{"paper":{"title":"Reversed inequality of the Herbst-type and the related Euler-Lagrange system","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tiantian Zhou, Yutian Lei","submitted_at":"2026-07-07T07:31:08Z","abstract_excerpt":"In 2008, Beckner (Proc. Amer. Math. Soc. 136(5), 1871-1885) proved two inequalities of the Herbst type, which are the critical forms of the Stein-Weiss inequality. In 2018, Chen et al. (Tran. Amer. Math. Soc. 370(12), 8429-8450) established the reversed Stein-Weiss inequality. In this paper, we are concerned about its critical case and give a reversed Herbst inequality. Namely,\n  $$\n  \\left|\\int_{\\mathbb{R}^n}\\int_{\\mathbb{R}^n}|x-y|^{\\alpha/q'-n}|y|^{\\alpha/q'}g(x)h(y)dxdy\\right|\n  \\geq C_{n,\\alpha,p,q'}\\|g\\|_{L^{q'}(\\mathbb{R}^n)}\\|h\\|_{L^p(\\mathbb{R}^n)}\n  $$ holds for any nonnegative funct"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.05928","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/2607.05928/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"}