{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:VE7DQGVOFDJRCW5NDWJZYMCMUH","short_pith_number":"pith:VE7DQGVO","canonical_record":{"source":{"id":"1702.02075","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-02-07T16:07:30Z","cross_cats_sorted":[],"title_canon_sha256":"03a4f88bb7596f3c547f9fc4858f1a1920844c7c311f3e2c6151b51ebe226b13","abstract_canon_sha256":"494b78564eadfa813c3321f70acd88fb9ffd82a1ea509e0dda2a41f35f79fe75"},"schema_version":"1.0"},"canonical_sha256":"a93e381aae28d3115bad1d939c304ca1e01f9eb26a0a4bdc73af816dc666f2af","source":{"kind":"arxiv","id":"1702.02075","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1702.02075","created_at":"2026-05-18T00:51:10Z"},{"alias_kind":"arxiv_version","alias_value":"1702.02075v1","created_at":"2026-05-18T00:51:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.02075","created_at":"2026-05-18T00:51:10Z"},{"alias_kind":"pith_short_12","alias_value":"VE7DQGVOFDJR","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"VE7DQGVOFDJRCW5N","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"VE7DQGVO","created_at":"2026-05-18T12:31:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:VE7DQGVOFDJRCW5NDWJZYMCMUH","target":"record","payload":{"canonical_record":{"source":{"id":"1702.02075","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-02-07T16:07:30Z","cross_cats_sorted":[],"title_canon_sha256":"03a4f88bb7596f3c547f9fc4858f1a1920844c7c311f3e2c6151b51ebe226b13","abstract_canon_sha256":"494b78564eadfa813c3321f70acd88fb9ffd82a1ea509e0dda2a41f35f79fe75"},"schema_version":"1.0"},"canonical_sha256":"a93e381aae28d3115bad1d939c304ca1e01f9eb26a0a4bdc73af816dc666f2af","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:10.396523Z","signature_b64":"6rlZwj1Cy4qktb5yY5AQC3OnY7ZQMw0kW+nC4Dr1fCJ9DwAjIhndv7kIGjWiisqa0rxjNmtSEn33RIjQA4qZBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a93e381aae28d3115bad1d939c304ca1e01f9eb26a0a4bdc73af816dc666f2af","last_reissued_at":"2026-05-18T00:51:10.395738Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:10.395738Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1702.02075","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":"2026-05-18T00:51:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AbPcPgeNezHzbZLF5ZpyNmLFmcGfNYolJNz+DLi7U3RSX2GF/of96hWhA5JxRcwxThJpFZaCZWKuXd29j7kEAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T17:33:02.909599Z"},"content_sha256":"e9ad51ec94e1a0fdf8cbc3b8b7501780ad29e92da498446bbb9038bc8b50c4ed","schema_version":"1.0","event_id":"sha256:e9ad51ec94e1a0fdf8cbc3b8b7501780ad29e92da498446bbb9038bc8b50c4ed"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:VE7DQGVOFDJRCW5NDWJZYMCMUH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fractional Sobolev Regularity for the Brouwer Degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Camillo De Lellis, Dominik Inauen","submitted_at":"2017-02-07T16:07:30Z","abstract_excerpt":"We prove that if $\\Omega\\subset \\mathbb R^n$ is a bounded open set and $n\\alpha> {\\rm dim}_b (\\partial \\Omega) = d$, then the Brouwer degree deg$(v,\\Omega,\\cdot)$ of any H\\\"older function $v\\in C^{0,\\alpha}\\left (\\Omega, \\mathbb R^{n}\\right)$ belongs to the Sobolev space $W^{\\beta, p} (\\mathbb R^n)$ for every $0\\leq \\beta < \\frac{n}{p} - \\frac{d}{\\alpha}$. This extends a summability result of Olbermann and in fact we get, as a byproduct, a more elementary proof of it. Moreover we show the optimality of the range of exponents in the following sense: for every $\\beta\\geq 0$ and $p\\geq 1$ with $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.02075","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":"2026-05-18T00:51:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U0RssUfmsszSq4amNZcnK+AIBogw2VIMQ0+TfljKa9Lh9VrX74IRgkTH2o09D8dwpKRLf5H8hwdPWemP/yI7Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T17:33:02.909973Z"},"content_sha256":"7278e0009e2eb7cc82be98d09be3518d5b872d95fb6021dd8537a01b58ca9cef","schema_version":"1.0","event_id":"sha256:7278e0009e2eb7cc82be98d09be3518d5b872d95fb6021dd8537a01b58ca9cef"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VE7DQGVOFDJRCW5NDWJZYMCMUH/bundle.json","state_url":"https://pith.science