{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:5QYKH6KAP4YJKUVCLEL5SOZL3I","short_pith_number":"pith:5QYKH6KA","schema_version":"1.0","canonical_sha256":"ec30a3f9407f309552a25917d93b2bda19111391e2ef8402508b78b777a60541","source":{"kind":"arxiv","id":"1505.07623","version":2},"attestation_state":"computed","paper":{"title":"Local and global sharp gradient estimates for weighted $p$-harmonic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Nguyen Duy Dat, Nguyen Thac Dung","submitted_at":"2015-05-28T10:11:30Z","abstract_excerpt":"Let $(M^n, g, e^{-f}dv)$ be a smooth metric measure space of dimensional $n$. Suppose that $v$ is a positive weighted $p$-eigenfunctions associated to the eigenvalues $\\lambda_{1,p}$ on $M$, namely $$ e^{f}div(e^{-f}|\\nabla v|^{p-2}\\nabla v)=-\\lambda_{1,p}v^{p-1}.$$ in the distribution sense. We first give a local gradient estimate for $v$ provided the $m$-dimmensional Bakry-\\'Emery curvature $Ric_f^{m}$ bounded from below. Consequently, we show that when $Ric_f^m\\geq0$ then $v$ is constant if $v$ is of sublinear growth. At the same time, we prove a Harnack inequality for weighted $p$-harmonic"},"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":"1505.07623","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-05-28T10:11:30Z","cross_cats_sorted":[],"title_canon_sha256":"ddae1226281921bd311ab0cbd3b909e3d59da84233640518fcdd02c58e6718a4","abstract_canon_sha256":"72eb939965207d0049b3a928cbdab3f8ddb0d49e55b00e882025fbe935689a31"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:18.322486Z","signature_b64":"/Jx3/41z4ipSf87XjpLdV8kXFcDebP8PKsuDumNeOfnaUjPnl6qIKgThhfXeaVvf1gFObJ7Q6dM9+JQCA9gnCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec30a3f9407f309552a25917d93b2bda19111391e2ef8402508b78b777a60541","last_reissued_at":"2026-05-18T01:26:18.321815Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:18.321815Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Local and global sharp gradient estimates for weighted $p$-harmonic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Nguyen Duy Dat, Nguyen Thac Dung","submitted_at":"2015-05-28T10:11:30Z","abstract_excerpt":"Let $(M^n, g, e^{-f}dv)$ be a smooth metric measure space of dimensional $n$. Suppose that $v$ is a positive weighted $p$-eigenfunctions associated to the eigenvalues $\\lambda_{1,p}$ on $M$, namely $$ e^{f}div(e^{-f}|\\nabla v|^{p-2}\\nabla v)=-\\lambda_{1,p}v^{p-1}.$$ in the distribution sense. We first give a local gradient estimate for $v$ provided the $m$-dimmensional Bakry-\\'Emery curvature $Ric_f^{m}$ bounded from below. Consequently, we show that when $Ric_f^m\\geq0$ then $v$ is constant if $v$ is of sublinear growth. At the same time, we prove a Harnack inequality for weighted $p$-harmonic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.07623","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1505.07623","created_at":"2026-05-18T01:26:18.321924+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.07623v2","created_at":"2026-05-18T01:26:18.321924+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.07623","created_at":"2026-05-18T01:26:18.321924+00:00"},{"alias_kind":"pith_short_12","alias_value":"5QYKH6KAP4YJ","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_16","alias_value":"5QYKH6KAP4YJKUVC","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_8","alias_value":"5QYKH6KA","created_at":"2026-05-18T12:29:05.191682+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/5QYKH6KAP4YJKUVCLEL5SOZL3I","json":"https://pith.science/pith/5QYKH6KAP4YJKUVCLEL5SOZL3I.json","graph_json":"https://pith.science/api/pith-number/5QYKH6KAP4YJKUVCLEL5SOZL3I/graph.json","events_json":"https://pith.science/api/pith-number/5QYKH6KAP4YJKUVCLEL5SOZL3I/events.json","paper":"https://pith.science/paper/5QYKH6KA"},"agent_actions":{"view_html":"https://pith.science/pith/5QYKH6KAP4YJKUVCLEL5SOZL3I","download_json":"https://pith.science/pith/5QYKH6KAP4YJKUVCLEL5SOZL3I.json","view_paper":"https://pith.science/paper/5QYKH6KA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.07623&json=true","fetch_graph":"https://pith.science/api/pith-number/5QYKH6KAP4YJKUVCLEL5SOZL3I/graph.json","fetch_events":"https://pith.science/api/pith-number/5QYKH6KAP4YJKUVCLEL5SOZL3I/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5QYKH6KAP4YJKUVCLEL5SOZL3I/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5QYKH6KAP4YJKUVCLEL5SOZL3I/action/storage_attestation","attest_author":"https://pith.science/pith/5QYKH6KAP4YJKUVCLEL5SOZL3I/action/author_attestation","sign_citation":"https://pith.science/pith/5QYKH6KAP4YJKUVCLEL5SOZL3I/action/citation_signature","submit_replication":"https://pith.science/pith/5QYKH6KAP4YJKUVCLEL5SOZL3I/action/replication_record"}},"created_at":"2026-05-18T01:26:18.321924+00:00","updated_at":"2026-05-18T01:26:18.321924+00:00"}