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We assume that each V_j depends on variables from a linear subspace VV_j of \\Rn, dim VV_j \\geq 3, and V_j belongs to L^q(VV_j) for certain q. We prove that there exist two distinct isomorphisms of H^1_L with the classical Hardy space. As a corollary we deduce a specific atomic characterization of H_L^1. We also prove that the space H_L^1 is described by means of the Riesz transforms R_{L,i} = \\partial_i L^{-1/2}."},"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":"1109.5626","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-09-26T16:25:55Z","cross_cats_sorted":[],"title_canon_sha256":"20df461e7ac1b3a7a79b4bf8f62dc64e3f12305c408d35e98b72a05777b4487a","abstract_canon_sha256":"737ec8ea85e4fc8b6956e36debd9842a966bc73a574c3a71218b406e42874be8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:12:19.070372Z","signature_b64":"Uc6BdnMiDF8HbTJxNewnAIUgoA8dUy3Ejquz4t+8AIe5HyhfGx7DHD2optKGSU9A/28hnM7gQTol0ipQMYlxDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe6b1e4ca2d206801f022676c890b0fc4f1135cb34cecce7bf80a2ea6dd326b6","last_reissued_at":"2026-05-18T04:12:19.069958Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:12:19.069958Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hardy spaces related to Schr\\\"odinger operators with potentials which are sums of L^p-functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Jacek Dziuba\\'nski, Marcin Preisner","submitted_at":"2011-09-26T16:25:55Z","abstract_excerpt":"We investigate the Hardy space H^1_L associated to the Schr\\\"odinger operator L=-\\Delta+V on R^n, where V=\\sum_{j=1}^d V_j. We assume that each V_j depends on variables from a linear subspace VV_j of \\Rn, dim VV_j \\geq 3, and V_j belongs to L^q(VV_j) for certain q. We prove that there exist two distinct isomorphisms of H^1_L with the classical Hardy space. As a corollary we deduce a specific atomic characterization of H_L^1. We also prove that the space H_L^1 is described by means of the Riesz transforms R_{L,i} = \\partial_i L^{-1/2}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.5626","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":"1109.5626","created_at":"2026-05-18T04:12:19.070025+00:00"},{"alias_kind":"arxiv_version","alias_value":"1109.5626v1","created_at":"2026-05-18T04:12:19.070025+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.5626","created_at":"2026-05-18T04:12:19.070025+00:00"},{"alias_kind":"pith_short_12","alias_value":"7ZVR4TFC2IDI","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_16","alias_value":"7ZVR4TFC2IDIAHYC","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_8","alias_value":"7ZVR4TFC","created_at":"2026-05-18T12:26:22.705136+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/7ZVR4TFC2IDIAHYCEZ3MREFQ7R","json":"https://pith.science/pith/7ZVR4TFC2IDIAHYCEZ3MREFQ7R.json","graph_json":"https://pith.science/api/pith-number/7ZVR4TFC2IDIAHYCEZ3MREFQ7R/graph.json","events_json":"https://pith.science/api/pith-number/7ZVR4TFC2IDIAHYCEZ3MREFQ7R/events.json","paper":"https://pith.science/paper/7ZVR4TFC"},"agent_actions":{"view_html":"https://pith.science/pith/7ZVR4TFC2IDIAHYCEZ3MREFQ7R","download_json":"https://pith.science/pith/7ZVR4TFC2IDIAHYCEZ3MREFQ7R.json","view_paper":"https://pith.science/paper/7ZVR4TFC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1109.5626&json=true","fetch_graph":"https://pith.science/api/pith-number/7ZVR4TFC2IDIAHYCEZ3MREFQ7R/graph.json","fetch_events":"https://pith.science/api/pith-number/7ZVR4TFC2IDIAHYCEZ3MREFQ7R/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7ZVR4TFC2IDIAHYCEZ3MREFQ7R/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7ZVR4TFC2IDIAHYCEZ3MREFQ7R/action/storage_attestation","attest_author":"https://pith.science/pith/7ZVR4TFC2IDIAHYCEZ3MREFQ7R/action/author_attestation","sign_citation":"https://pith.science/pith/7ZVR4TFC2IDIAHYCEZ3MREFQ7R/action/citation_signature","submit_replication":"https://pith.science/pith/7ZVR4TFC2IDIAHYCEZ3MREFQ7R/action/replication_record"}},"created_at":"2026-05-18T04:12:19.070025+00:00","updated_at":"2026-05-18T04:12:19.070025+00:00"}