{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:J2J7DS64O65LHNAFFLS6XV5WFZ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"a03a80e73b82a171fd2709ca1002af564b50982c0792554e8f78f9583a7b0d2d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-02-05T12:37:23Z","title_canon_sha256":"67370b88a08f41808df67c989949ad22946a31ea1e244f19656b3ac64b4dbf1e"},"schema_version":"1.0","source":{"id":"1502.01522","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.01522","created_at":"2026-05-18T01:31:29Z"},{"alias_kind":"arxiv_version","alias_value":"1502.01522v2","created_at":"2026-05-18T01:31:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.01522","created_at":"2026-05-18T01:31:29Z"},{"alias_kind":"pith_short_12","alias_value":"J2J7DS64O65L","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"J2J7DS64O65LHNAF","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"J2J7DS64","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:b1f011b87fef73b5ca4fdaa010675b2ee6b3fe1507889f6416db1d7c0506d3ce","target":"graph","created_at":"2026-05-18T01:31:29Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The Hardy-Littlewood inequalities for $m$-linear forms on $\\ell_{p}$ spaces are stated for $p>m$. In this paper, among other results, we investigate similar results for $1\\leq p\\leq m.$ Let $\\mathbb{K}$ be $% \\mathbb{R}$ or $\\mathbb{C}$ and $m\\geq 2$ be a positive integer. Our main results are the following sharp inequalities:\n  (i) If $\\left(r,p\\right) \\in \\left(\\lbrack 1,2]\\times \\lbrack 2,2m)\\right) \\cup \\left(\\lbrack 1,\\infty)\\times \\lbrack 2m,\\infty \\right)) $, then there is a constant $D_{m,r,p}^{\\mathbb{K}}>0$ (not depending on $% n $) such that \\begin{equation*} \\textstyle\\left(\\sum\\li","authors_text":"Daniel Pellegrino, Gustavo Araujo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-02-05T12:37:23Z","title":"Optimal Hardy-Littlewood type inequalities for $m$-linear forms on $\\ell_{p}$ spaces with $1\\leq p\\leq m$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.01522","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d03259824c1a2bd8a2f3a4b67baf0dc12631e1039be2134e4321c29cc2606035","target":"record","created_at":"2026-05-18T01:31:29Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"a03a80e73b82a171fd2709ca1002af564b50982c0792554e8f78f9583a7b0d2d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-02-05T12:37:23Z","title_canon_sha256":"67370b88a08f41808df67c989949ad22946a31ea1e244f19656b3ac64b4dbf1e"},"schema_version":"1.0","source":{"id":"1502.01522","kind":"arxiv","version":2}},"canonical_sha256":"4e93f1cbdc77bab3b4052ae5ebd7b62e63293fdd3c52ff0bc5da64fccae6c47f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4e93f1cbdc77bab3b4052ae5ebd7b62e63293fdd3c52ff0bc5da64fccae6c47f","first_computed_at":"2026-05-18T01:31:29.976554Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:31:29.976554Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"B47ebCkMRN5FUqJjOaJz03/woYN7v+AfHy6I7lh9nhRZX+cRmgyC8PgFHRGci2mvJ91BCxe2W1P2OnfgynfKBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:31:29.977001Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.01522","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d03259824c1a2bd8a2f3a4b67baf0dc12631e1039be2134e4321c29cc2606035","sha256:b1f011b87fef73b5ca4fdaa010675b2ee6b3fe1507889f6416db1d7c0506d3ce"],"state_sha256":"95b99a4b78f9f8d3ffe6341f6fe231f9f702f23ad3cb9cfdfaf370b044e98250"}