{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:VLYKARY2LAJGPWX5WH33YSY5HQ","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":"dc9f64a90c58597101070dc8ef2015e982ac228a81ae1bbb2e84a9d7020cab1a","cross_cats_sorted":["math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-04-14T22:18:08Z","title_canon_sha256":"576639eef3ece53f2c46a72f166c9b023e54c09f70c92aed52464833058f29de"},"schema_version":"1.0","source":{"id":"1004.2517","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.2517","created_at":"2026-05-18T04:19:53Z"},{"alias_kind":"arxiv_version","alias_value":"1004.2517v4","created_at":"2026-05-18T04:19:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.2517","created_at":"2026-05-18T04:19:53Z"},{"alias_kind":"pith_short_12","alias_value":"VLYKARY2LAJG","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"VLYKARY2LAJGPWX5","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"VLYKARY2","created_at":"2026-05-18T12:26:15Z"}],"graph_snapshots":[{"event_id":"sha256:25101c95abadf42b175b06cb7c19ebd3b752c50f3e5f413357f8211cbded984f","target":"graph","created_at":"2026-05-18T04:19:53Z","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 prove the upper and lower bounds for normal derivatives of spectral clusters $u=\\chi_{\\lambda}^s f$ of Dirichlet Laplacian $\\Delta_M$, $$c_s \\lambda\\|u\\|_{L^2(M)} \\leq \\| \\partial_{\\nu}u \\|_{L^2(\\partial M)} \\leq C_s \\lambda \\|u\\|_{L^2(M)} $$ where the upper bound is true for any Riemannian manifold, and the lower bound is true for some small $0<s<s_M$, where $s_M$ depends on the manifold only, provided that $M$ has no trapped geodesics (see Theorem \\ref{Thm3} for a precise statement), which generalizes the early results for single eigenfunctions by Hassell and Tao.","authors_text":"Xiangjin Xu","cross_cats":["math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-04-14T22:18:08Z","title":"Upper and lower bounds for normal derivatives of spectral clusters of Dirichlet Laplacian"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.2517","kind":"arxiv","version":4},"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:a9053b1a36339aa8cd85dc5b42c61bfd3a39be9e67adfa5b679aee8e1cccf063","target":"record","created_at":"2026-05-18T04:19:53Z","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":"dc9f64a90c58597101070dc8ef2015e982ac228a81ae1bbb2e84a9d7020cab1a","cross_cats_sorted":["math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-04-14T22:18:08Z","title_canon_sha256":"576639eef3ece53f2c46a72f166c9b023e54c09f70c92aed52464833058f29de"},"schema_version":"1.0","source":{"id":"1004.2517","kind":"arxiv","version":4}},"canonical_sha256":"aaf0a0471a581267dafdb1f7bc4b1d3c1299958d8048c44c7859ca9dbbd9d337","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aaf0a0471a581267dafdb1f7bc4b1d3c1299958d8048c44c7859ca9dbbd9d337","first_computed_at":"2026-05-18T04:19:53.496401Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:19:53.496401Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PGm9kTgL9t6nmJUnYzY/kO5Jvt1MWMnqs3asBh8R61QZHX/5aT+sbE6Zy39BAGitf37k6E0jXVdUefDMuQUTBA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:19:53.497260Z","signed_message":"canonical_sha256_bytes"},"source_id":"1004.2517","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a9053b1a36339aa8cd85dc5b42c61bfd3a39be9e67adfa5b679aee8e1cccf063","sha256:25101c95abadf42b175b06cb7c19ebd3b752c50f3e5f413357f8211cbded984f"],"state_sha256":"e2a3cc56baa1718e06ce5393454f68031eadb0cb6af5acf3c9fecac2579bd3d8"}