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In this paper, we mainly give some necessary and sufficient conditions for the boundedness of $[b,M_{\\alpha}]$ on variable Lebesgue spaces when $b$ belongs to Lipschitz or $BMO(\\rn)$ spaces, by which some new characterizations for certain subclasses of Lipschitz and $BMO(\\rn)$ spaces are obtained."},"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":"1901.06835","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-01-21T09:24:10Z","cross_cats_sorted":[],"title_canon_sha256":"9d6205f0c2048057472be605ce5ccb8e5a7a539ec4406ea9280f3606827382f2","abstract_canon_sha256":"666b8de2287976c54a6a393970ac5e7efffef4e4d879094172e426fefff7a32d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:49.005806Z","signature_b64":"eDdjIoHXnCo9Ch9jRbIdSY1fjcuQ5/xtWZZ3fIA89PpXTTXnJ+thvK7oUzi/WoyT/HnXAaoMXmzruaHn2T4+AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"833dc959fdcc47e2188cbe0b468d98697d1b6460d4c519fc50cee0fa7b4209f3","last_reissued_at":"2026-05-17T23:55:49.005174Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:49.005174Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some notes on commutators of the fractional maximal function on variable Lebesgue spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Jianglong Wu, Pu Zhang, Zengyan Si","submitted_at":"2019-01-21T09:24:10Z","abstract_excerpt":"Let $0<\\alpha<n$ and $M_{\\alpha}$ be the fractional maximal function. 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