{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:GSTCRWMHZMZEVG4ZHE7FBGZAOU","short_pith_number":"pith:GSTCRWMH","canonical_record":{"source":{"id":"1008.4510","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-08-26T13:48:26Z","cross_cats_sorted":[],"title_canon_sha256":"29a80d526f0005242349b8174b9fda35bfa6e4f056ae2d70ff37c1efb014029d","abstract_canon_sha256":"4aa156e25177c681f2d27f1ab48c7ed132378efdb603d610b0e197e9fa83c010"},"schema_version":"1.0"},"canonical_sha256":"34a628d987cb324a9b99393e509b2075034bc451b44d94139fb36d6c626dcdc9","source":{"kind":"arxiv","id":"1008.4510","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.4510","created_at":"2026-05-18T03:50:37Z"},{"alias_kind":"arxiv_version","alias_value":"1008.4510v3","created_at":"2026-05-18T03:50:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.4510","created_at":"2026-05-18T03:50:37Z"},{"alias_kind":"pith_short_12","alias_value":"GSTCRWMHZMZE","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"GSTCRWMHZMZEVG4Z","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"GSTCRWMH","created_at":"2026-05-18T12:26:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:GSTCRWMHZMZEVG4ZHE7FBGZAOU","target":"record","payload":{"canonical_record":{"source":{"id":"1008.4510","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-08-26T13:48:26Z","cross_cats_sorted":[],"title_canon_sha256":"29a80d526f0005242349b8174b9fda35bfa6e4f056ae2d70ff37c1efb014029d","abstract_canon_sha256":"4aa156e25177c681f2d27f1ab48c7ed132378efdb603d610b0e197e9fa83c010"},"schema_version":"1.0"},"canonical_sha256":"34a628d987cb324a9b99393e509b2075034bc451b44d94139fb36d6c626dcdc9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:50:37.810559Z","signature_b64":"q1tMagtufJBhdQhfITFq1UDRnVpm6LOUQRQnNOlWkKHXGhFy6sj5dRQFv2GZ+TY+0qWISJwOI3tmq7h3/1cZBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"34a628d987cb324a9b99393e509b2075034bc451b44d94139fb36d6c626dcdc9","last_reissued_at":"2026-05-18T03:50:37.809534Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:50:37.809534Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1008.4510","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:50:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SwiwE8wFR+XowmNTpH38r50/onIW1866LWu0pfzu9RY8XOaC6D9Q4jGbODusOkaLV70QWns9HLQYgWWWunscAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T05:09:58.299170Z"},"content_sha256":"6e06338803f638d3bbe9edd558e79b209304d9069c469dbc4fd3e66fa7bd8e23","schema_version":"1.0","event_id":"sha256:6e06338803f638d3bbe9edd558e79b209304d9069c469dbc4fd3e66fa7bd8e23"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:GSTCRWMHZMZEVG4ZHE7FBGZAOU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A simple and consistent definition of homogeneous Besov spaces on stratified Lie groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Hartmut F\\\"uhr","submitted_at":"2010-08-26T13:48:26Z","abstract_excerpt":"We introduce a general definition of homogeneous Besov spaces on a stratified Lie group $G$, based on a Littlewood-Paley-type decomposition of Schwartz functions with all moments vanishing. We show that under mild and intuitive conditions the spaces thus defined are independent of the decomposition employed. A corollary of this is that previously constructed versions of homogeneous Besov spaces on $G$, relying on the spectral calculus of a sub-Laplacian of the group, are consistent, i.e., independent of the choice of sub-Laplacian.\n  We further prove characterizations of homogeneous Besov spac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.4510","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:50:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9r57q3djK4xsQ5wmuRxyKvMJJ/3cygxq8sw9tRkXVVUq//0tr8fRiCJDmJEUzsnFxI2g8u+xz1S3s8NIyNlVBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T05:09:58.299771Z"},"content_sha256":"d8d959e1f9364d888dd5ca5eef78e67f06d05e69a8e60ae471093f20bc068edf","schema_version":"1.0","event_id":"sha256:d8d959e1f9364d888dd5ca5eef78e67f06d05e69a8e60ae471093f20bc068edf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GSTCRWMHZMZEVG4ZHE7FBGZAOU/bundle.json","state_url":"https://pith.science/pith/GSTCRWMHZMZEVG4