{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:SDF4GIQTJRJVQCVS4V4D3WP6RI","short_pith_number":"pith:SDF4GIQT","canonical_record":{"source":{"id":"1111.6952","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-11-29T20:02:46Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"f621b1f18a528a8324ec9e19c045405fe51dd0ec59e3b57f32a7dd442d4f70a2","abstract_canon_sha256":"cd2808a285855e9631450d2f65e8db218eb945d2815f79407c764625cd46a9a5"},"schema_version":"1.0"},"canonical_sha256":"90cbc322134c53580ab2e5783dd9fe8a219bd0474c4a759ae007164d81b95cda","source":{"kind":"arxiv","id":"1111.6952","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.6952","created_at":"2026-05-18T03:41:32Z"},{"alias_kind":"arxiv_version","alias_value":"1111.6952v1","created_at":"2026-05-18T03:41:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.6952","created_at":"2026-05-18T03:41:32Z"},{"alias_kind":"pith_short_12","alias_value":"SDF4GIQTJRJV","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"SDF4GIQTJRJVQCVS","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"SDF4GIQT","created_at":"2026-05-18T12:26:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:SDF4GIQTJRJVQCVS4V4D3WP6RI","target":"record","payload":{"canonical_record":{"source":{"id":"1111.6952","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-11-29T20:02:46Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"f621b1f18a528a8324ec9e19c045405fe51dd0ec59e3b57f32a7dd442d4f70a2","abstract_canon_sha256":"cd2808a285855e9631450d2f65e8db218eb945d2815f79407c764625cd46a9a5"},"schema_version":"1.0"},"canonical_sha256":"90cbc322134c53580ab2e5783dd9fe8a219bd0474c4a759ae007164d81b95cda","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:32.243053Z","signature_b64":"90RWNRFf9uZRvMMco3uGheCJ1YRVZiYfObzfW7Ni7cz3GeuodXP9mqwmzq24qucB0fgSUUwioj9i5LZvJOY/BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"90cbc322134c53580ab2e5783dd9fe8a219bd0474c4a759ae007164d81b95cda","last_reissued_at":"2026-05-18T03:41:32.242598Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:32.242598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.6952","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":"2026-05-18T03:41:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S1JbZG+iTvLYJyjTq54UAvRG1FEs4pKvmyz+yf2gLWjFbMvvlKQENp2y8CRh8A+pp/ARJmsjGG5uRlMSuG/cBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T00:26:03.397735Z"},"content_sha256":"77e5b92f0993d6302c05b389be2fe1f6e56bd71c57b6a3f6a4d3e705c3ee21bd","schema_version":"1.0","event_id":"sha256:77e5b92f0993d6302c05b389be2fe1f6e56bd71c57b6a3f6a4d3e705c3ee21bd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:SDF4GIQTJRJVQCVS4V4D3WP6RI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Sobolev embedding theorem for variable exponent spaces in the critical range","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Analia Silva, Julian Fernandez Bonder, Nicolas Saintier","submitted_at":"2011-11-29T20:02:46Z","abstract_excerpt":"In this paper we study the Sobolev embedding theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. The proof is based on a suitable refinement of the estimates in the Concentration--Compactness Theorem for variable exponents and an adaptation of a convexity argument due to P.L. Lions, F. Pacella and M. Tricarico."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.6952","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":"2026-05-18T03:41:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TyqQsOsl4OsG5yY8z3QcEGMwNBmmkgXXdo5ZSEBQiyYF8o/kXPFgxsMmSgqhK97whcgCU23+aXm1uaYFipozBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T00:26:03.398079Z"},"content_sha256":"ce62ae672a84f17d75a10378dd210284f0d21aff3e1a7d9f6a3668daa63bd7b1","schema_version":"1.0","event_id":"sha256:ce62ae672a84f17d75a10378dd210284f0d21aff3e1a7d9f6a3668daa63bd7b1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SDF4GIQTJRJVQCVS4V4D3WP6RI/bundle.json","state_url":"https://pith.science