{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:DMYX5Q5WUXR5RLRSVF2G4OTNUQ","short_pith_number":"pith:DMYX5Q5W","canonical_record":{"source":{"id":"1807.07597","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-19T18:34:37Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"0b47a9cdb240fd749f2f19a18f0d14b1f6937143cdd98fb4e67de4565fa6b5f8","abstract_canon_sha256":"b84412895b1f51b946df993209ebf817023f69ffacf2ab94cdfbcdaf4a45fd2e"},"schema_version":"1.0"},"canonical_sha256":"1b317ec3b6a5e3d8ae32a9746e3a6da413991e3e47ebcfffa3b6f5ad6bc78413","source":{"kind":"arxiv","id":"1807.07597","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.07597","created_at":"2026-05-18T00:10:16Z"},{"alias_kind":"arxiv_version","alias_value":"1807.07597v1","created_at":"2026-05-18T00:10:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.07597","created_at":"2026-05-18T00:10:16Z"},{"alias_kind":"pith_short_12","alias_value":"DMYX5Q5WUXR5","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"DMYX5Q5WUXR5RLRS","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"DMYX5Q5W","created_at":"2026-05-18T12:32:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:DMYX5Q5WUXR5RLRSVF2G4OTNUQ","target":"record","payload":{"canonical_record":{"source":{"id":"1807.07597","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-19T18:34:37Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"0b47a9cdb240fd749f2f19a18f0d14b1f6937143cdd98fb4e67de4565fa6b5f8","abstract_canon_sha256":"b84412895b1f51b946df993209ebf817023f69ffacf2ab94cdfbcdaf4a45fd2e"},"schema_version":"1.0"},"canonical_sha256":"1b317ec3b6a5e3d8ae32a9746e3a6da413991e3e47ebcfffa3b6f5ad6bc78413","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:16.152786Z","signature_b64":"p74bRnQtvVsOM6cqLStrK54Am8DP2QzUJynrUTUVzhFMWquKIYixd4QR7H7mcXCmgf3qQIm9bqtEMucF80kGCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1b317ec3b6a5e3d8ae32a9746e3a6da413991e3e47ebcfffa3b6f5ad6bc78413","last_reissued_at":"2026-05-18T00:10:16.152144Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:16.152144Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1807.07597","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:10:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j8rd8TMPOKut86JdcFvQiQP2OKoqHoISV7wgKadugFLTezAunnqNa2x0JDZOwy3BbiiseBzE+AMZZqLXSw95DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T15:55:00.771127Z"},"content_sha256":"21cf9060a6e79bb0508ea6fb63c84a6f33cf0b9f54f7f6b1791007222a87d8b0","schema_version":"1.0","event_id":"sha256:21cf9060a6e79bb0508ea6fb63c84a6f33cf0b9f54f7f6b1791007222a87d8b0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:DMYX5Q5WUXR5RLRSVF2G4OTNUQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$\\mathcal W^{\\alpha, p}$ and $C^{0,\\gamma}$ regularity of solutions to $(\\mu - \\Delta + b \\cdot \\nabla)u=f$ with form-bounded vector fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.AP","authors_text":"Damir Kinzebulatov","submitted_at":"2018-07-19T18:34:37Z","abstract_excerpt":"We consider the operator $-\\Delta +b \\cdot \\nabla$ with $b:\\mathbb R^d \\rightarrow \\mathbb R^d$ ($d \\geq 3$) in the class of form-bounded vector fields (containing vector fields having critical-order singularities), and characterize quantitative dependence of the $\\mathcal W^{1+\\frac{2}{q},p}$ ($2 \\leq p < q$) and the $C^{0,\\gamma}$ regularity of solutions to the corresponding elliptic equation in $L^p$ on the value of the form-bound of $b$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.07597","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:10:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9eFxTMiNC19HyPoMebjy+qhDLht7Y3TgG6JpRbBnzE+0Kk9n2/ZaITGl1v8LX3dW3ZF0e+7pOolcJIgpedUpBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T15:55:00.771866Z"},"content_sha256":"76df05e463323eeb5a60506b5b05eff01d38288cfbdc3e6f3224d52ce1db29bf","schema_version":"1.0","event_id":"sha256:76df05e463323eeb5a60506b5b05eff01d38288cfbdc3e6f3224d52ce1db29bf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DMYX5Q5WUXR5RLRSVF2G4OTNUQ/bundle.json","state_url":"https://pith