{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:JMTEBCKIGU6FQ5KCQNA2BNWBJM","short_pith_number":"pith:JMTEBCKI","canonical_record":{"source":{"id":"1512.03198","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-12-10T10:24:02Z","cross_cats_sorted":[],"title_canon_sha256":"d353a6cc66f809ee2e7d3b31a0ca82d5e12fafd0bd0e4972f1fa000c03866906","abstract_canon_sha256":"a88637eca625522100d3bc3d43350a1a06efdee05fc1117767be43f18f0bec34"},"schema_version":"1.0"},"canonical_sha256":"4b26408948353c5875428341a0b6c14b02cdb11e6afe3c033bc6df42899325ba","source":{"kind":"arxiv","id":"1512.03198","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.03198","created_at":"2026-05-18T01:24:37Z"},{"alias_kind":"arxiv_version","alias_value":"1512.03198v1","created_at":"2026-05-18T01:24:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.03198","created_at":"2026-05-18T01:24:37Z"},{"alias_kind":"pith_short_12","alias_value":"JMTEBCKIGU6F","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JMTEBCKIGU6FQ5KC","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JMTEBCKI","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:JMTEBCKIGU6FQ5KCQNA2BNWBJM","target":"record","payload":{"canonical_record":{"source":{"id":"1512.03198","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-12-10T10:24:02Z","cross_cats_sorted":[],"title_canon_sha256":"d353a6cc66f809ee2e7d3b31a0ca82d5e12fafd0bd0e4972f1fa000c03866906","abstract_canon_sha256":"a88637eca625522100d3bc3d43350a1a06efdee05fc1117767be43f18f0bec34"},"schema_version":"1.0"},"canonical_sha256":"4b26408948353c5875428341a0b6c14b02cdb11e6afe3c033bc6df42899325ba","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:37.671855Z","signature_b64":"bl+dmKBNBLgGhxpCgBwb9+28qh7k0ouIgOujvVYUw6y0XB7ACApd/DM/qyiyqC9Ekxr1rhx3T9LIWAZ4hJHxBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4b26408948353c5875428341a0b6c14b02cdb11e6afe3c033bc6df42899325ba","last_reissued_at":"2026-05-18T01:24:37.671330Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:37.671330Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.03198","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":"2026-05-18T01:24:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eHJaN7YYQzRTQhZL6ECRym0ZeG4nYAQQECcsZLGfQnY+AYsKJsaFWE/vcxZyElSEccLUrir6zCxBVq/03OE6DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T07:59:34.543809Z"},"content_sha256":"4bafdbcae686df80e290a6a3b0ad5b8d63bc3a2b8f8b27b98951cbd0674a5afc","schema_version":"1.0","event_id":"sha256:4bafdbcae686df80e290a6a3b0ad5b8d63bc3a2b8f8b27b98951cbd0674a5afc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:JMTEBCKIGU6FQ5KCQNA2BNWBJM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Banach algebras of weakly differentiable functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Andrea Cianchi, Lenka Slav\\'ikov\\'a, Lubo\\v{s} Pick","submitted_at":"2015-12-10T10:24:02Z","abstract_excerpt":"The question is addressed of when a Sobolev type space, built upon a general rearrangement-invariant norm, on an $n$-dimensional domain, is a Banach algebra under pointwise multiplication of functions. A sharp balance condition among the order of the Sobolev space, the strength of the norm, and the (ir)regularity of the domain is provided for the relevant Sobolev space to be a Banach algebra. The regularity of the domain is described in terms of its isoperimetric function. Related results on the boundedness of the multiplication operator into lower-order Sobolev type spaces are also establishe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.03198","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":"2026-05-18T01:24:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2kADHKlLjrQdDCvX16T7nQJdFSN3pl4Wp+SB4z3z9mENBCADkd8icuurOw13xJVT8BIchJdoPpIavHmGTRdAAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T07:59:34.544434Z"},"content_sha256":"f71f7f604cccfcc6d777da5f33d55485f474cc09ad7884462fb212517aa4a8d0","schema_version":"1.0","event_id":"sha256:f71f7f604cccfcc6d777da5f33d55485f474cc09ad7884462fb212517aa4a8d0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JMTEBCKIGU6FQ5KCQNA2BNWBJM/bundle.json","state_url":"https://pith.science/pith