{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:3X2WUX6WNIWOYLEIDGRFDZJTCZ","short_pith_number":"pith:3X2WUX6W","canonical_record":{"source":{"id":"1012.0689","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-12-03T10:23:08Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"94433746d212d5ce234d86d9286eccd8155e8ff90abb51b1a7be99206c0dd252","abstract_canon_sha256":"60d893e802936b9dc4490a0a00066871c81e61cabbf5e1a2048dfb0cf6874d93"},"schema_version":"1.0"},"canonical_sha256":"ddf56a5fd66a2cec2c8819a251e53316497874196f506561ad2ddb51c96c91fa","source":{"kind":"arxiv","id":"1012.0689","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.0689","created_at":"2026-05-18T04:34:09Z"},{"alias_kind":"arxiv_version","alias_value":"1012.0689v1","created_at":"2026-05-18T04:34:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.0689","created_at":"2026-05-18T04:34:09Z"},{"alias_kind":"pith_short_12","alias_value":"3X2WUX6WNIWO","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"3X2WUX6WNIWOYLEI","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"3X2WUX6W","created_at":"2026-05-18T12:26:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:3X2WUX6WNIWOYLEIDGRFDZJTCZ","target":"record","payload":{"canonical_record":{"source":{"id":"1012.0689","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-12-03T10:23:08Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"94433746d212d5ce234d86d9286eccd8155e8ff90abb51b1a7be99206c0dd252","abstract_canon_sha256":"60d893e802936b9dc4490a0a00066871c81e61cabbf5e1a2048dfb0cf6874d93"},"schema_version":"1.0"},"canonical_sha256":"ddf56a5fd66a2cec2c8819a251e53316497874196f506561ad2ddb51c96c91fa","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:34:09.587445Z","signature_b64":"NlGZanAlLt1GKXkLsblxF16rw9TUJURLzHATL5ZuW8W3xiTNZLE+OW8iacbNkIFcWDAwZxIqnz9Wb8G60iYUAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ddf56a5fd66a2cec2c8819a251e53316497874196f506561ad2ddb51c96c91fa","last_reissued_at":"2026-05-18T04:34:09.587069Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:34:09.587069Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1012.0689","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":"2026-05-18T04:34:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BWLmk6MYZuNoLYfYUvEOGNUEy7wNCPu3MtnG7F5ueOEP825rj/D8EbqWgaBv+QqmPFTQWxkO6sgQgWkFwukOCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T05:33:50.156287Z"},"content_sha256":"da57759d91ab766c396ec38b4f5b8157be09da8ee9f7b156e66508e8b6656dcd","schema_version":"1.0","event_id":"sha256:da57759d91ab766c396ec38b4f5b8157be09da8ee9f7b156e66508e8b6656dcd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:3X2WUX6WNIWOYLEIDGRFDZJTCZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The wave equation on Damek-Ricci spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"Jean-Philippe Anker (MAPMO), Maria Vallarino, Vittoria Pierfelice (MAPMO)","submitted_at":"2010-12-03T10:23:08Z","abstract_excerpt":"We study the dispersive properties of the wave equation associated with the shifted Laplace-Beltrami operator on Damek-Ricci spaces, and deduce Strichartz estimates for a large family of admissible pairs. As an application, we obtain global well-posedness results for the nonlinear wave equation."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.0689","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:34:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"paRKJEJklnd3s9AftpB4sbH3j+NyypJ61rqrRrQaJkvDyGHDwCASu5fybxJ3e0E5+pP0IQEUeajZEsI5gg6aAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T05:33:50.156650Z"},"content_sha256":"b948aa29b08569b0bc2565cafc54f46b85dab229dd82e38fe9fc0342b3dca17b","schema_version":"1.0","event_id":"sha256:b948aa29b08569b0bc2565cafc54f46b85dab229dd82e38fe9fc0342b3dca17b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3X2WUX6WNIWOYLEIDGRFDZJTCZ/bundle.json","state_url":"https://pith