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In present paper, we prove that if $b$ is a locally integrable function satisfying $$\\sup_{{\\rm balls}\\; B\\subset \\mathbb R^n} \\frac{\\log(e+ 1/|B|)}{(1+ |B|)^\\theta} \\frac{1}{|B|}\\int_{B} \\Big|f(x)- \\frac{1}{|B|}\\int_{B} f(y) dy\\Big|dx <\\infty$$ for some $\\theta\\in [0,\\infty)$, then the commutator $[b,T]$ is bounded on the local Hardy space $h^1(\\mathbb R^n)$ introduced by Goldberg \\cite{Go}.\n  A"},"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":"1406.7393","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-06-28T12:42:29Z","cross_cats_sorted":[],"title_canon_sha256":"c5c58eeebe01bbd691661a6005638dec0f4905fd184b2086f0b12ed4157b7434","abstract_canon_sha256":"b9e495c750b9d9a7ab235f44d51e3000fa2624c41dfefc6e5ffaf5b43d4250c2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:16.751850Z","signature_b64":"/cAySdQTZSgxlARuiLLApdDJ03akDR+30LzAnMODjMgoKJy3ctDy7WWolhHV36jeyUlQu3LF3b8F3+uQQL/qAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"30374553a7db844acd616667f321228f088f771cf2cbc96c4cc5fcbbf7fbc1cc","last_reissued_at":"2026-05-18T02:19:16.751148Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:16.751148Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An Hardy estimate for commutators of pseudo-differential operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ha Duy Hung, Luong Dang Ky","submitted_at":"2014-06-28T12:42:29Z","abstract_excerpt":"Let $T$ be a pseudo-differential operator whose symbol belongs to the H\\\"ormander class $S^m_{\\rho,\\delta}$ with $0\\leq \\delta<1, 0< \\rho\\leq 1, \\delta \\leq \\rho$ and $-(n+1)< m \\leq - (n+1)(1-\\rho)$. In present paper, we prove that if $b$ is a locally integrable function satisfying $$\\sup_{{\\rm balls}\\; B\\subset \\mathbb R^n} \\frac{\\log(e+ 1/|B|)}{(1+ |B|)^\\theta} \\frac{1}{|B|}\\int_{B} \\Big|f(x)- \\frac{1}{|B|}\\int_{B} f(y) dy\\Big|dx <\\infty$$ for some $\\theta\\in [0,\\infty)$, then the commutator $[b,T]$ is bounded on the local Hardy space $h^1(\\mathbb R^n)$ introduced by Goldberg \\cite{Go}.\n  A"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.7393","kind":"arxiv","version":3},"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":"1406.7393","created_at":"2026-05-18T02:19:16.751248+00:00"},{"alias_kind":"arxiv_version","alias_value":"1406.7393v3","created_at":"2026-05-18T02:19:16.751248+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.7393","created_at":"2026-05-18T02:19:16.751248+00:00"},{"alias_kind":"pith_short_12","alias_value":"GA3UKU5H3OCE","created_at":"2026-05-18T12:28:28.263976+00:00"},{"alias_kind":"pith_short_16","alias_value":"GA3UKU5H3OCEVTLB","created_at":"2026-05-18T12:28:28.263976+00:00"},{"alias_kind":"pith_short_8","alias_value":"GA3UKU5H","created_at":"2026-05-18T12:28:28.263976+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/GA3UKU5H3OCEVTLBMZT7GIJCR4","json":"https://pith.science/pith/GA3UKU5H3OCEVTLBMZT7GIJCR4.json","graph_json":"https://pith.science/api/pith-number/GA3UKU5H3OCEVTLBMZT7GIJCR4/graph.json","events_json":"https://pith.science/api/pith-number/GA3UKU5H3OCEVTLBMZT7GIJCR4/events.json","paper":"https://pith.science/paper/GA3UKU5H"},"agent_actions":{"view_html":"https://pith.science/pith/GA3UKU5H3OCEVTLBMZT7GIJCR4","download_json":"https://pith.science/pith/GA3UKU5H3OCEVTLBMZT7GIJCR4.json","view_paper":"https://pith.science/paper/GA3UKU5H","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1406.7393&json=true","fetch_graph":"https://pith.science/api/pith-number/GA3UKU5H3OCEVTLBMZT7GIJCR4/graph.json","fetch_events":"https://pith.science/api/pith-number/GA3UKU5H3OCEVTLBMZT7GIJCR4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GA3UKU5H3OCEVTLBMZT7GIJCR4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GA3UKU5H3OCEVTLBMZT7GIJCR4/action/storage_attestation","attest_author":"https://pith.science/pith/GA3UKU5H3OCEVTLBMZT7GIJCR4/action/author_attestation","sign_citation":"https://pith.science/pith/GA3UKU5H3OCEVTLBMZT7GIJCR4/action/citation_signature","submit_replication":"https://pith.science/pith/GA3UKU5H3OCEVTLBMZT7GIJCR4/action/replication_record"}},"created_at":"2026-05-18T02:19:16.751248+00:00","updated_at":"2026-05-18T02:19:16.751248+00:00"}