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We say that an L^1(R^d)-function f belongs to the Hardy space H_L^1 associated with L if sup_{t>0} |K_t f| belongs to L^1(R^d). We prove that f\\in H_L^1 if and only if R_j f \\in L^1(R^d) for j=1,...,d, where R_j= \\frac{d}{dx_j} L^{-1/2} are the Riesz transforms associated with L."},"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":"0910.1017","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-10-06T14:25:29Z","cross_cats_sorted":[],"title_canon_sha256":"8a823e46f77e79c5dc8510e0a5e989a9937d0e4082f5407fa2c0dbf577f6594d","abstract_canon_sha256":"3b197425247601959f4bee1d8f63a0f30a895b87cdfbc697f64c914f55d8da06"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:41.500827Z","signature_b64":"JxKL+U3u0UChHxXAVr2GWNlqdb5IN94oQWOAasQtG828HWrK2LMOa/LAJ382DJuIuGUWwENbv+30rHvaCWfGCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a86c74f48e86ba4c10bb8418e783fe32305d68a116c6e866fdc2ff04f19c7a23","last_reissued_at":"2026-05-18T04:31:41.500414Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:41.500414Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Riesz transform characterization of Hardy spaces associated with Schr\\\"odinger operators with compactly supported potentials","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":"2009-10-06T14:25:29Z","abstract_excerpt":"Let L=-\\Delta+V be a Schr\\\"odinger operator on R^d, d\\geq 3. We assume that V is a nonnegative, compactly supported potential that belongs to L^p(R^d), for some p>d/2. Let K_t be the semigroup generated by -L. We say that an L^1(R^d)-function f belongs to the Hardy space H_L^1 associated with L if sup_{t>0} |K_t f| belongs to L^1(R^d). We prove that f\\in H_L^1 if and only if R_j f \\in L^1(R^d) for j=1,...,d, where R_j= \\frac{d}{dx_j} L^{-1/2} are the Riesz transforms associated with L."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.1017","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":"0910.1017","created_at":"2026-05-18T04:31:41.500474+00:00"},{"alias_kind":"arxiv_version","alias_value":"0910.1017v1","created_at":"2026-05-18T04:31:41.500474+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.1017","created_at":"2026-05-18T04:31:41.500474+00:00"},{"alias_kind":"pith_short_12","alias_value":"VBWHJ5EOQ25E","created_at":"2026-05-18T12:26:02.257875+00:00"},{"alias_kind":"pith_short_16","alias_value":"VBWHJ5EOQ25EYEF3","created_at":"2026-05-18T12:26:02.257875+00:00"},{"alias_kind":"pith_short_8","alias_value":"VBWHJ5EO","created_at":"2026-05-18T12:26:02.257875+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/VBWHJ5EOQ25EYEF3QQMOPA76GI","json":"https://pith.science/pith/VBWHJ5EOQ25EYEF3QQMOPA76GI.json","graph_json":"https://pith.science/api/pith-number/VBWHJ5EOQ25EYEF3QQMOPA76GI/graph.json","events_json":"https://pith.science/api/pith-number/VBWHJ5EOQ25EYEF3QQMOPA76GI/events.json","paper":"https://pith.science/paper/VBWHJ5EO"},"agent_actions":{"view_html":"https://pith.science/pith/VBWHJ5EOQ25EYEF3QQMOPA76GI","download_json":"https://pith.science/pith/VBWHJ5EOQ25EYEF3QQMOPA76GI.json","view_paper":"https://pith.science/paper/VBWHJ5EO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0910.1017&json=true","fetch_graph":"https://pith.science/api/pith-number/VBWHJ5EOQ25EYEF3QQMOPA76GI/graph.json","fetch_events":"https://pith.science/api/pith-number/VBWHJ5EOQ25EYEF3QQMOPA76GI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VBWHJ5EOQ25EYEF3QQMOPA76GI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VBWHJ5EOQ25EYEF3QQMOPA76GI/action/storage_attestation","attest_author":"https://pith.science/pith/VBWHJ5EOQ25EYEF3QQMOPA76GI/action/author_attestation","sign_citation":"https://pith.science/pith/VBWHJ5EOQ25EYEF3QQMOPA76GI/action/citation_signature","submit_replication":"https://pith.science/pith/VBWHJ5EOQ25EYEF3QQMOPA76GI/action/replication_record"}},"created_at":"2026-05-18T04:31:41.500474+00:00","updated_at":"2026-05-18T04:31:41.500474+00:00"}