{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:IJ2YGQDAEG3IKVSTJHC4JTP4SZ","short_pith_number":"pith:IJ2YGQDA","schema_version":"1.0","canonical_sha256":"427583406021b685565349c5c4cdfc964593920eab21fa47ce9285611ee7eade","source":{"kind":"arxiv","id":"1310.2262","version":1},"attestation_state":"computed","paper":{"title":"A characterization of Hardy spaces associated with certain Schr\\\"odinger operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Jacek Dziuba\\'nski, Jacek Zienkiewicz","submitted_at":"2013-10-08T20:12:44Z","abstract_excerpt":"Let $\\{K_t\\}_{t>0}$ be the semigroup of linear operators generated by a Schr\\\"odinger operator $-L=\\Delta - V(x)$ on $\\mathbb R^d$, $d\\geq 3$, where $V(x)\\geq 0$ satisfies $\\Delta^{-1} V\\in L^\\infty$. We say that an $L^1$-function $f$ belongs to the Hardy space $H^1_L$ if the maximal function $\\mathcal M_L f(x) = \\sup_{t>0} |K_tf(x)|$ belongs to $L^1(\\mathbb R^d) $. We prove that the operator $(-\\Delta)^{1\\slash 2} L^{-1\\slash 2}$ is an isomorphism of the space $H^1_L$ with the classical Hardy space $H^1(\\mathbb R^d)$ whose inverse is $L^{1\\slash 2} (-\\Delta)^{-1\\slash 2}$. As a corollary we o"},"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":"1310.2262","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-10-08T20:12:44Z","cross_cats_sorted":[],"title_canon_sha256":"a651fd47daa1f21def839116504b8c2c4a84258fdd26fe5d7ecaf6e2b0f41042","abstract_canon_sha256":"b94f8dffb7964d5f538469513bf41b255d2340b362d61b1fe6eb59e8dbf8d040"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:10:57.481158Z","signature_b64":"voxfz0E8JIeJrnUwAOyM3NzAO8nS80RCwvhVHBcVFXAj8/3KzE9exOf+NGNAd9VjTzOgNcj/rQt5antWN0SXAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"427583406021b685565349c5c4cdfc964593920eab21fa47ce9285611ee7eade","last_reissued_at":"2026-05-18T03:10:57.480406Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:10:57.480406Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A characterization of Hardy spaces associated with certain Schr\\\"odinger operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Jacek Dziuba\\'nski, Jacek Zienkiewicz","submitted_at":"2013-10-08T20:12:44Z","abstract_excerpt":"Let $\\{K_t\\}_{t>0}$ be the semigroup of linear operators generated by a Schr\\\"odinger operator $-L=\\Delta - V(x)$ on $\\mathbb R^d$, $d\\geq 3$, where $V(x)\\geq 0$ satisfies $\\Delta^{-1} V\\in L^\\infty$. We say that an $L^1$-function $f$ belongs to the Hardy space $H^1_L$ if the maximal function $\\mathcal M_L f(x) = \\sup_{t>0} |K_tf(x)|$ belongs to $L^1(\\mathbb R^d) $. We prove that the operator $(-\\Delta)^{1\\slash 2} L^{-1\\slash 2}$ is an isomorphism of the space $H^1_L$ with the classical Hardy space $H^1(\\mathbb R^d)$ whose inverse is $L^{1\\slash 2} (-\\Delta)^{-1\\slash 2}$. As a corollary we o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.2262","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":"1310.2262","created_at":"2026-05-18T03:10:57.480546+00:00"},{"alias_kind":"arxiv_version","alias_value":"1310.2262v1","created_at":"2026-05-18T03:10:57.480546+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.2262","created_at":"2026-05-18T03:10:57.480546+00:00"},{"alias_kind":"pith_short_12","alias_value":"IJ2YGQDAEG3I","created_at":"2026-05-18T12:27:46.883200+00:00"},{"alias_kind":"pith_short_16","alias_value":"IJ2YGQDAEG3IKVST","created_at":"2026-05-18T12:27:46.883200+00:00"},{"alias_kind":"pith_short_8","alias_value":"IJ2YGQDA","created_at":"2026-05-18T12:27:46.883200+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/IJ2YGQDAEG3IKVSTJHC4JTP4SZ","json":"https://pith.science/pith/IJ2YGQDAEG3IKVSTJHC4JTP4SZ.json","graph_json":"https://pith.science/api/pith-number/IJ2YGQDAEG3IKVSTJHC4JTP4SZ/graph.json","events_json":"https://pith.science/api/pith-number/IJ2YGQDAEG3IKVSTJHC4JTP4SZ/events.json","paper":"https://pith.science/paper/IJ2YGQDA"},"agent_actions":{"view_html":"https://pith.science/pith/IJ2YGQDAEG3IKVSTJHC4JTP4SZ","download_json":"https://pith.science/pith/IJ2YGQDAEG3IKVSTJHC4JTP4SZ.json","view_paper":"https://pith.science/paper/IJ2YGQDA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1310.2262&json=true","fetch_graph":"https://pith.science/api/pith-number/IJ2YGQDAEG3IKVSTJHC4JTP4SZ/graph.json","fetch_events":"https://pith.science/api/pith-number/IJ2YGQDAEG3IKVSTJHC4JTP4SZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IJ2YGQDAEG3IKVSTJHC4JTP4SZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IJ2YGQDAEG3IKVSTJHC4JTP4SZ/action/storage_attestation","attest_author":"https://pith.science/pith/IJ2YGQDAEG3IKVSTJHC4JTP4SZ/action/author_attestation","sign_citation":"https://pith.science/pith/IJ2YGQDAEG3IKVSTJHC4JTP4SZ/action/citation_signature","submit_replication":"https://pith.science/pith/IJ2YGQDAEG3IKVSTJHC4JTP4SZ/action/replication_record"}},"created_at":"2026-05-18T03:10:57.480546+00:00","updated_at":"2026-05-18T03:10:57.480546+00:00"}