{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:AA22UYPTI6ILA7HKON3HSBIBUQ","short_pith_number":"pith:AA22UYPT","schema_version":"1.0","canonical_sha256":"0035aa61f34790b07cea7376790501a4116d7600876d9d9f3ff9c885ee45c882","source":{"kind":"arxiv","id":"1012.2438","version":1},"attestation_state":"computed","paper":{"title":"Sub-criticality of non-local Schr\\\"odinger systems with antisymmetric potentials and applications to half-harmonic maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Francesca Da Lio, Tristan Riviere","submitted_at":"2010-12-11T09:04:54Z","abstract_excerpt":"We consider nonlocal linear Schr\\\"odinger-type critical systems of the type\n  \\begin{equation}\\label{eqabstr}\n  \\Delta^{1/4} v=\\Omega\\, v~~~\\mbox{in $\\R\\,.$} \\\n  \\end{equation} where $\\Omega$ is antisymmetric potential in $L^2(\\R,so(m))$, $v$ is a ${\\R}^m$ valued map and $\\Omega\\, v$ denotes the matrix multiplication. We show that every solution $v\\in L^2(\\R,\\R^m)$ of \\rec{eqabstr} is in fact in $L^p_{loc}(\\R,\\R^m)$, for every $2\\le p<+\\infty$, in other words, we prove that the system (\\ref{eqabstr}) which is a-priori only critical in $L^2$ happens to have a subcritical behavior for antisymmet"},"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":"1012.2438","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-12-11T09:04:54Z","cross_cats_sorted":[],"title_canon_sha256":"1ced8f9dc1ef387f0f866e464b5bada202b0e2af326d6ae284b79dc41728f161","abstract_canon_sha256":"8d2328cda14d2fd73c5fb96228ef42727ee787503b3ca34e46df1e655b0ca8bd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:33:28.632301Z","signature_b64":"qAEsnk9VmscPKiFbi7Wyz64npjtSc9r2Du2kcVI1BnrZnTRlWPYeG9rKW4/grYdW5lC4+jmrDCwJzwHCSl/XBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0035aa61f34790b07cea7376790501a4116d7600876d9d9f3ff9c885ee45c882","last_reissued_at":"2026-05-18T04:33:28.631696Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:33:28.631696Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sub-criticality of non-local Schr\\\"odinger systems with antisymmetric potentials and applications to half-harmonic maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Francesca Da Lio, Tristan Riviere","submitted_at":"2010-12-11T09:04:54Z","abstract_excerpt":"We consider nonlocal linear Schr\\\"odinger-type critical systems of the type\n  \\begin{equation}\\label{eqabstr}\n  \\Delta^{1/4} v=\\Omega\\, v~~~\\mbox{in $\\R\\,.$} \\\n  \\end{equation} where $\\Omega$ is antisymmetric potential in $L^2(\\R,so(m))$, $v$ is a ${\\R}^m$ valued map and $\\Omega\\, v$ denotes the matrix multiplication. We show that every solution $v\\in L^2(\\R,\\R^m)$ of \\rec{eqabstr} is in fact in $L^p_{loc}(\\R,\\R^m)$, for every $2\\le p<+\\infty$, in other words, we prove that the system (\\ref{eqabstr}) which is a-priori only critical in $L^2$ happens to have a subcritical behavior for antisymmet"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.2438","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":"1012.2438","created_at":"2026-05-18T04:33:28.631773+00:00"},{"alias_kind":"arxiv_version","alias_value":"1012.2438v1","created_at":"2026-05-18T04:33:28.631773+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.2438","created_at":"2026-05-18T04:33:28.631773+00:00"},{"alias_kind":"pith_short_12","alias_value":"AA22UYPTI6IL","created_at":"2026-05-18T12:26:05.355336+00:00"},{"alias_kind":"pith_short_16","alias_value":"AA22UYPTI6ILA7HK","created_at":"2026-05-18T12:26:05.355336+00:00"},{"alias_kind":"pith_short_8","alias_value":"AA22UYPT","created_at":"2026-05-18T12:26:05.355336+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/AA22UYPTI6ILA7HKON3HSBIBUQ","json":"https://pith.science/pith/AA22UYPTI6ILA7HKON3HSBIBUQ.json","graph_json":"https://pith.science/api/pith-number/AA22UYPTI6ILA7HKON3HSBIBUQ/graph.json","events_json":"https://pith.science/api/pith-number/AA22UYPTI6ILA7HKON3HSBIBUQ/events.json","paper":"https://pith.science/paper/AA22UYPT"},"agent_actions":{"view_html":"https://pith.science/pith/AA22UYPTI6ILA7HKON3HSBIBUQ","download_json":"https://pith.science/pith/AA22UYPTI6ILA7HKON3HSBIBUQ.json","view_paper":"https://pith.science/paper/AA22UYPT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1012.2438&json=true","fetch_graph":"https://pith.science/api/pith-number/AA22UYPTI6ILA7HKON3HSBIBUQ/graph.json","fetch_events":"https://pith.science/api/pith-number/AA22UYPTI6ILA7HKON3HSBIBUQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AA22UYPTI6ILA7HKON3HSBIBUQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AA22UYPTI6ILA7HKON3HSBIBUQ/action/storage_attestation","attest_author":"https://pith.science/pith/AA22UYPTI6ILA7HKON3HSBIBUQ/action/author_attestation","sign_citation":"https://pith.science/pith/AA22UYPTI6ILA7HKON3HSBIBUQ/action/citation_signature","submit_replication":"https://pith.science/pith/AA22UYPTI6ILA7HKON3HSBIBUQ/action/replication_record"}},"created_at":"2026-05-18T04:33:28.631773+00:00","updated_at":"2026-05-18T04:33:28.631773+00:00"}