{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:26ZMWZUL75KHTQHZV6UG7Q425C","short_pith_number":"pith:26ZMWZUL","schema_version":"1.0","canonical_sha256":"d7b2cb668bff5479c0f9afa86fc39ae8aaef5c06e549af077c2001719d536133","source":{"kind":"arxiv","id":"1503.02716","version":2},"attestation_state":"computed","paper":{"title":"Sobolev spaces adapted to the Schr\\\"odinger operator with inverse-square potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"C. Miao, J. Zhang, J. Zheng, M. Visan, R. Killip","submitted_at":"2015-03-09T22:35:05Z","abstract_excerpt":"We study the $L^p$-theory for the Schr\\\"odinger operator $\\mathcal L_a$ with inverse-square potential $a|x|^{-2}$. Our main result describes when $L^p$-based Sobolev spaces defined in terms of the operator $(\\mathcal L_a)^{s/2}$ agree with those defined via $(-\\Delta)^{s/2}$. We consider all regularities $0<s<2$.\n  In order to make the paper self-contained, we also review (with proofs) multiplier theorems, Littlewood-Paley theory, and Hardy-type inequalities associated to the operator $\\mathcal L_a$."},"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":"1503.02716","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-03-09T22:35:05Z","cross_cats_sorted":[],"title_canon_sha256":"d0bf98352eb05acc6b6545496dcfe68f3fc93f2788eb648ad1197fd48e4da6c3","abstract_canon_sha256":"2b67f731b47c7c1d057ec2ef3452fe9632172193a8d13b4f4f28b4df206c2978"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:20.587689Z","signature_b64":"imK3TGMel8S3ab5g0gXnZMVWFK0Kv3Zn8Ql31sc9/1GYXU2KarprfZ7NwFEoIVUo0IMxlBGdTa9uvVogIERWBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d7b2cb668bff5479c0f9afa86fc39ae8aaef5c06e549af077c2001719d536133","last_reissued_at":"2026-05-18T01:17:20.587295Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:20.587295Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sobolev spaces adapted to the Schr\\\"odinger operator with inverse-square potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"C. Miao, J. Zhang, J. Zheng, M. Visan, R. Killip","submitted_at":"2015-03-09T22:35:05Z","abstract_excerpt":"We study the $L^p$-theory for the Schr\\\"odinger operator $\\mathcal L_a$ with inverse-square potential $a|x|^{-2}$. Our main result describes when $L^p$-based Sobolev spaces defined in terms of the operator $(\\mathcal L_a)^{s/2}$ agree with those defined via $(-\\Delta)^{s/2}$. We consider all regularities $0<s<2$.\n  In order to make the paper self-contained, we also review (with proofs) multiplier theorems, Littlewood-Paley theory, and Hardy-type inequalities associated to the operator $\\mathcal L_a$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.02716","kind":"arxiv","version":2},"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":"1503.02716","created_at":"2026-05-18T01:17:20.587365+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.02716v2","created_at":"2026-05-18T01:17:20.587365+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.02716","created_at":"2026-05-18T01:17:20.587365+00:00"},{"alias_kind":"pith_short_12","alias_value":"26ZMWZUL75KH","created_at":"2026-05-18T12:28:59.999130+00:00"},{"alias_kind":"pith_short_16","alias_value":"26ZMWZUL75KHTQHZ","created_at":"2026-05-18T12:28:59.999130+00:00"},{"alias_kind":"pith_short_8","alias_value":"26ZMWZUL","created_at":"2026-05-18T12:28:59.999130+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/26ZMWZUL75KHTQHZV6UG7Q425C","json":"https://pith.science/pith/26ZMWZUL75KHTQHZV6UG7Q425C.json","graph_json":"https://pith.science/api/pith-number/26ZMWZUL75KHTQHZV6UG7Q425C/graph.json","events_json":"https://pith.science/api/pith-number/26ZMWZUL75KHTQHZV6UG7Q425C/events.json","paper":"https://pith.science/paper/26ZMWZUL"},"agent_actions":{"view_html":"https://pith.science/pith/26ZMWZUL75KHTQHZV6UG7Q425C","download_json":"https://pith.science/pith/26ZMWZUL75KHTQHZV6UG7Q425C.json","view_paper":"https://pith.science/paper/26ZMWZUL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.02716&json=true","fetch_graph":"https://pith.science/api/pith-number/26ZMWZUL75KHTQHZV6UG7Q425C/graph.json","fetch_events":"https://pith.science/api/pith-number/26ZMWZUL75KHTQHZV6UG7Q425C/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/26ZMWZUL75KHTQHZV6UG7Q425C/action/timestamp_anchor","attest_storage":"https://pith.science/pith/26ZMWZUL75KHTQHZV6UG7Q425C/action/storage_attestation","attest_author":"https://pith.science/pith/26ZMWZUL75KHTQHZV6UG7Q425C/action/author_attestation","sign_citation":"https://pith.science/pith/26ZMWZUL75KHTQHZV6UG7Q425C/action/citation_signature","submit_replication":"https://pith.science/pith/26ZMWZUL75KHTQHZV6UG7Q425C/action/replication_record"}},"created_at":"2026-05-18T01:17:20.587365+00:00","updated_at":"2026-05-18T01:17:20.587365+00:00"}