{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:4OGHNG7T5Y46N5TM4RIBGHYMCZ","short_pith_number":"pith:4OGHNG7T","canonical_record":{"source":{"id":"1707.04018","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-07-13T08:08:52Z","cross_cats_sorted":[],"title_canon_sha256":"54068d7b2b5e285489da8e370e9f89b484c6f48e943d694a2e494714b449707c","abstract_canon_sha256":"bc9ad9eb563a62e3cdbeb35f08e5f045c5301aabd8066089f0f93f2bede1b4d4"},"schema_version":"1.0"},"canonical_sha256":"e38c769bf3ee39e6f66ce450131f0c1655f9cc6eb9150020b6a9b2e0c495e778","source":{"kind":"arxiv","id":"1707.04018","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.04018","created_at":"2026-05-18T00:21:46Z"},{"alias_kind":"arxiv_version","alias_value":"1707.04018v2","created_at":"2026-05-18T00:21:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.04018","created_at":"2026-05-18T00:21:46Z"},{"alias_kind":"pith_short_12","alias_value":"4OGHNG7T5Y46","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_16","alias_value":"4OGHNG7T5Y46N5TM","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_8","alias_value":"4OGHNG7T","created_at":"2026-05-18T12:31:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:4OGHNG7T5Y46N5TM4RIBGHYMCZ","target":"record","payload":{"canonical_record":{"source":{"id":"1707.04018","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-07-13T08:08:52Z","cross_cats_sorted":[],"title_canon_sha256":"54068d7b2b5e285489da8e370e9f89b484c6f48e943d694a2e494714b449707c","abstract_canon_sha256":"bc9ad9eb563a62e3cdbeb35f08e5f045c5301aabd8066089f0f93f2bede1b4d4"},"schema_version":"1.0"},"canonical_sha256":"e38c769bf3ee39e6f66ce450131f0c1655f9cc6eb9150020b6a9b2e0c495e778","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:46.220635Z","signature_b64":"O7mNFmT9AD0+RIsPtHE1KHTevBoB0DyG94LBb3kMt0VILUPG3BF/ZUd/wZjW9/CA2mjB+9/07gU6p+GQdzVaAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e38c769bf3ee39e6f66ce450131f0c1655f9cc6eb9150020b6a9b2e0c495e778","last_reissued_at":"2026-05-18T00:21:46.219921Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:46.219921Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1707.04018","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:21:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qb2wlcXQBD19nMh67Sy+8sr5IX8Cq7iKCHMSRLHPP+/Q08gODhlMmpifqCrILR+StIpsKaGioOicWB1OEezKAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T03:22:31.964404Z"},"content_sha256":"c9968b896c84f13788d4f1791bfef4241b8425da0bab2501eb5ba002a6307dbf","schema_version":"1.0","event_id":"sha256:c9968b896c84f13788d4f1791bfef4241b8425da0bab2501eb5ba002a6307dbf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:4OGHNG7T5Y46N5TM4RIBGHYMCZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hardy's inequality in a limiting case on general bounded domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Futoshi Takahashi, Jaeyoung Byeon","submitted_at":"2017-07-13T08:08:52Z","abstract_excerpt":"In this paper, we study Hardy's inequality in a limiting case: $$\n  \\int_{\\Omega} |\\nabla u |^N dx \\ge C_N(\\Omega) \\int_{\\Omega} \\frac{|u(x)|^N}{|x|^N \\left(\\log \\frac{R}{|x|} \\right)^N} dx $$ for functions $u \\in W^{1,N}_0(\\Omega)$, where $\\Omega$ is a bounded domain in $\\mathbb{R}^N$ with $R = \\sup_{x \\in \\Omega} |x|$. We study the (non-)attainability of the best constant $C_N(\\Omega)$ in several cases. We provide sufficient conditions that assure $C_N(\\Omega) > C_N(B_R)$ and $C_N(\\Omega)$ is attained, here $B_R$ is the $N$-dimensional ball with center the origin and radius $R$. Also we prov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.04018","kind":"arxiv","version":2},"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":"2026-05-18T00:21:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DwMzFU9iXgjqtbNi481KsK7Xwbme8y8U0YPQi0JK51s+tB3gD/E/i1mmO/8A75QBKv+xEMGgmZFPXGp2LYA5Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T03:22:31.964769Z"},"content_sha256":"28e4e42eb22e8826c3b3ac05528ab7e139b2726c5188d273457630fa6ad08c06","schema_version":"1.0","event_id":"sha256:28e4e42eb22e8826c3b3ac05528ab7e139b2726c5188d273457630fa6ad08c06"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4OGHNG7T5Y46N5TM4RIBGHYMCZ/bundle.json","state_url":"https://pith.science