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Then we prove that the space H^1_{at}(X) is also characterized by the Riesz transform Rf=\\sqrt{\\pi}\\frac{d}{dx}L^{-1/2}f in the sense that f\\in H^1_{at}(X) if and only if f,Rf \\in L^1((0,infty),x^alpha dx)."},"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":"1002.3319","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-02-17T17:27:42Z","cross_cats_sorted":[],"title_canon_sha256":"bab68189cd7e911f69a72695acc7daf38464a0129fa77d7a61b1dc8a4c29a81f","abstract_canon_sha256":"d82668cf9245b68c8923d5bce2c682e62122e840a05aa6fa3233997330c4f633"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:07:40.809884Z","signature_b64":"kEVJaBBb7L2YpVbbGhLuXGWAaBUWdAX/vk5Kr6niSPdTozljmeZppvSlRhX+Xgcl0meYzRDgXDxNJ9KfATTJCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"741345a2579180d771ae181938afd13e8b5321005b7d163a286289cb42484847","last_reissued_at":"2026-05-18T04:07:40.809412Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:07:40.809412Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Riesz transform characterization of H^1 spaces associated with certain Laguerre expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Marcin Preisner","submitted_at":"2010-02-17T17:27:42Z","abstract_excerpt":"For alpha>0 we consider the system l_k^{(alpha-1)/2}(x) of the Laguerre functions which are eigenfunctions of the differential operator Lf =-\\frac{d^2}{dx^2}f-\\frac{alpha}{x}\\frac{d}{dx}f+x^2 f. We define an atomic Hardy space H^1_{at}(X), which is a subspace of L^1((0,infty), x^alpha dx). Then we prove that the space H^1_{at}(X) is also characterized by the Riesz transform Rf=\\sqrt{\\pi}\\frac{d}{dx}L^{-1/2}f in the sense that f\\in H^1_{at}(X) if and only if f,Rf \\in L^1((0,infty),x^alpha dx)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.3319","kind":"arxiv","version":3},"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":"1002.3319","created_at":"2026-05-18T04:07:40.809485+00:00"},{"alias_kind":"arxiv_version","alias_value":"1002.3319v3","created_at":"2026-05-18T04:07:40.809485+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1002.3319","created_at":"2026-05-18T04:07:40.809485+00:00"},{"alias_kind":"pith_short_12","alias_value":"OQJULISXSGAN","created_at":"2026-05-18T12:26:12.377268+00:00"},{"alias_kind":"pith_short_16","alias_value":"OQJULISXSGANO4NO","created_at":"2026-05-18T12:26:12.377268+00:00"},{"alias_kind":"pith_short_8","alias_value":"OQJULISX","created_at":"2026-05-18T12:26:12.377268+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/OQJULISXSGANO4NODAMTRL6RH2","json":"https://pith.science/pith/OQJULISXSGANO4NODAMTRL6RH2.json","graph_json":"https://pith.science/api/pith-number/OQJULISXSGANO4NODAMTRL6RH2/graph.json","events_json":"https://pith.science/api/pith-number/OQJULISXSGANO4NODAMTRL6RH2/events.json","paper":"https://pith.science/paper/OQJULISX"},"agent_actions":{"view_html":"https://pith.science/pith/OQJULISXSGANO4NODAMTRL6RH2","download_json":"https://pith.science/pith/OQJULISXSGANO4NODAMTRL6RH2.json","view_paper":"https://pith.science/paper/OQJULISX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1002.3319&json=true","fetch_graph":"https://pith.science/api/pith-number/OQJULISXSGANO4NODAMTRL6RH2/graph.json","fetch_events":"https://pith.science/api/pith-number/OQJULISXSGANO4NODAMTRL6RH2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OQJULISXSGANO4NODAMTRL6RH2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OQJULISXSGANO4NODAMTRL6RH2/action/storage_attestation","attest_author":"https://pith.science/pith/OQJULISXSGANO4NODAMTRL6RH2/action/author_attestation","sign_citation":"https://pith.science/pith/OQJULISXSGANO4NODAMTRL6RH2/action/citation_signature","submit_replication":"https://pith.science/pith/OQJULISXSGANO4NODAMTRL6RH2/action/replication_record"}},"created_at":"2026-05-18T04:07:40.809485+00:00","updated_at":"2026-05-18T04:07:40.809485+00:00"}