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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"},"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":"1502.01522","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-02-05T12:37:23Z","cross_cats_sorted":[],"title_canon_sha256":"67370b88a08f41808df67c989949ad22946a31ea1e244f19656b3ac64b4dbf1e","abstract_canon_sha256":"a03a80e73b82a171fd2709ca1002af564b50982c0792554e8f78f9583a7b0d2d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:29.977001Z","signature_b64":"B47ebCkMRN5FUqJjOaJz03/woYN7v+AfHy6I7lh9nhRZX+cRmgyC8PgFHRGci2mvJ91BCxe2W1P2OnfgynfKBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4e93f1cbdc77bab3b4052ae5ebd7b62e63293fdd3c52ff0bc5da64fccae6c47f","last_reissued_at":"2026-05-18T01:31:29.976554Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:29.976554Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal Hardy-Littlewood type inequalities for $m$-linear forms on $\\ell_{p}$ spaces with $1\\leq p\\leq m$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Daniel Pellegrino, Gustavo Araujo","submitted_at":"2015-02-05T12:37:23Z","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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.01522","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":"1502.01522","created_at":"2026-05-18T01:31:29.976615+00:00"},{"alias_kind":"arxiv_version","alias_value":"1502.01522v2","created_at":"2026-05-18T01:31:29.976615+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.01522","created_at":"2026-05-18T01:31:29.976615+00:00"},{"alias_kind":"pith_short_12","alias_value":"J2J7DS64O65L","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_16","alias_value":"J2J7DS64O65LHNAF","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_8","alias_value":"J2J7DS64","created_at":"2026-05-18T12:29:27.538025+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/J2J7DS64O65LHNAFFLS6XV5WFZ","json":"https://pith.science/pith/J2J7DS64O65LHNAFFLS6XV5WFZ.json","graph_json":"https://pith.science/api/pith-number/J2J7DS64O65LHNAFFLS6XV5WFZ/graph.json","events_json":"https://pith.science/api/pith-number/J2J7DS64O65LHNAFFLS6XV5WFZ/events.json","paper":"https://pith.science/paper/J2J7DS64"},"agent_actions":{"view_html":"https://pith.science/pith/J2J7DS64O65LHNAFFLS6XV5WFZ","download_json":"https://pith.science/pith/J2J7DS64O65LHNAFFLS6XV5WFZ.json","view_paper":"https://pith.science/paper/J2J7DS64","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1502.01522&json=true","fetch_graph":"https://pith.science/api/pith-number/J2J7DS64O65LHNAFFLS6XV5WFZ/graph.json","fetch_events":"https://pith.science/api/pith-number/J2J7DS64O65LHNAFFLS6XV5WFZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/J2J7DS64O65LHNAFFLS6XV5WFZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/J2J7DS64O65LHNAFFLS6XV5WFZ/action/storage_attestation","attest_author":"https://pith.science/pith/J2J7DS64O65LHNAFFLS6XV5WFZ/action/author_attestation","sign_citation":"https://pith.science/pith/J2J7DS64O65LHNAFFLS6XV5WFZ/action/citation_signature","submit_replication":"https://pith.science/pith/J2J7DS64O65LHNAFFLS6XV5WFZ/action/replication_record"}},"created_at":"2026-05-18T01:31:29.976615+00:00","updated_at":"2026-05-18T01:31:29.976615+00:00"}