{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:MH7GABXEUF67EIVQ3F2UV3JNCC","short_pith_number":"pith:MH7GABXE","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"},"canonical_sha256":"61fe6006e4a17df222b0d9754aed2d108d32add9812aa821509b5dbc17055d8f","source":{"kind":"arxiv","id":"1203.6335","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.6335","created_at":"2026-05-18T02:19:17Z"},{"alias_kind":"arxiv_version","alias_value":"1203.6335v3","created_at":"2026-05-18T02:19:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.6335","created_at":"2026-05-18T02:19:17Z"},{"alias_kind":"pith_short_12","alias_value":"MH7GABXEUF67","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"MH7GABXEUF67EIVQ","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"MH7GABXE","created_at":"2026-05-18T12:27:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:MH7GABXEUF67EIVQ3F2UV3JNCC","target":"record","payload":{"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"},"canonical_sha256":"61fe6006e4a17df222b0d9754aed2d108d32add9812aa821509b5dbc17055d8f","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"},"source_kind":"arxiv","source_id":"1203.6335","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:19:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jGOVgt0VVZC0Wv+fPWS17ifo1nEAfXDieERZDd94nGVeronESlXyVa6rzBFI9b0mm8CaCg2fg9+Ie/xVOOIhAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T15:58:51.807645Z"},"content_sha256":"94c247744292e2970b80aa7003e9e22d13f8dfb735b185e0862b84f9bf2ded10","schema_version":"1.0","event_id":"sha256:94c247744292e2970b80aa7003e9e22d13f8dfb735b185e0862b84f9bf2ded10"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:MH7GABXEUF67EIVQ3F2UV3JNCC","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:19:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hb3RNFdVCQ1iUZ5lK6BLZatzNSkHwKI3Lm6KsoO3WhE80t2acYwQevoB6aiEqQQDaSt/u8ntaoecQ4MH8apWAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T15:58:51.808297Z"},"content_sha256":"91f460ba4770597d4d209f2ef8b6e0c96d8874e5a46334826a4a73eacb5c2d95","schema_version":"1.0","event_id":"sha256:91f460ba4770597d4d209f2ef8b6e0c96d8874e5a46334826a4a73eacb5c2d95"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MH7GABXEUF67EIVQ3F2UV3JNCC/bundle.json","state_url":"https://pith.science/pith/MH7GABXEUF67EIVQ3F2UV3JNCC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MH7GABXEUF67EIVQ3F2UV3JNCC/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-21T15:58:51Z","links":{"resolver":"https://pith.science/pith/MH7GABXEUF67EIVQ3F2UV3JNCC","bundle":"https://pith.science/pith/MH7GABXEUF67EIVQ3F2UV3JNCC/bundle.json","state":"https://pith.science/pith/MH7GABXEUF67EIVQ3F2UV3JNCC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MH7GABXEUF67EIVQ3F2UV3JNCC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:MH7GABXEUF67EIVQ3F2UV3JNCC","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"6ae786d3b6622b916ec497d3e2d65e8f2e00d0f5f7f1f18428dc5a4213dd4529","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-03-28T18:51:10Z","title_canon_sha256":"2acf9ef60fd3a8079aa4d6420dcfc36e571d6b58c383238a44406c4a31995901"},"schema_version":"1.0","source":{"id":"1203.6335","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.6335","created_at":"2026-05-18T02:19:17Z"},{"alias_kind":"arxiv_version","alias_value":"1203.6335v3","created_at":"2026-05-18T02:19:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.6335","created_at":"2026-05-18T02:19:17Z"},{"alias_kind":"pith_short_12","alias_value":"MH7GABXEUF67","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"MH7GABXEUF67EIVQ","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"MH7GABXE","created_at":"2026-05-18T12:27:14Z"}],"graph_snapshots":[{"event_id":"sha256:91f460ba4770597d4d209f2ef8b6e0c96d8874e5a46334826a4a73eacb5c2d95","target":"graph","created_at":"2026-05-18T02:19:17Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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)","authors_text":"Luong Dang Ky","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-03-28T18:51:10Z","title":"Endpoint estimates for commutators of singular integrals related to Schr\\\"odinger operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.6335","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:94c247744292e2970b80aa7003e9e22d13f8dfb735b185e0862b84f9bf2ded10","target":"record","created_at":"2026-05-18T02:19:17Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"6ae786d3b6622b916ec497d3e2d65e8f2e00d0f5f7f1f18428dc5a4213dd4529","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-03-28T18:51:10Z","title_canon_sha256":"2acf9ef60fd3a8079aa4d6420dcfc36e571d6b58c383238a44406c4a31995901"},"schema_version":"1.0","source":{"id":"1203.6335","kind":"arxiv","version":3}},"canonical_sha256":"61fe6006e4a17df222b0d9754aed2d108d32add9812aa821509b5dbc17055d8f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"61fe6006e4a17df222b0d9754aed2d108d32add9812aa821509b5dbc17055d8f","first_computed_at":"2026-05-18T02:19:17.135867Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:19:17.135867Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"91d8JhzrbOieYeEwolA37om7PTby667LhT0EkQws4EUlQTsy9FDxZH0JWfKh1MAXYaQLn+tZwqNZAsaf4BE+Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:19:17.136525Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.6335","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:94c247744292e2970b80aa7003e9e22d13f8dfb735b185e0862b84f9bf2ded10","sha256:91f460ba4770597d4d209f2ef8b6e0c96d8874e5a46334826a4a73eacb5c2d95"],"state_sha256":"a2adb2042df8c16e448fa6350dd8c651be28395dfaabcc13010e0752be1dc25d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7HwYqH0OXvwaCdpVHORJ9yI+EcSG4ywDrGURPzaLl2EWRybZaZPR05hZJWm0TExD09XxAdkRIdFElo4g6hr0Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-21T15:58:51.811128Z","bundle_sha256":"6277bed4ba0784bdd767d3065521bb632dbfd602191dde83b32b5fb4db883661"}}