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When $b\\in BMO(\\omega)$(weighted $BMO$ space) or $b\\in BMO$, the author obtain the necessary and sufficient conditions for the boundedness of $[b,L^{-\\alpha/2}]$ on weighted Morrey spaces respectively."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1203.4407","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/publicdomain/","primary_cat":"math.FA","submitted_at":"2012-03-20T12:24:18Z","cross_cats_sorted":[],"title_canon_sha256":"9e7ab172ffac200ebf5773daee34a078cabec490fdc8413ccbd86d02aee74dfa","abstract_canon_sha256":"cc95a8f879332fdf629e1dcc141864ad8d87a5fc9b9d8e6a5ce46a66644f1d31"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:59:28.761195Z","signature_b64":"4wnJCWnFrbJ6JX+Oy/SWFDkvE10cI7J2mN42xiIWhaiPqIYiAXCyEWo9omDv4veWyGvAhRzIcKHLmEwkbODfCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a8ae0fcd95eb86714a8a8f9929210a4aff057c721f3f57e4d62721a9fe33ccd","last_reissued_at":"2026-05-18T03:59:28.760399Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:59:28.760399Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Commutator Theorems for Fractional Integral Operators on Weighted Morrey Spaces","license":"http://creativecommons.org/licenses/publicdomain/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Zengyan Si","submitted_at":"2012-03-20T12:24:18Z","abstract_excerpt":"Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2(R^n)$ with Gaussican kernel bounds, and let $L^{-\\alpha/2}$ be the fractional integrals of $L$ for $0<\\alpha<n.$ For any locally integrable function $b$, The commutators associated with $L^{-\\alpha/2}$ are defined by $[b,L^{-\\alpha/2}](f)(x)=b(x)L^{-\\alpha/2}(f)(x)-L^{-\\alpha/2}(bf)(x)$. When $b\\in BMO(\\omega)$(weighted $BMO$ space) or $b\\in BMO$, the author obtain the necessary and sufficient conditions for the boundedness of $[b,L^{-\\alpha/2}]$ on weighted Morrey spaces respectively."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.4407","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1203.4407","created_at":"2026-05-18T03:59:28.760529+00:00"},{"alias_kind":"arxiv_version","alias_value":"1203.4407v2","created_at":"2026-05-18T03:59:28.760529+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.4407","created_at":"2026-05-18T03:59:28.760529+00:00"},{"alias_kind":"pith_short_12","alias_value":"JKFOB7GZL24G","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_16","alias_value":"JKFOB7GZL24GOFFI","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_8","alias_value":"JKFOB7GZ","created_at":"2026-05-18T12:27:11.947152+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JKFOB7GZL24GOFFIVD4ZFEQQUS","json":"https://pith.science/pith/JKFOB7GZL24GOFFIVD4ZFEQQUS.json","graph_json":"https://pith.science/api/pith-number/JKFOB7GZL24GOFFIVD4ZFEQQUS/graph.json","events_json":"https://pith.science/api/pith-number/JKFOB7GZL24GOFFIVD4ZFEQQUS/events.json","paper":"https://pith.science/paper/JKFOB7GZ"},"agent_actions":{"view_html":"https://pith.science/pith/JKFOB7GZL24GOFFIVD4ZFEQQUS","download_json":"https://pith.science/pith/JKFOB7GZL24GOFFIVD4ZFEQQUS.json","view_paper":"https://pith.science/paper/JKFOB7GZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1203.4407&json=true","fetch_graph":"https://pith.science/api/pith-number/JKFOB7GZL24GOFFIVD4ZFEQQUS/graph.json","fetch_events":"https://pith.science/api/pith-number/JKFOB7GZL24GOFFIVD4ZFEQQUS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JKFOB7GZL24GOFFIVD4ZFEQQUS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JKFOB7GZL24GOFFIVD4ZFEQQUS/action/storage_attestation","attest_author":"https://pith.science/pith/JKFOB7GZL24GOFFIVD4ZFEQQUS/action/author_attestation","sign_citation":"https://pith.science/pith/JKFOB7GZL24GOFFIVD4ZFEQQUS/action/citation_signature","submit_replication":"https://pith.science/pith/JKFOB7GZL24GOFFIVD4ZFEQQUS/action/replication_record"}},"created_at":"2026-05-18T03:59:28.760529+00:00","updated_at":"2026-05-18T03:59:28.760529+00:00"}