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The H1 is defined by square functions of P. A. Meyer's gradient form. Our argument does not rely on any geometric/metric structure of M nor on the kernel of the semigroups of operators. This abstract argument allows to extend our main results to the noncommutative setting, e.g. the case where L"},"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":"1204.5082","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-04-23T15:05:43Z","cross_cats_sorted":["math.FA","math.OA"],"title_canon_sha256":"f6d4df7a1ea6d009150e449b39c43ebbe5ba7122a15a035636f9c7667a87e01e","abstract_canon_sha256":"a68ee57baa2ed9963913facddaa1bffe61a29cddd3e3c8ee476639fe886692e2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:56:44.682027Z","signature_b64":"oJ2ONowANZoFRiHD5BtFFiGyyDgqy8ockPo8AH8VMs99FgAr9v2+rptmsy8sr9yFFPMr6DtUwN3wmkKeNlgmDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3ab20e3e347a1e4bf43d6dcf7088f00ed1717be76f5d0bf02243d013d4cea8c6","last_reissued_at":"2026-05-18T03:56:44.681551Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:56:44.681551Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An H1-BMO duality theory for semigroups of operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.OA"],"primary_cat":"math.CA","authors_text":"Tao Mei","submitted_at":"2012-04-23T15:05:43Z","abstract_excerpt":"Let (M,\\mu) be a sigma-finite measure space. 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