{"paper":{"title":"H\\\"{o}lder regularity and Liouville Theorem for the Schr\\\"{o}dinger equation with certain critical potentials, and applications to Dirichlet problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"Bo Li, Ji Li, Liangchuan Wu","submitted_at":"2024-10-04T13:27:56Z","abstract_excerpt":"Let $(X,d,\\mu)$ be a metric measure space satisfying a doubling property with the upper/lower dimension $Q\\ge n>1$, and admitting an $L^2$-Poincar\\'e inequality. In this article, we establish the H\\\"{o}lder continuity and a Liouville-type theorem for the (elliptic-type) Schr\\\"odinger equation $$\\mathbb L u(x,t)=-\\partial^2_{t}u(x,t)+\\mathcal L u(x,t)+V(x)u(x,t)=0,\\quad x\\in X,\\, t\\in\\mathbb R, $$ where $\\mathcal L$ is a non-negative operator generated by a Dirichlet form on $X$, and the non-negative potential $V$ is a Muckenhoupt weight belonging to the reverse H\\\"older class ${RH}_q(X)$ for s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.03418","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/2410.03418/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"}