{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:YQBB4CC6R2NBGFAWSFHCMUP4LM","short_pith_number":"pith:YQBB4CC6","schema_version":"1.0","canonical_sha256":"c4021e085e8e9a131416914e2651fc5b359135e9fb791db4507643ac10b0bac8","source":{"kind":"arxiv","id":"math/0606151","version":1},"attestation_state":"computed","paper":{"title":"Sharp two-sided heat kernel estimates for critical Schr\\\"odinger operators on bounded domains","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Achilles Tertikas, Luisa Moschini, Stathis Filippas","submitted_at":"2006-06-07T12:47:19Z","abstract_excerpt":"On a smooth bounded domain \\Omega \\subset R^N we consider the Schr\\\"odinger operators -\\Delta -V, with V being either the critical borderline potential V(x)=(N-2)^2/4 |x|^{-2} or V(x)=(1/4) dist (x,\\partial\\Omega)^{-2}, under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Scr\\\"odinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a serier of new inequalities such as improve"},"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":"math/0606151","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2006-06-07T12:47:19Z","cross_cats_sorted":[],"title_canon_sha256":"4c18c2fd1d1b4bd7afd6ce973cfccb1139776e21058d8a7141c504e0af65acb4","abstract_canon_sha256":"ab4cdff6aba369ebcbb3ff5e17ebdec7553922e1b47ddce1da1bbf56ff27f490"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:38:24.042125Z","signature_b64":"zjczwqg8wtT/yo48OMpJNa0PjwZDPwXgPEG8wYUSyb33y8r3fvnQpMXcC9lCjbh0aGng7+WwPyc13XpnHWaWBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c4021e085e8e9a131416914e2651fc5b359135e9fb791db4507643ac10b0bac8","last_reissued_at":"2026-05-18T01:38:24.041329Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:38:24.041329Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp two-sided heat kernel estimates for critical Schr\\\"odinger operators on bounded domains","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Achilles Tertikas, Luisa Moschini, Stathis Filippas","submitted_at":"2006-06-07T12:47:19Z","abstract_excerpt":"On a smooth bounded domain \\Omega \\subset R^N we consider the Schr\\\"odinger operators -\\Delta -V, with V being either the critical borderline potential V(x)=(N-2)^2/4 |x|^{-2} or V(x)=(1/4) dist (x,\\partial\\Omega)^{-2}, under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Scr\\\"odinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a serier of new inequalities such as improve"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0606151","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":""},"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":"math/0606151","created_at":"2026-05-18T01:38:24.041463+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0606151v1","created_at":"2026-05-18T01:38:24.041463+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0606151","created_at":"2026-05-18T01:38:24.041463+00:00"},{"alias_kind":"pith_short_12","alias_value":"YQBB4CC6R2NB","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_16","alias_value":"YQBB4CC6R2NBGFAW","created_at":"2026-05-18T12:25:54.717736+00:00"},{"alias_kind":"pith_short_8","alias_value":"YQBB4CC6","created_at":"2026-05-18T12:25:54.717736+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/YQBB4CC6R2NBGFAWSFHCMUP4LM","json":"https://pith.science/pith/YQBB4CC6R2NBGFAWSFHCMUP4LM.json","graph_json":"https://pith.science/api/pith-number/YQBB4CC6R2NBGFAWSFHCMUP4LM/graph.json","events_json":"https://pith.science/api/pith-number/YQBB4CC6R2NBGFAWSFHCMUP4LM/events.json","paper":"https://pith.science/paper/YQBB4CC6"},"agent_actions":{"view_html":"https://pith.science/pith/YQBB4CC6R2NBGFAWSFHCMUP4LM","download_json":"https://pith.science/pith/YQBB4CC6R2NBGFAWSFHCMUP4LM.json","view_paper":"https://pith.science/paper/YQBB4CC6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0606151&json=true","fetch_graph":"https://pith.science/api/pith-number/YQBB4CC6R2NBGFAWSFHCMUP4LM/graph.json","fetch_events":"https://pith.science/api/pith-number/YQBB4CC6R2NBGFAWSFHCMUP4LM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YQBB4CC6R2NBGFAWSFHCMUP4LM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YQBB4CC6R2NBGFAWSFHCMUP4LM/action/storage_attestation","attest_author":"https://pith.science/pith/YQBB4CC6R2NBGFAWSFHCMUP4LM/action/author_attestation","sign_citation":"https://pith.science/pith/YQBB4CC6R2NBGFAWSFHCMUP4LM/action/citation_signature","submit_replication":"https://pith.science/pith/YQBB4CC6R2NBGFAWSFHCMUP4LM/action/replication_record"}},"created_at":"2026-05-18T01:38:24.041463+00:00","updated_at":"2026-05-18T01:38:24.041463+00:00"}