/pith/4OGHNG7T5Y46N5TM4RIBGHYMCZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4OGHNG7T5Y46N5TM4RIBGHYMCZ/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":"2026-06-23T03:22:31Z","links":{"resolver":"https://pith.science/pith/4OGHNG7T5Y46N5TM4RIBGHYMCZ","bundle":"https://pith.science/pith/4OGHNG7T5Y46N5TM4RIBGHYMCZ/bundle.json","state":"https://pith.science/pith/4OGHNG7T5Y46N5TM4RIBGHYMCZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4OGHNG7T5Y46N5TM4RIBGHYMCZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:4OGHNG7T5Y46N5TM4RIBGHYMCZ","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":"bc9ad9eb563a62e3cdbeb35f08e5f045c5301aabd8066089f0f93f2bede1b4d4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-07-13T08:08:52Z","title_canon_sha256":"54068d7b2b5e285489da8e370e9f89b484c6f48e943d694a2e494714b449707c"},"schema_version":"1.0","source":{"id":"1707.04018","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.04018","created_at":"2026-05-18T00:21:46Z"},{"alias_kind":"arxiv_version","alias_value":"1707.04018v2","created_at":"2026-05-18T00:21:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.04018","created_at":"2026-05-18T00:21:46Z"},{"alias_kind":"pith_short_12","alias_value":"4OGHNG7T5Y46","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_16","alias_value":"4OGHNG7T5Y46N5TM","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_8","alias_value":"4OGHNG7T","created_at":"2026-05-18T12:31:00Z"}],"graph_snapshots":[{"event_id":"sha256:28e4e42eb22e8826c3b3ac05528ab7e139b2726c5188d273457630fa6ad08c06","target":"graph","created_at":"2026-05-18T00:21:46Z","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 Hardy's inequality in a limiting case: $$\n  \\int_{\\Omega} |\\nabla u |^N dx \\ge C_N(\\Omega) \\int_{\\Omega} \\frac{|u(x)|^N}{|x|^N \\left(\\log \\frac{R}{|x|} \\right)^N} dx $$ for functions $u \\in W^{1,N}_0(\\Omega)$, where $\\Omega$ is a bounded domain in $\\mathbb{R}^N$ with $R = \\sup_{x \\in \\Omega} |x|$. We study the (non-)attainability of the best constant $C_N(\\Omega)$ in several cases. We provide sufficient conditions that assure $C_N(\\Omega) > C_N(B_R)$ and $C_N(\\Omega)$ is attained, here $B_R$ is the $N$-dimensional ball with center the origin and radius $R$. Also we prov","authors_text":"Futoshi Takahashi, Jaeyoung Byeon","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-07-13T08:08:52Z","title":"Hardy's inequality in a limiting case on general bounded domains"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.04018","kind":"arxiv","version":2},"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:c9968b896c84f13788d4f1791bfef4241b8425da0bab2501eb5ba002a6307dbf","target":"record","created_at":"2026-05-18T00:21:46Z","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":"bc9ad9eb563a62e3cdbeb35f08e5f045c5301aabd8066089f0f93f2bede1b4d4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-07-13T08:08:52Z","title_canon_sha256":"54068d7b2b5e285489da8e370e9f89b484c6f48e943d694a2e494714b449707c"},"schema_version":"1.0","source":{"id":"1707.04018","kind":"arxiv","version":2}},"canonical_sha256":"e38c769bf3ee39e6f66ce450131f0c1655f9cc6eb9150020b6a9b2e0c495e778","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e38c769bf3ee39e6f66ce450131f0c1655f9cc6eb9150020b6a9b2e0c495e778","first_computed_at":"2026-05-18T00:21:46.219921Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:21:46.219921Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"O7mNFmT9AD0+RIsPtHE1KHTevBoB0DyG94LBb3kMt0VILUPG3BF/ZUd/wZjW9/CA2mjB+9/07gU6p+GQdzVaAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:21:46.220635Z","signed_message":"canonical_sha256_bytes"},"source_id":"1707.04018","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c9968b896c84f13788d4f1791bfef4241b8425da0bab2501eb5ba002a6307dbf","sha256:28e4e42eb22e8826c3b3ac05528ab7e139b2726c5188d273457630fa6ad08c06"],"state_sha256":"702bd737a0e4f38525c6599f65b88e7cb4a3270e85ff407a2f19bf331073961a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TskaTfIoQOwFeYPG8H2ZJaE9KOv/Q7SyDmGfqd81KxeZABx7zYJDow+DEB9PBSCcfA0vQVRjRINW5H6LEOw1Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T03:22:31.966778Z","bundle_sha256":"35f402061cc12ecd508e1339ad2989a7d87d3e175c10f36d3fdb99dc7b0cc0a8"}}