{"paper":{"title":"Necessary conditions for the boundedness of fractional operators on variable Lebesgue spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"David Cruz-Uribe, Troy Roberts","submitted_at":"2024-08-22T22:02:44Z","abstract_excerpt":"In this paper we prove necessary conditions for the boundedness of fractional operators on the variable Lebesgue spaces. More precisely, we find necessary conditions on an exponent function $\\pp$ for a fractional maximal operator $M_\\alpha$ or a non-degenerate fractional singular integral operator $T_\\alpha$, $0 \\leq \\alpha < n$, to satisfy weak $(\\pp,\\qq)$ inequalities or strong $(\\pp,\\qq)$ inequalities, with $\\qq$ being defined pointwise almost everywhere by\n  %\n  \\[ \\frac{1}{p(x)} - \\frac{1}{q(x)} = \\frac{\\alpha}{n}. \\]\n  %\n  We first prove preliminary results linking fractional averaging o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.12745","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2408.12745/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}