{"paper":{"title":"Sharp Broken-Power Lorentz Estimates for Fractional Powers of Radial Schr\\\"odinger Operators with Inverse-Square Asymptotics","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.CA","math.MP"],"primary_cat":"math.AP","authors_text":"Haochen Liu, Hongyan Zhou, Qinghao Yu","submitted_at":"2026-07-13T09:00:20Z","abstract_excerpt":"Let $H=-\\Delta+V(|x|)$ be a nonnegative radial Schr\\\"odinger operator on $\\mathbb{R}^d$, $d\\ge 2$, whose positive harmonic function satisfies $U(r)\\simeq r^{-\\sigma_0}$ for $0<r\\le 1$ and $U(r)\\simeq r^{-\\sigma_\\infty}$ for $r\\ge 1$, with $-d/2<\\sigma_0,\\sigma_\\infty<d/2$. Assuming the two-sided ground-state heat-kernel estimate of Ishige, Kabeya, and Ouhabaz, we determine the maximal open range in which the kernel of $H^{-s/2}$ admits the clean two-sided estimate $K_s^H(x,y)\\simeq |x-y|^{s-d}U(|x|)U(|y|)/[U(|x|+|x-y|)U(|y|+|x-y|)]$, namely $0<s<\\min\\{d,d-2\\sigma_0,d-2\\sigma_\\infty\\}$. In this"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.11280","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/2607.11280/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"}