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We then show that our sufficient conditions are applicable to the following cases: (i) $T$ is the Riesz transform or a square function associated with the Laplace-Beltrami operator on a doubling Riemannian manifold, (ii) $T$ is the Riesz transform associated with the magnetic Schr\\\"odinger operator on an Euclidean space, and (iii) $T ="},"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":"1110.1770","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-10-08T21:32:53Z","cross_cats_sorted":[],"title_canon_sha256":"985b76139e6dcd6bb31d04e9dbf1f48c6e71c8af794007e5c6797ad5a8c55af9","abstract_canon_sha256":"d3ebecc31b53e030dbbcbb57c3066fd6921145dcbb181ad728803938af328c7f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:38:23.847926Z","signature_b64":"BrVZWLZX3u8pY/3dmiYFUDGm5wmxEUa0OddjPjwkhVmwB+9PDk2AbNm4HFYbEt9h8Q3Iz/rRB1XeOQUrqSgpBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"087663172ce053d6d2b02752e1294c37fa0f08bcc90ee2908c3f892bd93e32e5","last_reissued_at":"2026-05-18T03:38:23.847327Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:38:23.847327Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Boundedness of singular integrals and their commutators with BMO functions on Hardy spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"The Anh Bui, Xuan Thinh Duong","submitted_at":"2011-10-08T21:32:53Z","abstract_excerpt":"In this paper, we establish sufficient conditions for a singular integral $T$ to be bounded from certain Hardy spaces $H^p_L$ to Lebesgue spaces $L^p$, $0< p \\le 1$, and for the commutator of $T$ and a BMO function to be weak-type bounded on Hardy space $H_L^1$. 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