{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:6IZ7OFADQXJDOTV7XI4CUK6K3I","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":"61e9a55664d1c22dfa5c3fa168be8119bea33f6d6d589fd75412eb1fa81b1f06","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-06-06T02:25:28Z","title_canon_sha256":"34cb92af4061eca0973616896752b7bcd9c1fa60b0bf144d14bac604ad1944b3"},"schema_version":"1.0","source":{"id":"1806.01982","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.01982","created_at":"2026-05-18T00:14:03Z"},{"alias_kind":"arxiv_version","alias_value":"1806.01982v1","created_at":"2026-05-18T00:14:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.01982","created_at":"2026-05-18T00:14:03Z"},{"alias_kind":"pith_short_12","alias_value":"6IZ7OFADQXJD","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_16","alias_value":"6IZ7OFADQXJDOTV7","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_8","alias_value":"6IZ7OFAD","created_at":"2026-05-18T12:32:08Z"}],"graph_snapshots":[{"event_id":"sha256:3f9857e49505887f5efd9e8fb93c2f5b81502879b472539acf145b920cb2ed22","target":"graph","created_at":"2026-05-18T00:14:03Z","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"},"paper":{"abstract_excerpt":"Given an arbitrary planar $\\infty$-harmonic function $u$, for each $\\alpha>0$ we establish a quantitative local $W^{1,2}$-estimate of $|Du|^\\alpha $, which is sharp as $\\alpha\\to0$. We also show that the distributional determinant of $u$ is a Radon measure enjoying some quantitative lower and upper bounds. As a by-product, for each $p>2$ we obtain some quantitative local $W^{1,p}$-estimates of $u$, and consequently, an $L^p$-Liouville property for $\\infty$-harmonic functions in whole plane.","authors_text":"Herbert Koch, Yi Ru-Ya Zhang, Yuan Zhou","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-06-06T02:25:28Z","title":"An asymtotic sharp Sobolev regularity for planar infinity harmonic functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.01982","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":"sha256:339ce7188c43cc89dbc01a533f6a924806d923e56af0e226e58519faeeb4589c","target":"record","created_at":"2026-05-18T00:14:03Z","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":"61e9a55664d1c22dfa5c3fa168be8119bea33f6d6d589fd75412eb1fa81b1f06","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-06-06T02:25:28Z","title_canon_sha256":"34cb92af4061eca0973616896752b7bcd9c1fa60b0bf144d14bac604ad1944b3"},"schema_version":"1.0","source":{"id":"1806.01982","kind":"arxiv","version":1}},"canonical_sha256":"f233f7140385d2374ebfba382a2bcada321f2b5b7036f24593499454561e1629","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f233f7140385d2374ebfba382a2bcada321f2b5b7036f24593499454561e1629","first_computed_at":"2026-05-18T00:14:03.152694Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:03.152694Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Bc2LmndMnI8mcZheXM3Nf5iTne5n1OYmVvGAtNOesacwIX67lORN6lIn5VIGEET1o8Op9YbfgRYPSn6jDDMvDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:03.153488Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.01982","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:339ce7188c43cc89dbc01a533f6a924806d923e56af0e226e58519faeeb4589c","sha256:3f9857e49505887f5efd9e8fb93c2f5b81502879b472539acf145b920cb2ed22"],"state_sha256":"b5bf757acd65cc04b22c216bf418dd6cf1beab18f2014b480626962f46756d23"}