{"paper":{"title":"Sharp reversed Hardy-Littlewood-Sobolev inequality on $\\mathbb R^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Qu\\^oc-Anh Ng\\^o, Van Hoang Nguyen","submitted_at":"2015-08-09T15:40:07Z","abstract_excerpt":"This is the first in our series of papers concerning some Hardy-Littlewood-Sobolev type inequalities. In the present paper, the main objective is to establish the following sharp reversed HLS inequality in the whole space $\\mathbb R^n$ \\[\\int_{\\mathbb R^n} \\int_{\\mathbb R^n} f(x) |x-y|^\\lambda g(y) dx dy \\geqslant \\mathscr C_{n,p,r} \\|f\\|_{L^p (\\mathbb R^n)}\\, \\|g\\|_{L^r (\\mathbb R^n)}\\] for any nonnegative functions $f\\in L^p(\\mathbb R^n)$, $g\\in L^r(\\mathbb R^n)$, and $p,r\\in (0,1)$, $\\lambda > 0$ such that $1/p + 1/r -\\lambda /n =2$. We will also explore some estimates for $\\mathscr C_{n,p,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.02041","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"}