{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:5CTQNJSCAX7ITRHTIW4YUZBZCS","short_pith_number":"pith:5CTQNJSC","schema_version":"1.0","canonical_sha256":"e8a706a64205fe89c4f345b98a643914a9350d145ebdedbfd679edc9271f93bb","source":{"kind":"arxiv","id":"1509.03204","version":1},"attestation_state":"computed","paper":{"title":"A weighted estimate for two dimensional Schrodinger, matrix schrodinger and wave equations with resonance of first kind at zero energy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ebru Toprak","submitted_at":"2015-09-10T15:57:20Z","abstract_excerpt":"We study the two dimensional Schr\\\"odinger operator, $H=-\\Delta+V$, in the weighted L^1(\\R^2) \\rightarrow L^{\\infty}(\\R^2) setting when there is a resonance of the first kind at zero energy. In particular, we show that if |V(x)|\\les \\la x \\ra ^{-3-} and there is only s-wave resonance at zero of H, then \\big\\| w^{-1} \\big( e^{itH}P_{ac} f - {\\f 1 t } F f \\big) \\big\\| _{\\infty} \\leq \\frac {C} {|t| (\\log|t|)^2 } \\|wf\\|_1 |t|>2, with w(x)=\\log^2(2+|x|). Here Ff=c \\psi\\la f,\\psi \\ra, where \\psi is an s-wave resonance function. We also extend this result to matrix Schr\\\"odinger and wave equations wi"},"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":"1509.03204","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-09-10T15:57:20Z","cross_cats_sorted":[],"title_canon_sha256":"ac8accef2413c1118cf0ac549bd2969f5b2658bda066144eb94b5f91dfe2a3eb","abstract_canon_sha256":"6176691c74e53d3c86c3c7a417ae14ae9dbdc45ced6c8d33ff397e321d661a6f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:50.627742Z","signature_b64":"pT8aYmRYUmmiAx23LoZe2f0UNbyGzLTtC8nXcQxn+8Pcv1LQq7gbs0lq1A7rjGFdUuq/W82sDy3hTTYO2+YsBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e8a706a64205fe89c4f345b98a643914a9350d145ebdedbfd679edc9271f93bb","last_reissued_at":"2026-05-18T00:03:50.627231Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:50.627231Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A weighted estimate for two dimensional Schrodinger, matrix schrodinger and wave equations with resonance of first kind at zero energy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ebru Toprak","submitted_at":"2015-09-10T15:57:20Z","abstract_excerpt":"We study the two dimensional Schr\\\"odinger operator, $H=-\\Delta+V$, in the weighted L^1(\\R^2) \\rightarrow L^{\\infty}(\\R^2) setting when there is a resonance of the first kind at zero energy. In particular, we show that if |V(x)|\\les \\la x \\ra ^{-3-} and there is only s-wave resonance at zero of H, then \\big\\| w^{-1} \\big( e^{itH}P_{ac} f - {\\f 1 t } F f \\big) \\big\\| _{\\infty} \\leq \\frac {C} {|t| (\\log|t|)^2 } \\|wf\\|_1 |t|>2, with w(x)=\\log^2(2+|x|). Here Ff=c \\psi\\la f,\\psi \\ra, where \\psi is an s-wave resonance function. We also extend this result to matrix Schr\\\"odinger and wave equations wi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.03204","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":"1509.03204","created_at":"2026-05-18T00:03:50.627293+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.03204v1","created_at":"2026-05-18T00:03:50.627293+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.03204","created_at":"2026-05-18T00:03:50.627293+00:00"},{"alias_kind":"pith_short_12","alias_value":"5CTQNJSCAX7I","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_16","alias_value":"5CTQNJSCAX7ITRHT","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_8","alias_value":"5CTQNJSC","created_at":"2026-05-18T12:29:05.191682+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/5CTQNJSCAX7ITRHTIW4YUZBZCS","json":"https://pith.science/pith/5CTQNJSCAX7ITRHTIW4YUZBZCS.json","graph_json":"https://pith.science/api/pith-number/5CTQNJSCAX7ITRHTIW4YUZBZCS/graph.json","events_json":"https://pith.science/api/pith-number/5CTQNJSCAX7ITRHTIW4YUZBZCS/events.json","paper":"https://pith.science/paper/5CTQNJSC"},"agent_actions":{"view_html":"https://pith.science/pith/5CTQNJSCAX7ITRHTIW4YUZBZCS","download_json":"https://pith.science/pith/5CTQNJSCAX7ITRHTIW4YUZBZCS.json","view_paper":"https://pith.science/paper/5CTQNJSC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.03204&json=true","fetch_graph":"https://pith.science/api/pith-number/5CTQNJSCAX7ITRHTIW4YUZBZCS/graph.json","fetch_events":"https://pith.science/api/pith-number/5CTQNJSCAX7ITRHTIW4YUZBZCS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5CTQNJSCAX7ITRHTIW4YUZBZCS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5CTQNJSCAX7ITRHTIW4YUZBZCS/action/storage_attestation","attest_author":"https://pith.science/pith/5CTQNJSCAX7ITRHTIW4YUZBZCS/action/author_attestation","sign_citation":"https://pith.science/pith/5CTQNJSCAX7ITRHTIW4YUZBZCS/action/citation_signature","submit_replication":"https://pith.science/pith/5CTQNJSCAX7ITRHTIW4YUZBZCS/action/replication_record"}},"created_at":"2026-05-18T00:03:50.627293+00:00","updated_at":"2026-05-18T00:03:50.627293+00:00"}