{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:QB6P5NVMYKGTHOE6KNVAQJNYEI","short_pith_number":"pith:QB6P5NVM","canonical_record":{"source":{"id":"1908.05642","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-15T17:18:24Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"4647e929acd08f5eec6f12e815776daf5f8e72b222e17304081c3ffe9111cddf","abstract_canon_sha256":"fdd13c3ccdefc0c1fdeb9102ad882fc7e7fe95270f8596d27de61eae3e204d7e"},"schema_version":"1.0"},"canonical_sha256":"807cfeb6acc28d33b89e536a0825b8220dbb8bfcc7af3538112c0785882de42f","source":{"kind":"arxiv","id":"1908.05642","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.05642","created_at":"2026-07-04T23:56:54Z"},{"alias_kind":"arxiv_version","alias_value":"1908.05642v1","created_at":"2026-07-04T23:56:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.05642","created_at":"2026-07-04T23:56:54Z"},{"alias_kind":"pith_short_12","alias_value":"QB6P5NVMYKGT","created_at":"2026-07-04T23:56:54Z"},{"alias_kind":"pith_short_16","alias_value":"QB6P5NVMYKGTHOE6","created_at":"2026-07-04T23:56:54Z"},{"alias_kind":"pith_short_8","alias_value":"QB6P5NVM","created_at":"2026-07-04T23:56:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:QB6P5NVMYKGTHOE6KNVAQJNYEI","target":"record","payload":{"canonical_record":{"source":{"id":"1908.05642","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-15T17:18:24Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"4647e929acd08f5eec6f12e815776daf5f8e72b222e17304081c3ffe9111cddf","abstract_canon_sha256":"fdd13c3ccdefc0c1fdeb9102ad882fc7e7fe95270f8596d27de61eae3e204d7e"},"schema_version":"1.0"},"canonical_sha256":"807cfeb6acc28d33b89e536a0825b8220dbb8bfcc7af3538112c0785882de42f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:56:54.583375Z","signature_b64":"TJDrzYEO3OhVigGo8+/MGVTEuoSo23JtBKcoYqgiLU5OufJF+IRtT6ohbZ/BpDZNeOLZ6leGvOR1CPzcvqFDCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"807cfeb6acc28d33b89e536a0825b8220dbb8bfcc7af3538112c0785882de42f","last_reissued_at":"2026-07-04T23:56:54.582890Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:56:54.582890Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.05642","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T23:56:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r+6HCcpPcnYCBRQayCqBdedy2+lEoGFLMjm9qNjs2RTXjpSqrnr8Kh0SDglNj7wdyAtcaiZyOsTwMvQmOX3KDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T20:34:48.478263Z"},"content_sha256":"46f6e1787df29da4a4ea2f8a2b2f63a36fb1e1443c7162f77730a87a2c66cb7b","schema_version":"1.0","event_id":"sha256:46f6e1787df29da4a4ea2f8a2b2f63a36fb1e1443c7162f77730a87a2c66cb7b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:QB6P5NVMYKGTHOE6KNVAQJNYEI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An Improved Compact Embedding Theorem for Degenerate Sobolev Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Dario D. Monticelli, Scott Rodney","submitted_at":"2019-08-15T17:18:24Z","abstract_excerpt":"This short note investigates the compact embedding of degenerate matrix weighted Sobolev spaces into weighted Lebesgue spaces. The Sobolev spaces explored are defined as the abstract completion of Lipschitz functions in a bounded domain $\\Omega$ with respect to the norm: $$\\|f\\|_{QH^{1,p}(v,\\mu;\\Omega)} = \\|f\\|_{L^p_v(\\Omega)} + \\|\\nabla f\\|_{\\mathcal{L}^p_Q(\\mu;\\Omega)}$$ where the weight $v$ is comparable to a power of the pointwise operator norm of the matrix valued function $Q=Q(x)$ in $\\Omega$. Following our main theorem, we give an explicit application where degeneracy is controlled thro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.05642","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.05642/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T23:56:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vuFn+dpuaGES07vu12DDMjcFqIYNvMI6L6GNyI67zZ+knVKfik/jbTTFEhPfPz5qbqgB9Wr/0t7JhLLHo7Z9DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T20:34:48.478757Z"},"content_sha256":"b65cd2f5d8d5e15e663329cd18ebac4e609b157ed9c1ae6c137ced1a3f5301d3","schema_version":"1.0","event_id":"sha256:b65cd2f5d8d5e15e663329cd18ebac4e609b157ed9c1ae6c137ced1a3f5301d3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QB6P5NVMYKGTHOE6KNVAQJNYEI/bundle.json","state_url":"https://pith.science/pith/QB6P5NVMYKGTHOE6KNVAQJNYEI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QB6P5NVMYKGTHOE6KNVAQJNYEI/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-16T20:34:48Z","links":{"resolver":"https://pith.science/pith/QB6P5NVMYKGTHOE6KNVAQJNYEI","bundle":"https://pith.science/pith/QB6P5NVMYKGTHOE6KNVAQJNYEI/bundle.json","state":"https://pith.science/pith/QB6P5NVMYKGTHOE6KNVAQJNYEI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QB6P5NVMYKGTHOE6KNVAQJNYEI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:QB6P5NVMYKGTHOE6KNVAQJNYEI","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":"fdd13c3ccdefc0c1fdeb9102ad882fc7e7fe95270f8596d27de61eae3e204d7e","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-15T17:18:24Z","title_canon_sha256":"4647e929acd08f5eec6f12e815776daf5f8e72b222e17304081c3ffe9111cddf"},"schema_version":"1.0","source":{"id":"1908.05642","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.05642","created_at":"2026-07-04T23:56:54Z"},{"alias_kind":"arxiv_version","alias_value":"1908.05642v1","created_at":"2026-07-04T23:56:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.05642","created_at":"2026-07-04T23:56:54Z"},{"alias_kind":"pith_short_12","alias_value":"QB6P5NVMYKGT","created_at":"2026-07-04T23:56:54Z"},{"alias_kind":"pith_short_16","alias_value":"QB6P5NVMYKGTHOE6","created_at":"2026-07-04T23:56:54Z"},{"alias_kind":"pith_short_8","alias_value":"QB6P5NVM","created_at":"2026-07-04T23:56:54Z"}],"graph_snapshots":[{"event_id":"sha256:b65cd2f5d8d5e15e663329cd18ebac4e609b157ed9c1ae6c137ced1a3f5301d3","target":"graph","created_at":"2026-07-04T23:56:54Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.05642/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This short note investigates the compact embedding of degenerate matrix weighted Sobolev spaces into weighted Lebesgue spaces. The Sobolev spaces explored are defined as the abstract completion of Lipschitz functions in a bounded domain $\\Omega$ with respect to the norm: $$\\|f\\|_{QH^{1,p}(v,\\mu;\\Omega)} = \\|f\\|_{L^p_v(\\Omega)} + \\|\\nabla f\\|_{\\mathcal{L}^p_Q(\\mu;\\Omega)}$$ where the weight $v$ is comparable to a power of the pointwise operator norm of the matrix valued function $Q=Q(x)$ in $\\Omega$. Following our main theorem, we give an explicit application where degeneracy is controlled thro","authors_text":"Dario D. Monticelli, Scott Rodney","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-15T17:18:24Z","title":"An Improved Compact Embedding Theorem for Degenerate Sobolev Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.05642","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":"sha256:46f6e1787df29da4a4ea2f8a2b2f63a36fb1e1443c7162f77730a87a2c66cb7b","target":"record","created_at":"2026-07-04T23:56:54Z","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":"fdd13c3ccdefc0c1fdeb9102ad882fc7e7fe95270f8596d27de61eae3e204d7e","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-15T17:18:24Z","title_canon_sha256":"4647e929acd08f5eec6f12e815776daf5f8e72b222e17304081c3ffe9111cddf"},"schema_version":"1.0","source":{"id":"1908.05642","kind":"arxiv","version":1}},"canonical_sha256":"807cfeb6acc28d33b89e536a0825b8220dbb8bfcc7af3538112c0785882de42f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"807cfeb6acc28d33b89e536a0825b8220dbb8bfcc7af3538112c0785882de42f","first_computed_at":"2026-07-04T23:56:54.582890Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T23:56:54.582890Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TJDrzYEO3OhVigGo8+/MGVTEuoSo23JtBKcoYqgiLU5OufJF+IRtT6ohbZ/BpDZNeOLZ6leGvOR1CPzcvqFDCw==","signature_status":"signed_v1","signed_at":"2026-07-04T23:56:54.583375Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.05642","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:46f6e1787df29da4a4ea2f8a2b2f63a36fb1e1443c7162f77730a87a2c66cb7b","sha256:b65cd2f5d8d5e15e663329cd18ebac4e609b157ed9c1ae6c137ced1a3f5301d3"],"state_sha256":"d89199f9fdffd75cabf828819d67558c89703000f108e01d60b0359d68cafba8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hG83aHf06nRP5Yq9en9CX3Cs8XjTqwEv0RwVLMndHVvCd6BsL3H0zcXLJxil62PCkxXITQoHg1FlWIvQh+J+Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T20:34:48.483099Z","bundle_sha256":"de2e6054ba047c6e962230adb04d5b7990086e3c4c0f9209d630946eb13699ff"}}