{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:EMFY6TOKPDNQ57QYF5N63UGCUB","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":"b8fe4b2dca2a8832ed9e8b70ebd4a3df7871589a932c11a690440182212e1fc0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-06-12T13:14:22Z","title_canon_sha256":"170d753fd7b63e04a62db1d6b6d67d04c4d65881010f206f715b7b0fe51677f7"},"schema_version":"1.0","source":{"id":"1906.05124","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.05124","created_at":"2026-05-17T23:43:29Z"},{"alias_kind":"arxiv_version","alias_value":"1906.05124v1","created_at":"2026-05-17T23:43:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.05124","created_at":"2026-05-17T23:43:29Z"},{"alias_kind":"pith_short_12","alias_value":"EMFY6TOKPDNQ","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_16","alias_value":"EMFY6TOKPDNQ57QY","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_8","alias_value":"EMFY6TOK","created_at":"2026-05-18T12:33:15Z"}],"graph_snapshots":[{"event_id":"sha256:c019b45e1278a2dee2ab9d9d526f2bcb9136fa59b1ff3155cd890aedaf034a42","target":"graph","created_at":"2026-05-17T23:43:29Z","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 $K$ be a locally compact hypergroup with a left invariant Haar measure. We show that the Liouville property and amenability are equivalent for $K$ when it is second countable. Suppose that $\\sigma$ is a non-degenerate probability measure on $K$, we show that there is no non-trivial $\\sigma$-harmonic function which is continuous and vanishing at infinity. Using this, we prove that the space $H_\\sigma^p(K)$ of all $\\sigma$-harmonic $L^p$-functions, is trivial for all $1\\leq p<\\infty$. Further, it is shown that $H_\\sigma^\\infty(K)$ contains only constant functions if and only if it is a subal","authors_text":"Jila Sohaei, Mehdi Nemati","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-06-12T13:14:22Z","title":"Amenability and harmonic $L^p$-functions on hypergroups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.05124","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:f2648ecdc74cfbba2f47e0c72610a3a817dc225e47d5551858669fb55affdb98","target":"record","created_at":"2026-05-17T23:43:29Z","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":"b8fe4b2dca2a8832ed9e8b70ebd4a3df7871589a932c11a690440182212e1fc0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-06-12T13:14:22Z","title_canon_sha256":"170d753fd7b63e04a62db1d6b6d67d04c4d65881010f206f715b7b0fe51677f7"},"schema_version":"1.0","source":{"id":"1906.05124","kind":"arxiv","version":1}},"canonical_sha256":"230b8f4dca78db0efe182f5bedd0c2a04954c682e86d4d42a3cb4560b7c8849b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"230b8f4dca78db0efe182f5bedd0c2a04954c682e86d4d42a3cb4560b7c8849b","first_computed_at":"2026-05-17T23:43:29.408019Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:43:29.408019Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"G2klAPpqGXRioAHfSKyApFsABSs4OXpQxxZO/As7Ee6QwZqRg/MT9pjejKO9SBXVgskz4S6EsdopQXMRPSidCw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:43:29.408513Z","signed_message":"canonical_sha256_bytes"},"source_id":"1906.05124","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f2648ecdc74cfbba2f47e0c72610a3a817dc225e47d5551858669fb55affdb98","sha256:c019b45e1278a2dee2ab9d9d526f2bcb9136fa59b1ff3155cd890aedaf034a42"],"state_sha256":"482def08f15ee13a566995d7ff47c4d54447b910b764518e249e38da377d9c7d"}