ZHE7FBGZAOU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GSTCRWMHZMZEVG4ZHE7FBGZAOU/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":"2026-05-26T05:09:58Z","links":{"resolver":"https://pith.science/pith/GSTCRWMHZMZEVG4ZHE7FBGZAOU","bundle":"https://pith.science/pith/GSTCRWMHZMZEVG4ZHE7FBGZAOU/bundle.json","state":"https://pith.science/pith/GSTCRWMHZMZEVG4ZHE7FBGZAOU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GSTCRWMHZMZEVG4ZHE7FBGZAOU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:GSTCRWMHZMZEVG4ZHE7FBGZAOU","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":"4aa156e25177c681f2d27f1ab48c7ed132378efdb603d610b0e197e9fa83c010","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-08-26T13:48:26Z","title_canon_sha256":"29a80d526f0005242349b8174b9fda35bfa6e4f056ae2d70ff37c1efb014029d"},"schema_version":"1.0","source":{"id":"1008.4510","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.4510","created_at":"2026-05-18T03:50:37Z"},{"alias_kind":"arxiv_version","alias_value":"1008.4510v3","created_at":"2026-05-18T03:50:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.4510","created_at":"2026-05-18T03:50:37Z"},{"alias_kind":"pith_short_12","alias_value":"GSTCRWMHZMZE","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"GSTCRWMHZMZEVG4Z","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"GSTCRWMH","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:d8d959e1f9364d888dd5ca5eef78e67f06d05e69a8e60ae471093f20bc068edf","target":"graph","created_at":"2026-05-18T03:50:37Z","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":"We introduce a general definition of homogeneous Besov spaces on a stratified Lie group $G$, based on a Littlewood-Paley-type decomposition of Schwartz functions with all moments vanishing. We show that under mild and intuitive conditions the spaces thus defined are independent of the decomposition employed. A corollary of this is that previously constructed versions of homogeneous Besov spaces on $G$, relying on the spectral calculus of a sub-Laplacian of the group, are consistent, i.e., independent of the choice of sub-Laplacian.\n  We further prove characterizations of homogeneous Besov spac","authors_text":"Hartmut F\\\"uhr","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-08-26T13:48:26Z","title":"A simple and consistent definition of homogeneous Besov spaces on stratified Lie groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.4510","kind":"arxiv","version":3},"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":"sha256:6e06338803f638d3bbe9edd558e79b209304d9069c469dbc4fd3e66fa7bd8e23","target":"record","created_at":"2026-05-18T03:50:37Z","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":"4aa156e25177c681f2d27f1ab48c7ed132378efdb603d610b0e197e9fa83c010","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-08-26T13:48:26Z","title_canon_sha256":"29a80d526f0005242349b8174b9fda35bfa6e4f056ae2d70ff37c1efb014029d"},"schema_version":"1.0","source":{"id":"1008.4510","kind":"arxiv","version":3}},"canonical_sha256":"34a628d987cb324a9b99393e509b2075034bc451b44d94139fb36d6c626dcdc9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"34a628d987cb324a9b99393e509b2075034bc451b44d94139fb36d6c626dcdc9","first_computed_at":"2026-05-18T03:50:37.809534Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:50:37.809534Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"q1tMagtufJBhdQhfITFq1UDRnVpm6LOUQRQnNOlWkKHXGhFy6sj5dRQFv2GZ+TY+0qWISJwOI3tmq7h3/1cZBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:50:37.810559Z","signed_message":"canonical_sha256_bytes"},"source_id":"1008.4510","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6e06338803f638d3bbe9edd558e79b209304d9069c469dbc4fd3e66fa7bd8e23","sha256:d8d959e1f9364d888dd5ca5eef78e67f06d05e69a8e60ae471093f20bc068edf"],"state_sha256":"9925b72d4aed8d5b6fdfd1f451d617321105398bbcefbf69927ebf96f0ffdeb9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mgCRT6HGd6TfEPyOJodzSj0mV+Zjvd4AQ/znWl/MtHCaUPmn2A+hMDeluwdGgaVsp/drHvS501vvcg2kImDCBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T05:09:58.303328Z","bundle_sha256":"5eb71bae33aa326171e5a107fe8c1808a0baa760c26e01cbd57fea4c512d7b70"}}