{"paper":{"title":"Necessary and sufficient conditions for boundedness of commutators of the general fractional integral operators on weighted Morrey spaces","license":"http://creativecommons.org/licenses/publicdomain/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Fayou Zhao, Zengyan Si","submitted_at":"2012-03-20T08:24:56Z","abstract_excerpt":"We prove that $b$ is in $Lip_{\\bz}(\\bz)$ if and only if the commutator $[b,L^{-\\alpha/2}]$ of the multiplication operator by $b$ and the general fractional integral operator $L^{-\\alpha/2}$ is bounded from the weighed Morrey space $L^{p,k}(\\omega)$ to $L^{q,kq/p}(\\omega^{1-(1-\\alpha/n)q},\\omega)$, where $0<\\beta<1$, $0<\\alpha+\\beta<n, 1<p<{n}/({\\alpha+\\beta})$, ${1}/{q}={1}/{p}-{(\\alpha+\\beta)}/{n},$ $0\\leq k<{p}/{q},$ $\\omega^{{q}/{p}}\\in A_1$ and $ r_\\omega> \\frac{1-k}{p/q-k},$ and here $r_\\omega$ denotes the critical index of $\\omega$ for the reverse H\\\"{o}lder condition."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.4337","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"}