{"paper":{"title":"The Cauchy Singular Integral Operator on Weighted Variable Lebesgue Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.FA","authors_text":"Alexei Yu. Karlovich, Ilya M. Spitkovsky","submitted_at":"2012-02-10T10:06:16Z","abstract_excerpt":"Let $p:\\R\\to(1,\\infty)$ be a globally log-H\\\"older continuous variable exponent and $w:\\R\\to[0,\\infty]$ be a weight. We prove that the Cauchy singular integral operator $S$ is bounded on the weighted variable Lebesgue space $L^{p(\\cdot)}(\\R,w)=\\{f:fw\\in L^{p(\\cdot)}(\\R)\\}$ if and only if the weight $w$ satisfies \\[ \\sup_{-\\infty<a<b<\\infty} \\frac{1}{b-a}\\|w\\chi_{(a,b)}\\|_{p(\\cdot)}\\|w^{-1}\\chi_{(a,b)}\\|_{p'(\\cdot)}<\\infty \\quad (1/p(x)+1/p'(x)=1). \\]"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.2226","kind":"arxiv","version":1},"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"}