{"paper":{"title":"Sharp and Endpoint Two-Weight Fractional Integral Estimates for Schr\"odinger Operators with Inverse-Square Potentials","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.AP","math.MP"],"primary_cat":"math.CA","authors_text":"Haochen Liu, Hongyan Zhou, Qinghao Yu","submitted_at":"2026-07-10T16:35:43Z","abstract_excerpt":"Let $H_a=-\\Delta+a|x|^{-2}$ be the Friedrichs extension on $L^2(\\mathbb{R}^d)$, where $d\\ge 3$ and $-(d-2)^2/4\\le a<0$ lies in the attractive Hardy range. Starting from the known positive two-sided comparison for the kernel of $H_a^{-s/2}$, we determine the complete strong non-endpoint mapping range for two power weights. If $\\sigma=(d-2-\\sqrt{(d-2)^2+4a})/2$ and $0<s<d-2\\sigma$, then [\n||x|^{-\\beta}H_a^{-s/2}f|{L^q} \\lesssim ||x|^\\alpha f|{L^p} ]\nholds for $1<p,q<\\infty$ precisely under the exponent ordering, scaling, sum, and origin conditions stated in the main theorem. At either origin-cri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.09585","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/2607.09585/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"}