{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:FHM2KAOQOET5LPKBQVVQQJHLLB","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":"96dc7c92e701d6f59745ad66ea480db8da9d59a35425a9e33c7eada62ee022ae","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2024-01-05T14:52:22Z","title_canon_sha256":"420eecdc0e13d7bfda8453f877bf47653ef9ea6c49d3569ca8540df193305ecf"},"schema_version":"1.0","source":{"id":"2401.04129","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2401.04129","created_at":"2026-07-05T08:21:29Z"},{"alias_kind":"arxiv_version","alias_value":"2401.04129v1","created_at":"2026-07-05T08:21:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2401.04129","created_at":"2026-07-05T08:21:29Z"},{"alias_kind":"pith_short_12","alias_value":"FHM2KAOQOET5","created_at":"2026-07-05T08:21:29Z"},{"alias_kind":"pith_short_16","alias_value":"FHM2KAOQOET5LPKB","created_at":"2026-07-05T08:21:29Z"},{"alias_kind":"pith_short_8","alias_value":"FHM2KAOQ","created_at":"2026-07-05T08:21:29Z"}],"graph_snapshots":[{"event_id":"sha256:ddd2f5af629ef71d146519fd9c68aceb2ef9febc2caa744796a750e15fa49306","target":"graph","created_at":"2026-07-05T08:21:29Z","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/2401.04129/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The best constant and extremal functions are well known of the following Caffarelli-Kohn-Nirenberg inequality \\[ \\int_{\\mathbb{R}^N}|\\nabla u|^p\\frac{\\mathrm{d}x}{|x|^{\\mu}}\\geq \\mathcal{S} \\left(\\int_{\\mathbb{R}^N}|u|^r\\frac{\\mathrm{d}x}{|x|^s} \\right)^{\\frac{p}{r}}, \\quad \\mbox{for all}\\quad u\\in C^\\infty_c(\\mathbb{R}^N), \\] where $1<p<p+\\mu<N$, $\\frac{\\mu}{p}\\leq \\frac{s}{r}<\\frac{\\mu}{p}+1$, $r=\\frac{p(N-s)}{N-p-\\mu}$. An important task is investigating the stability of extremals for this inequality. Firstly, we give the classification to the linearized problem related to the extremals whi","authors_text":"Shengbing Deng, Xingliang Tian","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2024-01-05T14:52:22Z","title":"Gradient stability of Caffarelli-Kohn-Nirenberg inequality involving weighted p-Laplace"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.04129","kind":"arxiv","version":1},"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":"sha256:0d925ba8a7e6d961ada838816fc3b4fcf7ff1a77b2598e869891f6cc77a08d48","target":"record","created_at":"2026-07-05T08:21:29Z","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":"96dc7c92e701d6f59745ad66ea480db8da9d59a35425a9e33c7eada62ee022ae","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2024-01-05T14:52:22Z","title_canon_sha256":"420eecdc0e13d7bfda8453f877bf47653ef9ea6c49d3569ca8540df193305ecf"},"schema_version":"1.0","source":{"id":"2401.04129","kind":"arxiv","version":1}},"canonical_sha256":"29d9a501d07127d5bd41856b0824eb587c38229a7af1bd3edaa8b9497e106f8c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"29d9a501d07127d5bd41856b0824eb587c38229a7af1bd3edaa8b9497e106f8c","first_computed_at":"2026-07-05T08:21:29.419755Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:21:29.419755Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ws8Z+GaMvJQmVreXE15m+nCKFDfPMPQ4vEsw/oZAPshBrxZYuTJWkabRVLXV1uih/ln6sxtxE5UjijxewBFXBg==","signature_status":"signed_v1","signed_at":"2026-07-05T08:21:29.420226Z","signed_message":"canonical_sha256_bytes"},"source_id":"2401.04129","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0d925ba8a7e6d961ada838816fc3b4fcf7ff1a77b2598e869891f6cc77a08d48","sha256:ddd2f5af629ef71d146519fd9c68aceb2ef9febc2caa744796a750e15fa49306"],"state_sha256":"0fa3a2d847d47423cd4dd9dc6dc386007b5d0c2af571b41ed8defc12769a51bf"}