/pith/VE7DQGVOFDJRCW5NDWJZYMCMUH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VE7DQGVOFDJRCW5NDWJZYMCMUH/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":"2026-06-01T17:33:02Z","links":{"resolver":"https://pith.science/pith/VE7DQGVOFDJRCW5NDWJZYMCMUH","bundle":"https://pith.science/pith/VE7DQGVOFDJRCW5NDWJZYMCMUH/bundle.json","state":"https://pith.science/pith/VE7DQGVOFDJRCW5NDWJZYMCMUH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VE7DQGVOFDJRCW5NDWJZYMCMUH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:VE7DQGVOFDJRCW5NDWJZYMCMUH","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":"494b78564eadfa813c3321f70acd88fb9ffd82a1ea509e0dda2a41f35f79fe75","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-02-07T16:07:30Z","title_canon_sha256":"03a4f88bb7596f3c547f9fc4858f1a1920844c7c311f3e2c6151b51ebe226b13"},"schema_version":"1.0","source":{"id":"1702.02075","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1702.02075","created_at":"2026-05-18T00:51:10Z"},{"alias_kind":"arxiv_version","alias_value":"1702.02075v1","created_at":"2026-05-18T00:51:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.02075","created_at":"2026-05-18T00:51:10Z"},{"alias_kind":"pith_short_12","alias_value":"VE7DQGVOFDJR","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"VE7DQGVOFDJRCW5N","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"VE7DQGVO","created_at":"2026-05-18T12:31:49Z"}],"graph_snapshots":[{"event_id":"sha256:7278e0009e2eb7cc82be98d09be3518d5b872d95fb6021dd8537a01b58ca9cef","target":"graph","created_at":"2026-05-18T00:51:10Z","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 prove that if $\\Omega\\subset \\mathbb R^n$ is a bounded open set and $n\\alpha> {\\rm dim}_b (\\partial \\Omega) = d$, then the Brouwer degree deg$(v,\\Omega,\\cdot)$ of any H\\\"older function $v\\in C^{0,\\alpha}\\left (\\Omega, \\mathbb R^{n}\\right)$ belongs to the Sobolev space $W^{\\beta, p} (\\mathbb R^n)$ for every $0\\leq \\beta < \\frac{n}{p} - \\frac{d}{\\alpha}$. This extends a summability result of Olbermann and in fact we get, as a byproduct, a more elementary proof of it. Moreover we show the optimality of the range of exponents in the following sense: for every $\\beta\\geq 0$ and $p\\geq 1$ with $\\","authors_text":"Camillo De Lellis, Dominik Inauen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-02-07T16:07:30Z","title":"Fractional Sobolev Regularity for the Brouwer Degree"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.02075","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:e9ad51ec94e1a0fdf8cbc3b8b7501780ad29e92da498446bbb9038bc8b50c4ed","target":"record","created_at":"2026-05-18T00:51:10Z","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":"494b78564eadfa813c3321f70acd88fb9ffd82a1ea509e0dda2a41f35f79fe75","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-02-07T16:07:30Z","title_canon_sha256":"03a4f88bb7596f3c547f9fc4858f1a1920844c7c311f3e2c6151b51ebe226b13"},"schema_version":"1.0","source":{"id":"1702.02075","kind":"arxiv","version":1}},"canonical_sha256":"a93e381aae28d3115bad1d939c304ca1e01f9eb26a0a4bdc73af816dc666f2af","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a93e381aae28d3115bad1d939c304ca1e01f9eb26a0a4bdc73af816dc666f2af","first_computed_at":"2026-05-18T00:51:10.395738Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:51:10.395738Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6rlZwj1Cy4qktb5yY5AQC3OnY7ZQMw0kW+nC4Dr1fCJ9DwAjIhndv7kIGjWiisqa0rxjNmtSEn33RIjQA4qZBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:51:10.396523Z","signed_message":"canonical_sha256_bytes"},"source_id":"1702.02075","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e9ad51ec94e1a0fdf8cbc3b8b7501780ad29e92da498446bbb9038bc8b50c4ed","sha256:7278e0009e2eb7cc82be98d09be3518d5b872d95fb6021dd8537a01b58ca9cef"],"state_sha256":"b413e5c549241cbf29f16ab94d4951ec2b71e6b2f91d45b315b3b8f24e2bcf8a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7vTTB4SpvFqJ/SuGLJuPUR8AFCNi96asiv+OLCuXmPjnqXI3ae2YYiniTaEl4pbeO696ibB8LMrQseNDVyUMBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T17:33:02.912155Z","bundle_sha256":"4af851b525188bb4eb7563ce2a34ca4298595a60dffc3fe008bf88adc128cd05"}}