{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:YREIZDNPAO5GNUALBG3PFFDEOE","short_pith_number":"pith:YREIZDNP","canonical_record":{"source":{"id":"1703.01005","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-03-03T00:59:32Z","cross_cats_sorted":[],"title_canon_sha256":"e254699d626831751a27808f7707243e57a964dda2adafd1b51ae8516535e62b","abstract_canon_sha256":"7baf6360438b7917fa573bf1b1b66aa1f792f11c5c6365b7de27807cacfa50df"},"schema_version":"1.0"},"canonical_sha256":"c4488c8daf03ba66d00b09b6f294647104c6e7db5573be5f758b12d223dc073a","source":{"kind":"arxiv","id":"1703.01005","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.01005","created_at":"2026-05-18T00:49:37Z"},{"alias_kind":"arxiv_version","alias_value":"1703.01005v1","created_at":"2026-05-18T00:49:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.01005","created_at":"2026-05-18T00:49:37Z"},{"alias_kind":"pith_short_12","alias_value":"YREIZDNPAO5G","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_16","alias_value":"YREIZDNPAO5GNUAL","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_8","alias_value":"YREIZDNP","created_at":"2026-05-18T12:31:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:YREIZDNPAO5GNUALBG3PFFDEOE","target":"record","payload":{"canonical_record":{"source":{"id":"1703.01005","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-03-03T00:59:32Z","cross_cats_sorted":[],"title_canon_sha256":"e254699d626831751a27808f7707243e57a964dda2adafd1b51ae8516535e62b","abstract_canon_sha256":"7baf6360438b7917fa573bf1b1b66aa1f792f11c5c6365b7de27807cacfa50df"},"schema_version":"1.0"},"canonical_sha256":"c4488c8daf03ba66d00b09b6f294647104c6e7db5573be5f758b12d223dc073a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:37.256589Z","signature_b64":"bIHK/UoDJzGnHgSwMxsZliNDumNBAX4dg8KIy4pvgGO+OCoZPCSntpsC7IkI4uXMWSGf3p+N52cvulp3G0eSAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c4488c8daf03ba66d00b09b6f294647104c6e7db5573be5f758b12d223dc073a","last_reissued_at":"2026-05-18T00:49:37.255787Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:37.255787Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1703.01005","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":"2026-05-18T00:49:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"h3fYFdlZCadEq+r2t5AHaf/Fh0Nxjg7+QIXxyWswCnefSmg3KRBtOfIKUmD3aO/ZSe9aHFb17PbnXDd2NU4pCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T02:51:22.451408Z"},"content_sha256":"6073ae190163d52121a19d8814fd79b6313a96d7295f513b9bb5ddefc0e90cca","schema_version":"1.0","event_id":"sha256:6073ae190163d52121a19d8814fd79b6313a96d7295f513b9bb5ddefc0e90cca"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:YREIZDNPAO5GNUALBG3PFFDEOE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Concentration-compactness principle for Trudinger-Moser inequalities on Heisenberg Groups and existence of ground state solutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guozhen Lu, Jungang Li, Maochun Zhu","submitted_at":"2017-03-03T00:59:32Z","abstract_excerpt":"Let $\\mathbb{H}^{n}=\\mathbb{C}^{n}\\times\\mathbb{R}$ be the $n$-dimensional Heisenberg group, $Q=2n+2$ be the homogeneous dimension of $\\mathbb{H}^{n}$. We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces of \\ P. L. Lions to the setting of the Heisenberg group $\\mathbb{H}^{n}$. Furthermore, we also obtain the corresponding concentration-compactness principle for the Sobolev space $HW^{1,Q}\\left( \\mathbb{H}^{n}\\right) $ on the entire Heisenberg group $\\mathbb{H}^{n}$.\n  Our results improve the sharp Trudinger-Moser inequality on domains of finit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.01005","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":"2026-05-18T00:49:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FR48wZmTRxy/TsuAzqU3l/IE7Dl/HHZeK0qby/l3C3cK2AVZbDsVd5AD9SYUozEFbPn9L/qSpOPtOk/zyLoADg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T02:51:22.451761Z"},"content_sha256":"8c7fbfb0193caa882c4bdcd6cd72f68ba8bee7c3b1c9c09b3626122d404cdf72","schema_version":"1.0","event_id":"sha256:8c7fbfb0193caa882c4bdcd6cd72f68ba8bee7c3b1c9c09b3626122d404cdf72"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YREIZDNPAO5GNUALBG3PFFDEOE/bundle.json","state_url":"https://pith.science/pith