{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:5KR3OJVR6GPYKYIDQR5CAY7N56","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":"55ccb5357d7d77c19931e8cdb0242162b246e5d7d0223f34a709fb307e0ea5dc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-06-06T20:36:27Z","title_canon_sha256":"877905e16680a91e94eb8a2d654f501f6da72196c113e7b8ce6ac2b4e7ce1a0d"},"schema_version":"1.0","source":{"id":"2006.03966","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2006.03966","created_at":"2026-07-05T01:08:34Z"},{"alias_kind":"arxiv_version","alias_value":"2006.03966v1","created_at":"2026-07-05T01:08:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.03966","created_at":"2026-07-05T01:08:34Z"},{"alias_kind":"pith_short_12","alias_value":"5KR3OJVR6GPY","created_at":"2026-07-05T01:08:34Z"},{"alias_kind":"pith_short_16","alias_value":"5KR3OJVR6GPYKYID","created_at":"2026-07-05T01:08:34Z"},{"alias_kind":"pith_short_8","alias_value":"5KR3OJVR","created_at":"2026-07-05T01:08:34Z"}],"graph_snapshots":[{"event_id":"sha256:8496d8c91bd193d13c267eda4f4af01d41da2eb426cfc4c9378322a2ba4cdd9f","target":"graph","created_at":"2026-07-05T01:08:34Z","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/2006.03966/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we review some results over the last 10-15 years on elliptic and parabolic equations with discontinuous coefficients. We begin with an approach given by N. V. Krylov to parabolic equations in the whole space with VMO$_x$ coefficients. We then discuss some subsequent development including elliptic and parabolic equations with coefficients which are allowed to be merely measurable in one or two space directions, weighted $L_p$ estimates with Muckenhoupt ($A_p$) weights, non-local elliptic and parabolic equations, as well as fully nonlinear elliptic and parabolic equations.","authors_text":"Hongjie Dong","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-06-06T20:36:27Z","title":"Recent progress in the $L_p$ theory for elliptic and parabolic equations with discontinuous coefficients"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.03966","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":"sha256:8483224fa059359b62fee0c5f384a883b7d52df96d30a5e8b61eac39a785a31d","target":"record","created_at":"2026-07-05T01:08:34Z","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":"55ccb5357d7d77c19931e8cdb0242162b246e5d7d0223f34a709fb307e0ea5dc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-06-06T20:36:27Z","title_canon_sha256":"877905e16680a91e94eb8a2d654f501f6da72196c113e7b8ce6ac2b4e7ce1a0d"},"schema_version":"1.0","source":{"id":"2006.03966","kind":"arxiv","version":1}},"canonical_sha256":"eaa3b726b1f19f856103847a2063edef883fc85c012a249210e248392964764b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eaa3b726b1f19f856103847a2063edef883fc85c012a249210e248392964764b","first_computed_at":"2026-07-05T01:08:34.583960Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:08:34.583960Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WeLKC8ItIiMuY86Al4dSWRJuExBNhd7zAoJ6+2xDS5QuKj+j18U5josYM9uH4JRM/mb7xRC7MesBGF2pJW8/Bw==","signature_status":"signed_v1","signed_at":"2026-07-05T01:08:34.584514Z","signed_message":"canonical_sha256_bytes"},"source_id":"2006.03966","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8483224fa059359b62fee0c5f384a883b7d52df96d30a5e8b61eac39a785a31d","sha256:8496d8c91bd193d13c267eda4f4af01d41da2eb426cfc4c9378322a2ba4cdd9f"],"state_sha256":"29c122b5253a8ed731edfb5e6b3bdf0e88f9d7b3bddd13851e5252ba09343cc4"}