.science/pith/DMYX5Q5WUXR5RLRSVF2G4OTNUQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DMYX5Q5WUXR5RLRSVF2G4OTNUQ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-06T15:55:00Z","links":{"resolver":"https://pith.science/pith/DMYX5Q5WUXR5RLRSVF2G4OTNUQ","bundle":"https://pith.science/pith/DMYX5Q5WUXR5RLRSVF2G4OTNUQ/bundle.json","state":"https://pith.science/pith/DMYX5Q5WUXR5RLRSVF2G4OTNUQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DMYX5Q5WUXR5RLRSVF2G4OTNUQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:DMYX5Q5WUXR5RLRSVF2G4OTNUQ","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":"b84412895b1f51b946df993209ebf817023f69ffacf2ab94cdfbcdaf4a45fd2e","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-19T18:34:37Z","title_canon_sha256":"0b47a9cdb240fd749f2f19a18f0d14b1f6937143cdd98fb4e67de4565fa6b5f8"},"schema_version":"1.0","source":{"id":"1807.07597","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.07597","created_at":"2026-05-18T00:10:16Z"},{"alias_kind":"arxiv_version","alias_value":"1807.07597v1","created_at":"2026-05-18T00:10:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.07597","created_at":"2026-05-18T00:10:16Z"},{"alias_kind":"pith_short_12","alias_value":"DMYX5Q5WUXR5","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"DMYX5Q5WUXR5RLRS","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"DMYX5Q5W","created_at":"2026-05-18T12:32:19Z"}],"graph_snapshots":[{"event_id":"sha256:76df05e463323eeb5a60506b5b05eff01d38288cfbdc3e6f3224d52ce1db29bf","target":"graph","created_at":"2026-05-18T00:10:16Z","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":"We consider the operator $-\\Delta +b \\cdot \\nabla$ with $b:\\mathbb R^d \\rightarrow \\mathbb R^d$ ($d \\geq 3$) in the class of form-bounded vector fields (containing vector fields having critical-order singularities), and characterize quantitative dependence of the $\\mathcal W^{1+\\frac{2}{q},p}$ ($2 \\leq p < q$) and the $C^{0,\\gamma}$ regularity of solutions to the corresponding elliptic equation in $L^p$ on the value of the form-bound of $b$.","authors_text":"Damir Kinzebulatov","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-19T18:34:37Z","title":"$\\mathcal W^{\\alpha, p}$ and $C^{0,\\gamma}$ regularity of solutions to $(\\mu - \\Delta + b \\cdot \\nabla)u=f$ with form-bounded vector fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.07597","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":"sha256:21cf9060a6e79bb0508ea6fb63c84a6f33cf0b9f54f7f6b1791007222a87d8b0","target":"record","created_at":"2026-05-18T00:10:16Z","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":"b84412895b1f51b946df993209ebf817023f69ffacf2ab94cdfbcdaf4a45fd2e","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-19T18:34:37Z","title_canon_sha256":"0b47a9cdb240fd749f2f19a18f0d14b1f6937143cdd98fb4e67de4565fa6b5f8"},"schema_version":"1.0","source":{"id":"1807.07597","kind":"arxiv","version":1}},"canonical_sha256":"1b317ec3b6a5e3d8ae32a9746e3a6da413991e3e47ebcfffa3b6f5ad6bc78413","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1b317ec3b6a5e3d8ae32a9746e3a6da413991e3e47ebcfffa3b6f5ad6bc78413","first_computed_at":"2026-05-18T00:10:16.152144Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:16.152144Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"p74bRnQtvVsOM6cqLStrK54Am8DP2QzUJynrUTUVzhFMWquKIYixd4QR7H7mcXCmgf3qQIm9bqtEMucF80kGCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:16.152786Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.07597","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:21cf9060a6e79bb0508ea6fb63c84a6f33cf0b9f54f7f6b1791007222a87d8b0","sha256:76df05e463323eeb5a60506b5b05eff01d38288cfbdc3e6f3224d52ce1db29bf"],"state_sha256":"428b93efcbd9c816f3b7efdd162cbaee539448c3cc83038283aed0b77eeea73c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tRHge6d+ibhCGZnQWd9GElKDdkyuksaavfTvfJlvwGVVd+pBRK7WO4boykiZSG53tW1jk1/bio8YI/9+b1gcAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T15:55:00.778571Z","bundle_sha256":"e653d966b2ca5015602f739f532828ba3ba3b95e8d405eae3b5de606fd249bf6"}}