{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:J34AEHOI2MKGOF3IYALZCVSETF","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":"3d39ba8b0d64fd1f41b8b18122fcb9fed1fe776f502e7aca5758620619fce459","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-05T10:22:07Z","title_canon_sha256":"e74969f8591b24f3277bcb282033b86629b0df572e938f3f08b4266ebead7eee"},"schema_version":"1.0","source":{"id":"1908.01547","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.01547","created_at":"2026-07-04T23:51:47Z"},{"alias_kind":"arxiv_version","alias_value":"1908.01547v2","created_at":"2026-07-04T23:51:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.01547","created_at":"2026-07-04T23:51:47Z"},{"alias_kind":"pith_short_12","alias_value":"J34AEHOI2MKG","created_at":"2026-07-04T23:51:47Z"},{"alias_kind":"pith_short_16","alias_value":"J34AEHOI2MKGOF3I","created_at":"2026-07-04T23:51:47Z"},{"alias_kind":"pith_short_8","alias_value":"J34AEHOI","created_at":"2026-07-04T23:51:47Z"}],"graph_snapshots":[{"event_id":"sha256:7e50cf7600aeabaa70a423fa7f96bf20444f373aac2de41a81b239c226be99e5","target":"graph","created_at":"2026-07-04T23:51:47Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.01547/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Hongjie Dong, Peng Fa, Yi Ru-Ya Zhang, Yuan Zhou","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-05T10:22:07Z","title":"Second order regularity for elliptic and parabolic equations involving $p$-Laplacian via a fundamental inequality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.01547","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:bf9742601faa2c586bec63ff2ab43dfc30a027d9644490603e50aff673874cb0","target":"record","created_at":"2026-07-04T23:51:47Z","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":"3d39ba8b0d64fd1f41b8b18122fcb9fed1fe776f502e7aca5758620619fce459","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-08-05T10:22:07Z","title_canon_sha256":"e74969f8591b24f3277bcb282033b86629b0df572e938f3f08b4266ebead7eee"},"schema_version":"1.0","source":{"id":"1908.01547","kind":"arxiv","version":2}},"canonical_sha256":"4ef8021dc8d314671768c017915644996ed0c75cecd1deaa7bcb7900bc1991f2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4ef8021dc8d314671768c017915644996ed0c75cecd1deaa7bcb7900bc1991f2","first_computed_at":"2026-07-04T23:51:47.356274Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T23:51:47.356274Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HS9LoDs4NALQOfch0q1Q7SlGQQkK6rAtM0azdx9Ez6v9kytiFIpOUg2BwBQmch8QQL7hlfBZb3SPGhdRscURCQ==","signature_status":"signed_v1","signed_at":"2026-07-04T23:51:47.356699Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.01547","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bf9742601faa2c586bec63ff2ab43dfc30a027d9644490603e50aff673874cb0","sha256:7e50cf7600aeabaa70a423fa7f96bf20444f373aac2de41a81b239c226be99e5"],"state_sha256":"ce13dc8445b94f2416fadb817aca7f3e6cdae7d20a51c3e23a5f3dfb08c0bddd"}