{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:OKB7OHDYYJ6SJFCHBKPL2UDIHF","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"277eb7170a6b1d76be3ea4155ed55747cf560d8970f3c127a55348d78a455a42","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.AP","submitted_at":"2013-04-17T00:39:58Z","title_canon_sha256":"cf4244306ef2dd25c27c8d5abb42e288ff29a2768afce69ccf9597d2dbec3f18"},"schema_version":"1.0","source":{"id":"1304.4653","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.4653","created_at":"2026-05-18T03:27:44Z"},{"alias_kind":"arxiv_version","alias_value":"1304.4653v1","created_at":"2026-05-18T03:27:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.4653","created_at":"2026-05-18T03:27:44Z"},{"alias_kind":"pith_short_12","alias_value":"OKB7OHDYYJ6S","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"OKB7OHDYYJ6SJFCH","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"OKB7OHDY","created_at":"2026-05-18T12:27:54Z"}],"graph_snapshots":[{"event_id":"sha256:78b4b7064454046d9dbb23262a292d90b03f09034315a602a5393956970b925a","target":"graph","created_at":"2026-05-18T03:27:44Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper, we give the H\\\"ormander's $L^2$ theorem for Dirac operator over an open subset $\\Omega\\in\\R^{n+1}$ with Clifford algebra. Some sufficient condition on the existence of the weak solutions for Dirac operator has been found in the sense of Clifford analysis. In particular, if $\\Omega$ is bounded, then we prove that for any $f$ in $L^2$ space with value in Clifford algebra, there exists a weak solution of Dirac operator such that $$\\bar{D}u=f$$ with $u$ in the $L^2$ space as well. The method is based on H\\\"ormander's $L^2$ existence theorem in complex analysis and the $L^2$ weighted","authors_text":"Yang Liu, Yifei Pan, Zhihua Chen","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.AP","submitted_at":"2013-04-17T00:39:58Z","title":"A variant of H\\\"ormander's $L^2$ theorem for Dirac operator in Clifford analysis"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.4653","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:8127701f7ffbea0eb0f2c6ee0f44c287ee5bbef8951ebb49ccf17557cc1af24a","target":"record","created_at":"2026-05-18T03:27:44Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"277eb7170a6b1d76be3ea4155ed55747cf560d8970f3c127a55348d78a455a42","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.AP","submitted_at":"2013-04-17T00:39:58Z","title_canon_sha256":"cf4244306ef2dd25c27c8d5abb42e288ff29a2768afce69ccf9597d2dbec3f18"},"schema_version":"1.0","source":{"id":"1304.4653","kind":"arxiv","version":1}},"canonical_sha256":"7283f71c78c27d2494470a9ebd506839794b117ab96ab3e259f55224d987106e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7283f71c78c27d2494470a9ebd506839794b117ab96ab3e259f55224d987106e","first_computed_at":"2026-05-18T03:27:44.062672Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:27:44.062672Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zwzZfuz2xcQzwwWbqyLgUsCCqWhtQiU+hf1aJzFA5BwWGvoaJ2lDQFvMnZ75Q0bCxrk+KYplGwnGCzQcVB7GCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:27:44.063189Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.4653","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8127701f7ffbea0eb0f2c6ee0f44c287ee5bbef8951ebb49ccf17557cc1af24a","sha256:78b4b7064454046d9dbb23262a292d90b03f09034315a602a5393956970b925a"],"state_sha256":"629b278e235e902984df044c11bfb13bdff38c955f5414f9ec37c52a2993dfff"}