"math.AP","authors_text":"Alexander Volberg, Fedor Nazarov, Vladimir Eiderman","submitted_at":"2011-09-10T21:11:35Z","abstract_excerpt":"In this paper, we prove that for $s\\in(1,2)$ there exists no totally lower irregular finite positive Borel measure $\\mu$ in $\\R^2$ with\\break $\\mathcal H^s(\\supp\\mu)<+\\infty$ such that $\\|R\\mu\\|\\ci{L^\\infty(m_2)}<+\\infty$, where $R\\mu=\\mu\\ast\\frac{x}{|x|^{s+1}}$ and $m_2$ is the Lebesgue measure in $\\R^2$. Combined with known results of Prat and Vihtil\\\"a, this shows that for any non-integer $s\\in(0,2)$ and any finite positive Borel measure in $\\R^2$ with $\\mathcal H^s(\\supp\\mu)<+\\infty$, we have $\\|R\\mu\\|\\ci{L^\\infty(m_2)}=\\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.2260","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"}