/YREIZDNPAO5GNUALBG3PFFDEOE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YREIZDNPAO5GNUALBG3PFFDEOE/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":"2026-06-04T02:51:22Z","links":{"resolver":"https://pith.science/pith/YREIZDNPAO5GNUALBG3PFFDEOE","bundle":"https://pith.science/pith/YREIZDNPAO5GNUALBG3PFFDEOE/bundle.json","state":"https://pith.science/pith/YREIZDNPAO5GNUALBG3PFFDEOE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YREIZDNPAO5GNUALBG3PFFDEOE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:YREIZDNPAO5GNUALBG3PFFDEOE","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":"7baf6360438b7917fa573bf1b1b66aa1f792f11c5c6365b7de27807cacfa50df","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-03-03T00:59:32Z","title_canon_sha256":"e254699d626831751a27808f7707243e57a964dda2adafd1b51ae8516535e62b"},"schema_version":"1.0","source":{"id":"1703.01005","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.01005","created_at":"2026-05-18T00:49:37Z"},{"alias_kind":"arxiv_version","alias_value":"1703.01005v1","created_at":"2026-05-18T00:49:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.01005","created_at":"2026-05-18T00:49:37Z"},{"alias_kind":"pith_short_12","alias_value":"YREIZDNPAO5G","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_16","alias_value":"YREIZDNPAO5GNUAL","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_8","alias_value":"YREIZDNP","created_at":"2026-05-18T12:31:56Z"}],"graph_snapshots":[{"event_id":"sha256:8c7fbfb0193caa882c4bdcd6cd72f68ba8bee7c3b1c9c09b3626122d404cdf72","target":"graph","created_at":"2026-05-18T00:49:37Z","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":"Let $\\mathbb{H}^{n}=\\mathbb{C}^{n}\\times\\mathbb{R}$ be the $n$-dimensional Heisenberg group, $Q=2n+2$ be the homogeneous dimension of $\\mathbb{H}^{n}$. We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces of \\ P. L. Lions to the setting of the Heisenberg group $\\mathbb{H}^{n}$. Furthermore, we also obtain the corresponding concentration-compactness principle for the Sobolev space $HW^{1,Q}\\left( \\mathbb{H}^{n}\\right) $ on the entire Heisenberg group $\\mathbb{H}^{n}$.\n  Our results improve the sharp Trudinger-Moser inequality on domains of finit","authors_text":"Guozhen Lu, Jungang Li, Maochun Zhu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-03-03T00:59:32Z","title":"Concentration-compactness principle for Trudinger-Moser inequalities on Heisenberg Groups and existence of ground state solutions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.01005","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":"sha256:6073ae190163d52121a19d8814fd79b6313a96d7295f513b9bb5ddefc0e90cca","target":"record","created_at":"2026-05-18T00:49:37Z","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":"7baf6360438b7917fa573bf1b1b66aa1f792f11c5c6365b7de27807cacfa50df","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-03-03T00:59:32Z","title_canon_sha256":"e254699d626831751a27808f7707243e57a964dda2adafd1b51ae8516535e62b"},"schema_version":"1.0","source":{"id":"1703.01005","kind":"arxiv","version":1}},"canonical_sha256":"c4488c8daf03ba66d00b09b6f294647104c6e7db5573be5f758b12d223dc073a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c4488c8daf03ba66d00b09b6f294647104c6e7db5573be5f758b12d223dc073a","first_computed_at":"2026-05-18T00:49:37.255787Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:49:37.255787Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bIHK/UoDJzGnHgSwMxsZliNDumNBAX4dg8KIy4pvgGO+OCoZPCSntpsC7IkI4uXMWSGf3p+N52cvulp3G0eSAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:49:37.256589Z","signed_message":"canonical_sha256_bytes"},"source_id":"1703.01005","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6073ae190163d52121a19d8814fd79b6313a96d7295f513b9bb5ddefc0e90cca","sha256:8c7fbfb0193caa882c4bdcd6cd72f68ba8bee7c3b1c9c09b3626122d404cdf72"],"state_sha256":"f26bd7763ba5601dd84936812b4544df70ef6192c827b4edf2457fd6762c64d8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zm3+rTOSAvDoaijevkTEYickbjqdG4Kgx4HgEI3jVmYXF/HUbgxTVIcBX+uei8DMGAAfN2smoY65j4Bvbp6ZAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T02:51:22.453648Z","bundle_sha256":"0ad3037e61376cc1de7186aa00aadc17b36a97187e80e5655281ec377b2735c3"}}