{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:Q3BNA4UQFKD3KMRUV5LDINICSL","short_pith_number":"pith:Q3BNA4UQ","schema_version":"1.0","canonical_sha256":"86c2d072902a87b53234af5634350292de4653515ae2f425b81859027f3a9533","source":{"kind":"arxiv","id":"1204.6030","version":2},"attestation_state":"computed","paper":{"title":"Musielak-Orlicz Spaces that are Isomorphic to Subspaces of L_1","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Joscha Prochno","submitted_at":"2012-04-26T19:57:44Z","abstract_excerpt":"In this note we prove that $\\frac{1}{n!} \\sum_{\\pi} (\\sum_{i=1}^n |x_i a_{i,\\pi(i)} |^2)^{1/2}$ is equivalent to a Musielak-Orlicz norm $\\norm{x}_{\\sum M_i}$. We also obtain the inverse result, i.e., given the Orlicz functions, we provide a formula for the choice of the matrix that generates the corresponding Musielak-Orlicz norm. As a consequence, we obtain the embedding of 2-concave Musielak-Orlicz spaces into L_1."},"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":"1204.6030","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-04-26T19:57:44Z","cross_cats_sorted":[],"title_canon_sha256":"9a2e496fc92dd71f06a304fca51b4af51be9de05603fac52b7a7bf1df13b3790","abstract_canon_sha256":"87659dbac46094d54d1c1acc49a97137a84f08afbe3fc5d4d08b5a185e6a4223"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:26:23.234374Z","signature_b64":"EWcqBWabS1fW3gH5WiDFbagcI8mbb+bEe6bCOjEhT6zDGD8KzlUhceywCNxBMo76VoqgN77/qJKsYeQKwS6rAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"86c2d072902a87b53234af5634350292de4653515ae2f425b81859027f3a9533","last_reissued_at":"2026-05-18T03:26:23.233624Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:26:23.233624Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Musielak-Orlicz Spaces that are Isomorphic to Subspaces of L_1","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Joscha Prochno","submitted_at":"2012-04-26T19:57:44Z","abstract_excerpt":"In this note we prove that $\\frac{1}{n!} \\sum_{\\pi} (\\sum_{i=1}^n |x_i a_{i,\\pi(i)} |^2)^{1/2}$ is equivalent to a Musielak-Orlicz norm $\\norm{x}_{\\sum M_i}$. We also obtain the inverse result, i.e., given the Orlicz functions, we provide a formula for the choice of the matrix that generates the corresponding Musielak-Orlicz norm. As a consequence, we obtain the embedding of 2-concave Musielak-Orlicz spaces into L_1."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.6030","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":"1204.6030","created_at":"2026-05-18T03:26:23.233733+00:00"},{"alias_kind":"arxiv_version","alias_value":"1204.6030v2","created_at":"2026-05-18T03:26:23.233733+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.6030","created_at":"2026-05-18T03:26:23.233733+00:00"},{"alias_kind":"pith_short_12","alias_value":"Q3BNA4UQFKD3","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_16","alias_value":"Q3BNA4UQFKD3KMRU","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_8","alias_value":"Q3BNA4UQ","created_at":"2026-05-18T12:27:18.751474+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/Q3BNA4UQFKD3KMRUV5LDINICSL","json":"https://pith.science/pith/Q3BNA4UQFKD3KMRUV5LDINICSL.json","graph_json":"https://pith.science/api/pith-number/Q3BNA4UQFKD3KMRUV5LDINICSL/graph.json","events_json":"https://pith.science/api/pith-number/Q3BNA4UQFKD3KMRUV5LDINICSL/events.json","paper":"https://pith.science/paper/Q3BNA4UQ"},"agent_actions":{"view_html":"https://pith.science/pith/Q3BNA4UQFKD3KMRUV5LDINICSL","download_json":"https://pith.science/pith/Q3BNA4UQFKD3KMRUV5LDINICSL.json","view_paper":"https://pith.science/paper/Q3BNA4UQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1204.6030&json=true","fetch_graph":"https://pith.science/api/pith-number/Q3BNA4UQFKD3KMRUV5LDINICSL/graph.json","fetch_events":"https://pith.science/api/pith-number/Q3BNA4UQFKD3KMRUV5LDINICSL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Q3BNA4UQFKD3KMRUV5LDINICSL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Q3BNA4UQFKD3KMRUV5LDINICSL/action/storage_attestation","attest_author":"https://pith.science/pith/Q3BNA4UQFKD3KMRUV5LDINICSL/action/author_attestation","sign_citation":"https://pith.science/pith/Q3BNA4UQFKD3KMRUV5LDINICSL/action/citation_signature","submit_replication":"https://pith.science/pith/Q3BNA4UQFKD3KMRUV5LDINICSL/action/replication_record"}},"created_at":"2026-05-18T03:26:23.233733+00:00","updated_at":"2026-05-18T03:26:23.233733+00:00"}