{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:MH7GABXEUF67EIVQ3F2UV3JNCC","short_pith_number":"pith:MH7GABXE","schema_version":"1.0","canonical_sha256":"61fe6006e4a17df222b0d9754aed2d108d32add9812aa821509b5dbc17055d8f","source":{"kind":"arxiv","id":"1203.6335","version":3},"attestation_state":"computed","paper":{"title":"Endpoint estimates for commutators of singular integrals related to Schr\\\"odinger operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Luong Dang Ky","submitted_at":"2012-03-28T18:51:10Z","abstract_excerpt":"Let $L= -\\Delta+ V$ be a Schr\\\"odinger operator on $\\mathbb R^d$, $d\\geq 3$, where $V$ is a nonnegative potential, $V\\ne 0$, and belongs to the reverse H\\\"older class $RH_{d/2}$. In this paper, we study the commutators $[b,T]$ for $T$ in a class $\\mathcal K_L$ of sublinear operators containing the fundamental operators in harmonic analysis related to $L$. More precisely, when $T\\in \\mathcal K_L$, we prove that there exists a bounded subbilinear operator $\\mathfrak R= \\mathfrak R_T: H^1_L(\\mathbb R^d)\\times BMO(\\mathbb R^d)\\to L^1(\\mathbb R^d)$ such that $|T(\\mathfrak S(f,b))|- \\mathfrak R(f,b)"},"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":"1203.6335","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-03-28T18:51:10Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"2acf9ef60fd3a8079aa4d6420dcfc36e571d6b58c383238a44406c4a31995901","abstract_canon_sha256":"6ae786d3b6622b916ec497d3e2d65e8f2e00d0f5f7f1f18428dc5a4213dd4529"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:17.136525Z","signature_b64":"91d8JhzrbOieYeEwolA37om7PTby667LhT0EkQws4EUlQTsy9FDxZH0JWfKh1MAXYaQLn+tZwqNZAsaf4BE+Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"61fe6006e4a17df222b0d9754aed2d108d32add9812aa821509b5dbc17055d8f","last_reissued_at":"2026-05-18T02:19:17.135867Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:17.135867Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Endpoint estimates for commutators of singular integrals related to Schr\\\"odinger operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Luong Dang Ky","submitted_at":"2012-03-28T18:51:10Z","abstract_excerpt":"Let $L= -\\Delta+ V$ be a Schr\\\"odinger operator on $\\mathbb R^d$, $d\\geq 3$, where $V$ is a nonnegative potential, $V\\ne 0$, and belongs to the reverse H\\\"older class $RH_{d/2}$. In this paper, we study the commutators $[b,T]$ for $T$ in a class $\\mathcal K_L$ of sublinear operators containing the fundamental operators in harmonic analysis related to $L$. More precisely, when $T\\in \\mathcal K_L$, we prove that there exists a bounded subbilinear operator $\\mathfrak R= \\mathfrak R_T: H^1_L(\\mathbb R^d)\\times BMO(\\mathbb R^d)\\to L^1(\\mathbb R^d)$ such that $|T(\\mathfrak S(f,b))|- \\mathfrak R(f,b)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.6335","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":"1203.6335","created_at":"2026-05-18T02:19:17.135965+00:00"},{"alias_kind":"arxiv_version","alias_value":"1203.6335v3","created_at":"2026-05-18T02:19:17.135965+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.6335","created_at":"2026-05-18T02:19:17.135965+00:00"},{"alias_kind":"pith_short_12","alias_value":"MH7GABXEUF67","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_16","alias_value":"MH7GABXEUF67EIVQ","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_8","alias_value":"MH7GABXE","created_at":"2026-05-18T12:27:14.488303+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/MH7GABXEUF67EIVQ3F2UV3JNCC","json":"https://pith.science/pith/MH7GABXEUF67EIVQ3F2UV3JNCC.json","graph_json":"https://pith.science/api/pith-number/MH7GABXEUF67EIVQ3F2UV3JNCC/graph.json","events_json":"https://pith.science/api/pith-number/MH7GABXEUF67EIVQ3F2UV3JNCC/events.json","paper":"https://pith.science/paper/MH7GABXE"},"agent_actions":{"view_html":"https://pith.science/pith/MH7GABXEUF67EIVQ3F2UV3JNCC","download_json":"https://pith.science/pith/MH7GABXEUF67EIVQ3F2UV3JNCC.json","view_paper":"https://pith.science/paper/MH7GABXE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1203.6335&json=true","fetch_graph":"https://pith.science/api/pith-number/MH7GABXEUF67EIVQ3F2UV3JNCC/graph.json","fetch_events":"https://pith.science/api/pith-number/MH7GABXEUF67EIVQ3F2UV3JNCC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MH7GABXEUF67EIVQ3F2UV3JNCC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MH7GABXEUF67EIVQ3F2UV3JNCC/action/storage_attestation","attest_author":"https://pith.science/pith/MH7GABXEUF67EIVQ3F2UV3JNCC/action/author_attestation","sign_citation":"https://pith.science/pith/MH7GABXEUF67EIVQ3F2UV3JNCC/action/citation_signature","submit_replication":"https://pith.science/pith/MH7GABXEUF67EIVQ3F2UV3JNCC/action/replication_record"}},"created_at":"2026-05-18T02:19:17.135965+00:00","updated_at":"2026-05-18T02:19:17.135965+00:00"}