.science/pith/3X2WUX6WNIWOYLEIDGRFDZJTCZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3X2WUX6WNIWOYLEIDGRFDZJTCZ/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":"2026-06-03T05:33:50Z","links":{"resolver":"https://pith.science/pith/3X2WUX6WNIWOYLEIDGRFDZJTCZ","bundle":"https://pith.science/pith/3X2WUX6WNIWOYLEIDGRFDZJTCZ/bundle.json","state":"https://pith.science/pith/3X2WUX6WNIWOYLEIDGRFDZJTCZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3X2WUX6WNIWOYLEIDGRFDZJTCZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:3X2WUX6WNIWOYLEIDGRFDZJTCZ","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":"60d893e802936b9dc4490a0a00066871c81e61cabbf5e1a2048dfb0cf6874d93","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-12-03T10:23:08Z","title_canon_sha256":"94433746d212d5ce234d86d9286eccd8155e8ff90abb51b1a7be99206c0dd252"},"schema_version":"1.0","source":{"id":"1012.0689","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.0689","created_at":"2026-05-18T04:34:09Z"},{"alias_kind":"arxiv_version","alias_value":"1012.0689v1","created_at":"2026-05-18T04:34:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.0689","created_at":"2026-05-18T04:34:09Z"},{"alias_kind":"pith_short_12","alias_value":"3X2WUX6WNIWO","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"3X2WUX6WNIWOYLEI","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"3X2WUX6W","created_at":"2026-05-18T12:26:03Z"}],"graph_snapshots":[{"event_id":"sha256:b948aa29b08569b0bc2565cafc54f46b85dab229dd82e38fe9fc0342b3dca17b","target":"graph","created_at":"2026-05-18T04:34:09Z","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 study the dispersive properties of the wave equation associated with the shifted Laplace-Beltrami operator on Damek-Ricci spaces, and deduce Strichartz estimates for a large family of admissible pairs. As an application, we obtain global well-posedness results for the nonlinear wave equation.","authors_text":"Jean-Philippe Anker (MAPMO), Maria Vallarino, Vittoria Pierfelice (MAPMO)","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-12-03T10:23:08Z","title":"The wave equation on Damek-Ricci spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.0689","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:da57759d91ab766c396ec38b4f5b8157be09da8ee9f7b156e66508e8b6656dcd","target":"record","created_at":"2026-05-18T04:34:09Z","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":"60d893e802936b9dc4490a0a00066871c81e61cabbf5e1a2048dfb0cf6874d93","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-12-03T10:23:08Z","title_canon_sha256":"94433746d212d5ce234d86d9286eccd8155e8ff90abb51b1a7be99206c0dd252"},"schema_version":"1.0","source":{"id":"1012.0689","kind":"arxiv","version":1}},"canonical_sha256":"ddf56a5fd66a2cec2c8819a251e53316497874196f506561ad2ddb51c96c91fa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ddf56a5fd66a2cec2c8819a251e53316497874196f506561ad2ddb51c96c91fa","first_computed_at":"2026-05-18T04:34:09.587069Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:34:09.587069Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NlGZanAlLt1GKXkLsblxF16rw9TUJURLzHATL5ZuW8W3xiTNZLE+OW8iacbNkIFcWDAwZxIqnz9Wb8G60iYUAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:34:09.587445Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.0689","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:da57759d91ab766c396ec38b4f5b8157be09da8ee9f7b156e66508e8b6656dcd","sha256:b948aa29b08569b0bc2565cafc54f46b85dab229dd82e38fe9fc0342b3dca17b"],"state_sha256":"4413bad9a2f5cca3dde441be0e57608a83860d4885d2e46a14bc0ca18e210cf4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w3CY9ThUQD6c5uZPYZqu1pZ3BUhtrUAjUZhBYPbBd22RJNjjzuuFnN+s9Q6Mt1slFDzSVY0hYF9Dii91i3kUDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T05:33:50.158594Z","bundle_sha256":"b7d02ec2316665a9c480611239f66fd78f365fbf631417184e644e8f82a29c52"}}