/pith/SDF4GIQTJRJVQCVS4V4D3WP6RI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SDF4GIQTJRJVQCVS4V4D3WP6RI/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":"2026-06-04T00:26:03Z","links":{"resolver":"https://pith.science/pith/SDF4GIQTJRJVQCVS4V4D3WP6RI","bundle":"https://pith.science/pith/SDF4GIQTJRJVQCVS4V4D3WP6RI/bundle.json","state":"https://pith.science/pith/SDF4GIQTJRJVQCVS4V4D3WP6RI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SDF4GIQTJRJVQCVS4V4D3WP6RI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:SDF4GIQTJRJVQCVS4V4D3WP6RI","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":"cd2808a285855e9631450d2f65e8db218eb945d2815f79407c764625cd46a9a5","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-11-29T20:02:46Z","title_canon_sha256":"f621b1f18a528a8324ec9e19c045405fe51dd0ec59e3b57f32a7dd442d4f70a2"},"schema_version":"1.0","source":{"id":"1111.6952","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.6952","created_at":"2026-05-18T03:41:32Z"},{"alias_kind":"arxiv_version","alias_value":"1111.6952v1","created_at":"2026-05-18T03:41:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.6952","created_at":"2026-05-18T03:41:32Z"},{"alias_kind":"pith_short_12","alias_value":"SDF4GIQTJRJV","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"SDF4GIQTJRJVQCVS","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"SDF4GIQT","created_at":"2026-05-18T12:26:41Z"}],"graph_snapshots":[{"event_id":"sha256:ce62ae672a84f17d75a10378dd210284f0d21aff3e1a7d9f6a3668daa63bd7b1","target":"graph","created_at":"2026-05-18T03:41:32Z","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":"In this paper we study the Sobolev embedding theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. The proof is based on a suitable refinement of the estimates in the Concentration--Compactness Theorem for variable exponents and an adaptation of a convexity argument due to P.L. Lions, F. Pacella and M. Tricarico.","authors_text":"Analia Silva, Julian Fernandez Bonder, Nicolas Saintier","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-11-29T20:02:46Z","title":"On the Sobolev embedding theorem for variable exponent spaces in the critical range"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.6952","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":"sha256:77e5b92f0993d6302c05b389be2fe1f6e56bd71c57b6a3f6a4d3e705c3ee21bd","target":"record","created_at":"2026-05-18T03:41:32Z","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":"cd2808a285855e9631450d2f65e8db218eb945d2815f79407c764625cd46a9a5","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-11-29T20:02:46Z","title_canon_sha256":"f621b1f18a528a8324ec9e19c045405fe51dd0ec59e3b57f32a7dd442d4f70a2"},"schema_version":"1.0","source":{"id":"1111.6952","kind":"arxiv","version":1}},"canonical_sha256":"90cbc322134c53580ab2e5783dd9fe8a219bd0474c4a759ae007164d81b95cda","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"90cbc322134c53580ab2e5783dd9fe8a219bd0474c4a759ae007164d81b95cda","first_computed_at":"2026-05-18T03:41:32.242598Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:41:32.242598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"90RWNRFf9uZRvMMco3uGheCJ1YRVZiYfObzfW7Ni7cz3GeuodXP9mqwmzq24qucB0fgSUUwioj9i5LZvJOY/BA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:41:32.243053Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.6952","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:77e5b92f0993d6302c05b389be2fe1f6e56bd71c57b6a3f6a4d3e705c3ee21bd","sha256:ce62ae672a84f17d75a10378dd210284f0d21aff3e1a7d9f6a3668daa63bd7b1"],"state_sha256":"acc0a6e450b92be9503e07d6b3014788be6b6b0d0273d2670824577bd9d11c88"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZprXtktJl88LmYR5KcSeK23tFXiaElQ8+QcL67ZxVczW+xUsG1vS3esFN7RSwoEpwAd+WQ8nGMpOh5ygObFiDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T00:26:03.400971Z","bundle_sha256":"c06552f5ba6623b1738d0d30af40b6e1b77b20b764129b6e7f16b9d2b08b9811"}}