/JMTEBCKIGU6FQ5KCQNA2BNWBJM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JMTEBCKIGU6FQ5KCQNA2BNWBJM/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":"2026-05-25T07:59:34Z","links":{"resolver":"https://pith.science/pith/JMTEBCKIGU6FQ5KCQNA2BNWBJM","bundle":"https://pith.science/pith/JMTEBCKIGU6FQ5KCQNA2BNWBJM/bundle.json","state":"https://pith.science/pith/JMTEBCKIGU6FQ5KCQNA2BNWBJM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JMTEBCKIGU6FQ5KCQNA2BNWBJM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JMTEBCKIGU6FQ5KCQNA2BNWBJM","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":"a88637eca625522100d3bc3d43350a1a06efdee05fc1117767be43f18f0bec34","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-12-10T10:24:02Z","title_canon_sha256":"d353a6cc66f809ee2e7d3b31a0ca82d5e12fafd0bd0e4972f1fa000c03866906"},"schema_version":"1.0","source":{"id":"1512.03198","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.03198","created_at":"2026-05-18T01:24:37Z"},{"alias_kind":"arxiv_version","alias_value":"1512.03198v1","created_at":"2026-05-18T01:24:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.03198","created_at":"2026-05-18T01:24:37Z"},{"alias_kind":"pith_short_12","alias_value":"JMTEBCKIGU6F","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JMTEBCKIGU6FQ5KC","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JMTEBCKI","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:f71f7f604cccfcc6d777da5f33d55485f474cc09ad7884462fb212517aa4a8d0","target":"graph","created_at":"2026-05-18T01:24:37Z","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":"The question is addressed of when a Sobolev type space, built upon a general rearrangement-invariant norm, on an $n$-dimensional domain, is a Banach algebra under pointwise multiplication of functions. A sharp balance condition among the order of the Sobolev space, the strength of the norm, and the (ir)regularity of the domain is provided for the relevant Sobolev space to be a Banach algebra. The regularity of the domain is described in terms of its isoperimetric function. Related results on the boundedness of the multiplication operator into lower-order Sobolev type spaces are also establishe","authors_text":"Andrea Cianchi, Lenka Slav\\'ikov\\'a, Lubo\\v{s} Pick","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-12-10T10:24:02Z","title":"Banach algebras of weakly differentiable functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.03198","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":"sha256:4bafdbcae686df80e290a6a3b0ad5b8d63bc3a2b8f8b27b98951cbd0674a5afc","target":"record","created_at":"2026-05-18T01:24:37Z","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":"a88637eca625522100d3bc3d43350a1a06efdee05fc1117767be43f18f0bec34","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-12-10T10:24:02Z","title_canon_sha256":"d353a6cc66f809ee2e7d3b31a0ca82d5e12fafd0bd0e4972f1fa000c03866906"},"schema_version":"1.0","source":{"id":"1512.03198","kind":"arxiv","version":1}},"canonical_sha256":"4b26408948353c5875428341a0b6c14b02cdb11e6afe3c033bc6df42899325ba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4b26408948353c5875428341a0b6c14b02cdb11e6afe3c033bc6df42899325ba","first_computed_at":"2026-05-18T01:24:37.671330Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:24:37.671330Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bl+dmKBNBLgGhxpCgBwb9+28qh7k0ouIgOujvVYUw6y0XB7ACApd/DM/qyiyqC9Ekxr1rhx3T9LIWAZ4hJHxBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:24:37.671855Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.03198","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4bafdbcae686df80e290a6a3b0ad5b8d63bc3a2b8f8b27b98951cbd0674a5afc","sha256:f71f7f604cccfcc6d777da5f33d55485f474cc09ad7884462fb212517aa4a8d0"],"state_sha256":"0bfd4d6a3c23d12829ce72d53a4b2ef88b6e14d06d7a3321aa4c66c24bb338b0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T+W9r64uUbEj4+mA+jGCV4CMpGIy/D+1iMvohe1wW5SsUXplO9JXU+bH4puhqdR4MGMC+c9/zcM7uELSIfvuCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T07:59:34.547007Z","bundle_sha256":"0d728611bbf658839e94db15242bc03e903ec75d0b18e43f7f6a807c313b05ec"}}