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Let $\\chi_\\lambda$ be the unit band spectral projection operator associated with the Neumann Laplacian and $f$ a square integrable function on $M$. We show the following gradient estimate for $\\chi_\\lambda\\,f$ as $\\lambda\\geq 1$: $\\|\\nabla\\ \\chi_\\l\\ f\\|_\\infty\\leq C\\l \\|\\chi_\\l\\f\\|_\\infty+\\l^{-1}\\|\\Delta\\ \\chi_\\l\\ f\\|_\\infty$, whe"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1306.4033","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-06-17T22:06:25Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"8d1991112549d4942ba68fca237abd64533b6697f905ca5059edf34d67658bcf","abstract_canon_sha256":"59817c3712f59a53588e417c49bd7cad4c467d9c10c30c8e19de6f1e1c9c5218"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:20:44.614695Z","signature_b64":"HU63xBhVhkeHBYZ/dYESy8ghSuO4/ARrrJoAAC00YP+R9foYndG9QhCG99vdrduNYrTjjjOSIGKP+WO7p5UrCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"53f6de3a80b82c01d26d54cce00c4815aa1406a80409d430505a7e2b14d8a634","last_reissued_at":"2026-05-18T03:20:44.613782Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:20:44.613782Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gradient estimate of a Neumann eigenfunction on a compact manifold with boundary","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.SP","authors_text":"Bin Xu, Jingchen Hu, Yiqian Shi","submitted_at":"2013-06-17T22:06:25Z","abstract_excerpt":"Let $e_\\l(x)$ be a Neumann eigenfunction with respect to the positive Laplacian $\\Delta$ on a compact Riemannian manifold $M$ with boundary such that $\\Delta\\, e_\\l=\\l^2 e_\\l$ in the interior of $M$ and the normal derivative of $e_\\l$ vanishes on the boundary of $M$. Let $\\chi_\\lambda$ be the unit band spectral projection operator associated with the Neumann Laplacian and $f$ a square integrable function on $M$. We show the following gradient estimate for $\\chi_\\lambda\\,f$ as $\\lambda\\geq 1$: $\\|\\nabla\\ \\chi_\\l\\ f\\|_\\infty\\leq C\\l \\|\\chi_\\l\\f\\|_\\infty+\\l^{-1}\\|\\Delta\\ \\chi_\\l\\ f\\|_\\infty$, whe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.4033","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1306.4033","created_at":"2026-05-18T03:20:44.613946+00:00"},{"alias_kind":"arxiv_version","alias_value":"1306.4033v1","created_at":"2026-05-18T03:20:44.613946+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.4033","created_at":"2026-05-18T03:20:44.613946+00:00"},{"alias_kind":"pith_short_12","alias_value":"KP3N4OUAXAWA","created_at":"2026-05-18T12:27:51.066281+00:00"},{"alias_kind":"pith_short_16","alias_value":"KP3N4OUAXAWADUTN","created_at":"2026-05-18T12:27:51.066281+00:00"},{"alias_kind":"pith_short_8","alias_value":"KP3N4OUA","created_at":"2026-05-18T12:27:51.066281+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KP3N4OUAXAWADUTNKTGOADCICW","json":"https://pith.science/pith/KP3N4OUAXAWADUTNKTGOADCICW.json","graph_json":"https://pith.science/api/pith-number/KP3N4OUAXAWADUTNKTGOADCICW/graph.json","events_json":"https://pith.science/api/pith-number/KP3N4OUAXAWADUTNKTGOADCICW/events.json","paper":"https://pith.science/paper/KP3N4OUA"},"agent_actions":{"view_html":"https://pith.science/pith/KP3N4OUAXAWADUTNKTGOADCICW","download_json":"https://pith.science/pith/KP3N4OUAXAWADUTNKTGOADCICW.json","view_paper":"https://pith.science/paper/KP3N4OUA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1306.4033&json=true","fetch_graph":"https://pith.science/api/pith-number/KP3N4OUAXAWADUTNKTGOADCICW/graph.json","fetch_events":"https://pith.science/api/pith-number/KP3N4OUAXAWADUTNKTGOADCICW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KP3N4OUAXAWADUTNKTGOADCICW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KP3N4OUAXAWADUTNKTGOADCICW/action/storage_attestation","attest_author":"https://pith.science/pith/KP3N4OUAXAWADUTNKTGOADCICW/action/author_attestation","sign_citation":"https://pith.science/pith/KP3N4OUAXAWADUTNKTGOADCICW/action/citation_signature","submit_replication":"https://pith.science/pith/KP3N4OUAXAWADUTNKTGOADCICW/action/replication_record"}},"created_at":"2026-05-18T03:20:44.613946+00:00","updated_at":"2026-05-18T03:20:44.613946+00:00"}