{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:IFEFO6UNUWYYIR7RRAAEY6BNLX","short_pith_number":"pith:IFEFO6UN","schema_version":"1.0","canonical_sha256":"4148577a8da5b18447f188004c782d5dcff2ea8433f0f0b4b6001e57dfefb368","source":{"kind":"arxiv","id":"2410.03418","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2410.03418","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-10-04T13:27:56Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"14e0549efc4def8ab07e1013074f81fe0ed55c87a287979eeed73196e8ee10e9","abstract_canon_sha256":"f0192151e4e453916af241bf74e98c059775dec888291f27aa280dc3ece4cda2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:16:00.768479Z","signature_b64":"nw+6IvN6QrqNIcIZQ4tamKUE2THBLJkeQoXvoQVCjmGgmh6rZGzFEXa9OqlVphArKTeF/Z2mzKY/57SNpLuaAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4148577a8da5b18447f188004c782d5dcff2ea8433f0f0b4b6001e57dfefb368","last_reissued_at":"2026-07-05T09:16:00.767980Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:16:00.767980Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2410.03418","created_at":"2026-07-05T09:16:00.768040+00:00"},{"alias_kind":"arxiv_version","alias_value":"2410.03418v1","created_at":"2026-07-05T09:16:00.768040+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.03418","created_at":"2026-07-05T09:16:00.768040+00:00"},{"alias_kind":"pith_short_12","alias_value":"IFEFO6UNUWYY","created_at":"2026-07-05T09:16:00.768040+00:00"},{"alias_kind":"pith_short_16","alias_value":"IFEFO6UNUWYYIR7R","created_at":"2026-07-05T09:16:00.768040+00:00"},{"alias_kind":"pith_short_8","alias_value":"IFEFO6UN","created_at":"2026-07-05T09:16:00.768040+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IFEFO6UNUWYYIR7RRAAEY6BNLX","json":"https://pith.science/pith/IFEFO6UNUWYYIR7RRAAEY6BNLX.json","graph_json":"https://pith.science/api/pith-number/IFEFO6UNUWYYIR7RRAAEY6BNLX/graph.json","events_json":"https://pith.science/api/pith-number/IFEFO6UNUWYYIR7RRAAEY6BNLX/events.json","paper":"https://pith.science/paper/IFEFO6UN"},"agent_actions":{"view_html":"https://pith.science/pith/IFEFO6UNUWYYIR7RRAAEY6BNLX","download_json":"https://pith.science/pith/IFEFO6UNUWYYIR7RRAAEY6BNLX.json","view_paper":"https://pith.science/paper/IFEFO6UN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2410.03418&json=true","fetch_graph":"https://pith.science/api/pith-number/IFEFO6UNUWYYIR7RRAAEY6BNLX/graph.json","fetch_events":"https://pith.science/api/pith-number/IFEFO6UNUWYYIR7RRAAEY6BNLX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IFEFO6UNUWYYIR7RRAAEY6BNLX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IFEFO6UNUWYYIR7RRAAEY6BNLX/action/storage_attestation","attest_author":"https://pith.science/pith/IFEFO6UNUWYYIR7RRAAEY6BNLX/action/author_attestation","sign_citation":"https://pith.science/pith/IFEFO6UNUWYYIR7RRAAEY6BNLX/action/citation_signature","submit_replication":"https://pith.science/pith/IFEFO6UNUWYYIR7RRAAEY6BNLX/action/replication_record"}},"created_at":"2026-07-05T09:16:00.768040+00:00","updated_at":"2026-07-05T09:16:00.768040+00:00"}