{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:B62JARZVQTODLSZKW76XMDH5OA","short_pith_number":"pith:B62JARZV","schema_version":"1.0","canonical_sha256":"0fb490473584dc35cb2ab7fd760cfd701a90ff35deb0527d0d09703e6976c05b","source":{"kind":"arxiv","id":"1409.4612","version":1},"attestation_state":"computed","paper":{"title":"Atomic decompositions for Hardy spaces related to Schr\\\"odinger operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Marcin Preisner","submitted_at":"2014-09-16T12:46:38Z","abstract_excerpt":"Let L_U = -Delta+U be a Schr\\\"odinger operator on R^d, where U\\in L^1_{loc}(R^d) is a non-negative potential and d\\geq 3. The Hardy space H^1(L_U) is defined in terms of the maximal function for the semigroup K_{t,U} = exp(-t L_U), namely H^1(L_U) = {f\\in L^1(R^d): \\|f\\|_{H^1(L_U)}:= \\|sup_{t>0} |K_{t,U} f| \\|_{L^1(R^d)} < \\infty. Assume that U=V+W, where V\\geq 0 satisfies the global Kato condition sup_{x\\in R^d} \\int_{R^d} V(y)|x-y|^{2-d} < \\infty. We prove that, under certain assumptions on W\\geq 0, the space H^1(L_U) admits an atomic decomposition of local type. An atom a for H^1(L_U) is ei"},"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":"1409.4612","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-09-16T12:46:38Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"765dbf02a89a35a860e509765b0b03e6620e15f9c246e438cb65ba4eabd6361a","abstract_canon_sha256":"f7e4ed74a6850d2cb1f944e64a3faafb56865eb358335512d9d141f6419a337e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:45.239261Z","signature_b64":"RUr4FID4xrf2DZK9j3MGXELPhltWjiLqhXUw2uM3kdlw+68XRs7vFOPRCqW14Dvva/5v6TLboPvYNPgV3/pIDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0fb490473584dc35cb2ab7fd760cfd701a90ff35deb0527d0d09703e6976c05b","last_reissued_at":"2026-05-18T02:42:45.238782Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:45.238782Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Atomic decompositions for Hardy spaces related to Schr\\\"odinger operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Marcin Preisner","submitted_at":"2014-09-16T12:46:38Z","abstract_excerpt":"Let L_U = -Delta+U be a Schr\\\"odinger operator on R^d, where U\\in L^1_{loc}(R^d) is a non-negative potential and d\\geq 3. The Hardy space H^1(L_U) is defined in terms of the maximal function for the semigroup K_{t,U} = exp(-t L_U), namely H^1(L_U) = {f\\in L^1(R^d): \\|f\\|_{H^1(L_U)}:= \\|sup_{t>0} |K_{t,U} f| \\|_{L^1(R^d)} < \\infty. Assume that U=V+W, where V\\geq 0 satisfies the global Kato condition sup_{x\\in R^d} \\int_{R^d} V(y)|x-y|^{2-d} < \\infty. We prove that, under certain assumptions on W\\geq 0, the space H^1(L_U) admits an atomic decomposition of local type. An atom a for H^1(L_U) is ei"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.4612","kind":"arxiv","version":1},"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":"1409.4612","created_at":"2026-05-18T02:42:45.238849+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.4612v1","created_at":"2026-05-18T02:42:45.238849+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.4612","created_at":"2026-05-18T02:42:45.238849+00:00"},{"alias_kind":"pith_short_12","alias_value":"B62JARZVQTOD","created_at":"2026-05-18T12:28:22.404517+00:00"},{"alias_kind":"pith_short_16","alias_value":"B62JARZVQTODLSZK","created_at":"2026-05-18T12:28:22.404517+00:00"},{"alias_kind":"pith_short_8","alias_value":"B62JARZV","created_at":"2026-05-18T12:28:22.404517+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/B62JARZVQTODLSZKW76XMDH5OA","json":"https://pith.science/pith/B62JARZVQTODLSZKW76XMDH5OA.json","graph_json":"https://pith.science/api/pith-number/B62JARZVQTODLSZKW76XMDH5OA/graph.json","events_json":"https://pith.science/api/pith-number/B62JARZVQTODLSZKW76XMDH5OA/events.json","paper":"https://pith.science/paper/B62JARZV"},"agent_actions":{"view_html":"https://pith.science/pith/B62JARZVQTODLSZKW76XMDH5OA","download_json":"https://pith.science/pith/B62JARZVQTODLSZKW76XMDH5OA.json","view_paper":"https://pith.science/paper/B62JARZV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.4612&json=true","fetch_graph":"https://pith.science/api/pith-number/B62JARZVQTODLSZKW76XMDH5OA/graph.json","fetch_events":"https://pith.science/api/pith-number/B62JARZVQTODLSZKW76XMDH5OA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/B62JARZVQTODLSZKW76XMDH5OA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/B62JARZVQTODLSZKW76XMDH5OA/action/storage_attestation","attest_author":"https://pith.science/pith/B62JARZVQTODLSZKW76XMDH5OA/action/author_attestation","sign_citation":"https://pith.science/pith/B62JARZVQTODLSZKW76XMDH5OA/action/citation_signature","submit_replication":"https://pith.science/pith/B62JARZVQTODLSZKW76XMDH5OA/action/replication_record"}},"created_at":"2026-05-18T02:42:45.238849+00:00","updated_at":"2026-05-18T02:42:45.238849+00:00"}