{"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"}