{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:IDPK665DS4XRZPUC7A7ETBKFSM","short_pith_number":"pith:IDPK665D","schema_version":"1.0","canonical_sha256":"40deaf7ba3972f1cbe82f83e498545931c0d233266e637369352a9ccb4549342","source":{"kind":"arxiv","id":"1606.01064","version":2},"attestation_state":"computed","paper":{"title":"Hardy spaces for semigroups with Gaussian bounds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.FA","authors_text":"Jacek Dziuba\\'nski, Marcin Preisner","submitted_at":"2016-06-03T12:40:47Z","abstract_excerpt":"Let T_t=e^{-tL} be a semigroup of self-adjoint linear operators acting on L^2(X,mu), where (X,d mu) is a space of homogeneous type. We assume that T_t has an integral kernel T_t(x,y) which satisfies the upper and lower Gaussian bounds: \\frac{C_1}{mu(B(x,\\sqrt{t}))} \\exp(-c_1d(x,y)^2/t)\\leq T_t(x,y) \\leq \\frac{C_2}{\\mu(B(x,\\sqrt{t}))} \\exp(-c_2 d(x,y)^2/t). By definition, f belongs to H^1_L if \\| f\\|_{H^1_L}=\\|\\sup_{t>0}|T_t f(x)|\\|_{L^1(X,\\mu)} <\\infty. We prove that there is a function \\omega(x), 0<c \\leq \\omega(x) \\leq C, such that H^1_L admits an atomic decomposition with atoms satisfying: "},"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":"1606.01064","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-06-03T12:40:47Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"f48b74f4ad550649fb0c6dab221e9d7a2f51fcb185e62eb179a6a01db3d6dc68","abstract_canon_sha256":"1b630b8d3e6ba8219de873a90c65900aa6b8bd2c4c74d22bbf36aa7b6fbb53c0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:34.831329Z","signature_b64":"yAgv5uddfpCU33oaC5V7aibdczXmDmFxZ8zhBk7m30cnQdrU0bzkWG+YFQpyTYkrVz3cOFehHyVBSsrSJRoZDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"40deaf7ba3972f1cbe82f83e498545931c0d233266e637369352a9ccb4549342","last_reissued_at":"2026-05-18T00:45:34.830457Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:34.830457Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hardy spaces for semigroups with Gaussian bounds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.FA","authors_text":"Jacek Dziuba\\'nski, Marcin Preisner","submitted_at":"2016-06-03T12:40:47Z","abstract_excerpt":"Let T_t=e^{-tL} be a semigroup of self-adjoint linear operators acting on L^2(X,mu), where (X,d mu) is a space of homogeneous type. We assume that T_t has an integral kernel T_t(x,y) which satisfies the upper and lower Gaussian bounds: \\frac{C_1}{mu(B(x,\\sqrt{t}))} \\exp(-c_1d(x,y)^2/t)\\leq T_t(x,y) \\leq \\frac{C_2}{\\mu(B(x,\\sqrt{t}))} \\exp(-c_2 d(x,y)^2/t). By definition, f belongs to H^1_L if \\| f\\|_{H^1_L}=\\|\\sup_{t>0}|T_t f(x)|\\|_{L^1(X,\\mu)} <\\infty. We prove that there is a function \\omega(x), 0<c \\leq \\omega(x) \\leq C, such that H^1_L admits an atomic decomposition with atoms satisfying: "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.01064","kind":"arxiv","version":2},"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":"1606.01064","created_at":"2026-05-18T00:45:34.830828+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.01064v2","created_at":"2026-05-18T00:45:34.830828+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.01064","created_at":"2026-05-18T00:45:34.830828+00:00"},{"alias_kind":"pith_short_12","alias_value":"IDPK665DS4XR","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_16","alias_value":"IDPK665DS4XRZPUC","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_8","alias_value":"IDPK665D","created_at":"2026-05-18T12:30:22.444734+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/IDPK665DS4XRZPUC7A7ETBKFSM","json":"https://pith.science/pith/IDPK665DS4XRZPUC7A7ETBKFSM.json","graph_json":"https://pith.science/api/pith-number/IDPK665DS4XRZPUC7A7ETBKFSM/graph.json","events_json":"https://pith.science/api/pith-number/IDPK665DS4XRZPUC7A7ETBKFSM/events.json","paper":"https://pith.science/paper/IDPK665D"},"agent_actions":{"view_html":"https://pith.science/pith/IDPK665DS4XRZPUC7A7ETBKFSM","download_json":"https://pith.science/pith/IDPK665DS4XRZPUC7A7ETBKFSM.json","view_paper":"https://pith.science/paper/IDPK665D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.01064&json=true","fetch_graph":"https://pith.science/api/pith-number/IDPK665DS4XRZPUC7A7ETBKFSM/graph.json","fetch_events":"https://pith.science/api/pith-number/IDPK665DS4XRZPUC7A7ETBKFSM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IDPK665DS4XRZPUC7A7ETBKFSM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IDPK665DS4XRZPUC7A7ETBKFSM/action/storage_attestation","attest_author":"https://pith.science/pith/IDPK665DS4XRZPUC7A7ETBKFSM/action/author_attestation","sign_citation":"https://pith.science/pith/IDPK665DS4XRZPUC7A7ETBKFSM/action/citation_signature","submit_replication":"https://pith.science/pith/IDPK665DS4XRZPUC7A7ETBKFSM/action/replication_record"}},"created_at":"2026-05-18T00:45:34.830828+00:00","updated_at":"2026-05-18T00:45:34.830828+00:00"}