{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:LC44623ILX2B6C6CN73F4HZVCM","short_pith_number":"pith:LC44623I","canonical_record":{"source":{"id":"math-ph/0603080","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2006-03-29T22:57:44Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"abc1a6a4b5ad5539d040eaec9dc9b16bec57fa2e37a6d6fd0ac54b2980ea2935","abstract_canon_sha256":"af4840feffa4139db4d828e5d8b1e37336b3ca40651a4a4744b72e38325baccc"},"schema_version":"1.0"},"canonical_sha256":"58b9cf6b685df41f0bc26ff65e1f3513092dc4bc0e4c1330739ccef2da43af21","source":{"kind":"arxiv","id":"math-ph/0603080","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math-ph/0603080","created_at":"2026-05-18T02:56:52Z"},{"alias_kind":"arxiv_version","alias_value":"math-ph/0603080v2","created_at":"2026-05-18T02:56:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math-ph/0603080","created_at":"2026-05-18T02:56:52Z"},{"alias_kind":"pith_short_12","alias_value":"LC44623ILX2B","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"LC44623ILX2B6C6C","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"LC44623I","created_at":"2026-05-18T12:25:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:LC44623ILX2B6C6CN73F4HZVCM","target":"record","payload":{"canonical_record":{"source":{"id":"math-ph/0603080","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2006-03-29T22:57:44Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"abc1a6a4b5ad5539d040eaec9dc9b16bec57fa2e37a6d6fd0ac54b2980ea2935","abstract_canon_sha256":"af4840feffa4139db4d828e5d8b1e37336b3ca40651a4a4744b72e38325baccc"},"schema_version":"1.0"},"canonical_sha256":"58b9cf6b685df41f0bc26ff65e1f3513092dc4bc0e4c1330739ccef2da43af21","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:56:52.390347Z","signature_b64":"witBtw8r4PIFe4UmgXLH2aZzzp0P/7IkGqqKmZ0Zw43elMzqF943uTof4lYYe/cgoMCPhUL536yzth0gQj49Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"58b9cf6b685df41f0bc26ff65e1f3513092dc4bc0e4c1330739ccef2da43af21","last_reissued_at":"2026-05-18T02:56:52.389859Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:56:52.389859Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math-ph/0603080","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:56:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DcOssCIJP7arKy+0cSJxTpTwak/dw71Pmm8GTROrlRhiYOw8WUUY+8uvRZk7Am8PfHkGvaBmX3qyYi4wWltlCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T02:32:45.485178Z"},"content_sha256":"c132d575ba26655e496a9f1d248ea8acf9a9fc22b7f9f56f24bfdef3cde455e0","schema_version":"1.0","event_id":"sha256:c132d575ba26655e496a9f1d248ea8acf9a9fc22b7f9f56f24bfdef3cde455e0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:LC44623ILX2B6C6CN73F4HZVCM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Semiclassical Lp estimates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Daniel Tataru, Herbert Koch, Maciej Zworski","submitted_at":"2006-03-29T22:57:44Z","abstract_excerpt":"The purpose of this paper is to use semiclassical analysis to unify and generalize Lp estimates on high energy eigenfunctions and spectral clusters. In our approach these estimates do not depend on ellipticity and order, and apply to operators which are selfadjoint only at the principal level. They are estimates on weakly approximate solutions to semiclassical pseudodifferential equations. The revision corrects an exponent in the main theorems."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0603080","kind":"arxiv","version":2},"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":"2026-05-18T02:56:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A9JjsuwkCwD87mCxuOnw10x8B6bM3crJFWTqEHKpWEoazLThiHKowzFUkDxXHmxnCI9MMnJLX4FkhUWrQ8u9DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T02:32:45.485570Z"},"content_sha256":"96cc3c254f09bdadd028824a6e8788cbbdf5cfae5bee1b5bad10b7a4a945c977","schema_version":"1.0","event_id":"sha256:96cc3c254f09bdadd028824a6e8788cbbdf5cfae5bee1b5bad10b7a4a945c977"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LC44623ILX2B6C6CN73F4HZVCM/bundle.json","state_url":"https://pith.science/pith/LC44623ILX2B6C6CN73F4HZVCM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LC44623ILX2B6C6CN73F4HZVCM/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":"2026-05-22T02:32:45Z","links":{"resolver":"https://pith.science/pith/LC44623ILX2B6C6CN73F4HZVCM","bundle":"https://pith.science/pith/LC44623ILX2B6C6CN73F4HZVCM/bundle.json","state":"https://pith.science/pith/LC44623ILX2B6C6CN73F4HZVCM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LC44623ILX2B6C6CN73F4HZVCM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:LC44623ILX2B6C6CN73F4HZVCM","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":"af4840feffa4139db4d828e5d8b1e37336b3ca40651a4a4744b72e38325baccc","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2006-03-29T22:57:44Z","title_canon_sha256":"abc1a6a4b5ad5539d040eaec9dc9b16bec57fa2e37a6d6fd0ac54b2980ea2935"},"schema_version":"1.0","source":{"id":"math-ph/0603080","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math-ph/0603080","created_at":"2026-05-18T02:56:52Z"},{"alias_kind":"arxiv_version","alias_value":"math-ph/0603080v2","created_at":"2026-05-18T02:56:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math-ph/0603080","created_at":"2026-05-18T02:56:52Z"},{"alias_kind":"pith_short_12","alias_value":"LC44623ILX2B","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"LC44623ILX2B6C6C","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"LC44623I","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:96cc3c254f09bdadd028824a6e8788cbbdf5cfae5bee1b5bad10b7a4a945c977","target":"graph","created_at":"2026-05-18T02:56:52Z","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":"The purpose of this paper is to use semiclassical analysis to unify and generalize Lp estimates on high energy eigenfunctions and spectral clusters. In our approach these estimates do not depend on ellipticity and order, and apply to operators which are selfadjoint only at the principal level. They are estimates on weakly approximate solutions to semiclassical pseudodifferential equations. The revision corrects an exponent in the main theorems.","authors_text":"Daniel Tataru, Herbert Koch, Maciej Zworski","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2006-03-29T22:57:44Z","title":"Semiclassical Lp estimates"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0603080","kind":"arxiv","version":2},"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:c132d575ba26655e496a9f1d248ea8acf9a9fc22b7f9f56f24bfdef3cde455e0","target":"record","created_at":"2026-05-18T02:56:52Z","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":"af4840feffa4139db4d828e5d8b1e37336b3ca40651a4a4744b72e38325baccc","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2006-03-29T22:57:44Z","title_canon_sha256":"abc1a6a4b5ad5539d040eaec9dc9b16bec57fa2e37a6d6fd0ac54b2980ea2935"},"schema_version":"1.0","source":{"id":"math-ph/0603080","kind":"arxiv","version":2}},"canonical_sha256":"58b9cf6b685df41f0bc26ff65e1f3513092dc4bc0e4c1330739ccef2da43af21","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"58b9cf6b685df41f0bc26ff65e1f3513092dc4bc0e4c1330739ccef2da43af21","first_computed_at":"2026-05-18T02:56:52.389859Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:56:52.389859Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"witBtw8r4PIFe4UmgXLH2aZzzp0P/7IkGqqKmZ0Zw43elMzqF943uTof4lYYe/cgoMCPhUL536yzth0gQj49Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:56:52.390347Z","signed_message":"canonical_sha256_bytes"},"source_id":"math-ph/0603080","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c132d575ba26655e496a9f1d248ea8acf9a9fc22b7f9f56f24bfdef3cde455e0","sha256:96cc3c254f09bdadd028824a6e8788cbbdf5cfae5bee1b5bad10b7a4a945c977"],"state_sha256":"fc206ec39ccacb173eadcd5104531df51891a9940d68baec09c56b7bb80305b1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EJGxWfGBxevh21Uuo04IE5AcItB6q4pQTYbJOdWEpGXOi1rac0m4QTGTEZpO9ze6cRYnhZc5cLgpd76Lc60CAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-22T02:32:45.487851Z","bundle_sha256":"145186612648e3b4f9410d0b8096066ddc586c251ba9029158fc202ebcb447de"}}