{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:J34AEHOI2MKGOF3IYALZCVSETF","short_pith_number":"pith:J34AEHOI","schema_version":"1.0","canonical_sha256":"4ef8021dc8d314671768c017915644996ed0c75cecd1deaa7bcb7900bc1991f2","source":{"kind":"arxiv","id":"1908.01547","version":2},"attestation_state":"computed","paper":{"title":"Second order regularity for elliptic and parabolic equations involving $p$-Laplacian via a fundamental inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hongjie Dong, Peng Fa, Yi Ru-Ya Zhang, Yuan Zhou","submitted_at":"2019-08-05T10:22:07Z","abstract_excerpt":"Denote by $\\Delta$ the Laplacian and by $\\Delta_\\infty $ the $\\infty$-Laplacian. A fundamental inequality is proved for the algebraic structure of $\\Delta v\\Delta_\\infty v$: for every $v\\in C^\\infty$, $$\\ | { |D^2vDv|^2} - {\\Delta v \\Delta_\\infty v } -\\frac12[|D^2v|^2-(\\Delta v)^2]|Dv|^2\\ | \\le \\frac{n-2}2 [|D^2v|^2{|Dv|^2}- |D^2vDv|^2 ]. $$ Based on this, we prove the following results:\n  1. For any $p$-harmonic functions $u$, $p\\in(1,2)\\cup(2,\\infty)$, we have $$|Du|^{\\frac{p-\\gamma}2}Du\\in W^{1,2}_{\\rm loc},$$ with $\\gamma<\\min\\{p+\\frac{n-1}{n},3+\\frac{p-1}{n-1}\\}$. As a by-product, when $p"},"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":"1908.01547","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-05T10:22:07Z","cross_cats_sorted":[],"title_canon_sha256":"e74969f8591b24f3277bcb282033b86629b0df572e938f3f08b4266ebead7eee","abstract_canon_sha256":"3d39ba8b0d64fd1f41b8b18122fcb9fed1fe776f502e7aca5758620619fce459"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:51:47.356699Z","signature_b64":"HS9LoDs4NALQOfch0q1Q7SlGQQkK6rAtM0azdx9Ez6v9kytiFIpOUg2BwBQmch8QQL7hlfBZb3SPGhdRscURCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4ef8021dc8d314671768c017915644996ed0c75cecd1deaa7bcb7900bc1991f2","last_reissued_at":"2026-07-04T23:51:47.356274Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:51:47.356274Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Second order regularity for elliptic and parabolic equations involving $p$-Laplacian via a fundamental inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hongjie Dong, Peng Fa, Yi Ru-Ya Zhang, Yuan Zhou","submitted_at":"2019-08-05T10:22:07Z","abstract_excerpt":"Denote by $\\Delta$ the Laplacian and by $\\Delta_\\infty $ the $\\infty$-Laplacian. A fundamental inequality is proved for the algebraic structure of $\\Delta v\\Delta_\\infty v$: for every $v\\in C^\\infty$, $$\\ | { |D^2vDv|^2} - {\\Delta v \\Delta_\\infty v } -\\frac12[|D^2v|^2-(\\Delta v)^2]|Dv|^2\\ | \\le \\frac{n-2}2 [|D^2v|^2{|Dv|^2}- |D^2vDv|^2 ]. $$ Based on this, we prove the following results:\n  1. For any $p$-harmonic functions $u$, $p\\in(1,2)\\cup(2,\\infty)$, we have $$|Du|^{\\frac{p-\\gamma}2}Du\\in W^{1,2}_{\\rm loc},$$ with $\\gamma<\\min\\{p+\\frac{n-1}{n},3+\\frac{p-1}{n-1}\\}$. As a by-product, when $p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.01547","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.01547/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"1908.01547","created_at":"2026-07-04T23:51:47.356343+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.01547v2","created_at":"2026-07-04T23:51:47.356343+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.01547","created_at":"2026-07-04T23:51:47.356343+00:00"},{"alias_kind":"pith_short_12","alias_value":"J34AEHOI2MKG","created_at":"2026-07-04T23:51:47.356343+00:00"},{"alias_kind":"pith_short_16","alias_value":"J34AEHOI2MKGOF3I","created_at":"2026-07-04T23:51:47.356343+00:00"},{"alias_kind":"pith_short_8","alias_value":"J34AEHOI","created_at":"2026-07-04T23:51:47.356343+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/J34AEHOI2MKGOF3IYALZCVSETF","json":"https://pith.science/pith/J34AEHOI2MKGOF3IYALZCVSETF.json","graph_json":"https://pith.science/api/pith-number/J34AEHOI2MKGOF3IYALZCVSETF/graph.json","events_json":"https://pith.science/api/pith-number/J34AEHOI2MKGOF3IYALZCVSETF/events.json","paper":"https://pith.science/paper/J34AEHOI"},"agent_actions":{"view_html":"https://pith.science/pith/J34AEHOI2MKGOF3IYALZCVSETF","download_json":"https://pith.science/pith/J34AEHOI2MKGOF3IYALZCVSETF.json","view_paper":"https://pith.science/paper/J34AEHOI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.01547&json=true","fetch_graph":"https://pith.science/api/pith-number/J34AEHOI2MKGOF3IYALZCVSETF/graph.json","fetch_events":"https://pith.science/api/pith-number/J34AEHOI2MKGOF3IYALZCVSETF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/J34AEHOI2MKGOF3IYALZCVSETF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/J34AEHOI2MKGOF3IYALZCVSETF/action/storage_attestation","attest_author":"https://pith.science/pith/J34AEHOI2MKGOF3IYALZCVSETF/action/author_attestation","sign_citation":"https://pith.science/pith/J34AEHOI2MKGOF3IYALZCVSETF/action/citation_signature","submit_replication":"https://pith.science/pith/J34AEHOI2MKGOF3IYALZCVSETF/action/replication_record"}},"created_at":"2026-07-04T23:51:47.356343+00:00","updated_at":"2026-07-04T23:51:47.356343+00:00"}