{"paper":{"title":"All conditions for Stein-Weiss inequalities are necessary","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.FA","authors_text":"Qu\\^oc Anh Ng\\^o","submitted_at":"2021-10-27T07:10:46Z","abstract_excerpt":"The famous Stein-Weiss inequality on $\\mathbf R^n \\times \\mathbf R^n$, also known as the doubly weighted Hardy-Littlewood-Sobolev inequality, asserts that \\[ \\Big| \\iint_{\\mathbf R^n \\times \\mathbf R^n} \\frac{f(x) g(y)}{|x|^\\alpha |x-y|^\\lambda |y|^\\beta} dx dy \\Big| \\lesssim \\| f \\| _{L^p(\\mathbf R^n)} \\| g\\| _{L^r(\\mathbf R^n)} \\] holds for any $f\\in L^p(\\mathbf R^n)$ and $g\\in L^r(\\mathbf R^n)$ under several conditions on the parameters $n$, $p$, $r$, $\\alpha$, $\\beta$, and $\\lambda$. Extending the above inequality to either different domains rather than $\\mathbf R^n \\times \\mathbf R^n$ or "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.14220","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2110.14220/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}