{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:RLBHKF6ETVCS6BPXDPNHBBERWH","short_pith_number":"pith:RLBHKF6E","canonical_record":{"source":{"id":"1308.3607","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-08-16T11:31:50Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"9794a5af6eb9f64c8e8c3178589669414096195dc75626fbabeac6899e06d85e","abstract_canon_sha256":"30ae95c8c15f5249b2da2e8db45827a45e5c63b689efcd7a579f874c72f289eb"},"schema_version":"1.0"},"canonical_sha256":"8ac27517c49d452f05f71bda708491b1c2d9afd5949a93637eee5737b9b5c702","source":{"kind":"arxiv","id":"1308.3607","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.3607","created_at":"2026-05-18T01:19:00Z"},{"alias_kind":"arxiv_version","alias_value":"1308.3607v2","created_at":"2026-05-18T01:19:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.3607","created_at":"2026-05-18T01:19:00Z"},{"alias_kind":"pith_short_12","alias_value":"RLBHKF6ETVCS","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"RLBHKF6ETVCS6BPX","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"RLBHKF6E","created_at":"2026-05-18T12:27:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:RLBHKF6ETVCS6BPXDPNHBBERWH","target":"record","payload":{"canonical_record":{"source":{"id":"1308.3607","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-08-16T11:31:50Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"9794a5af6eb9f64c8e8c3178589669414096195dc75626fbabeac6899e06d85e","abstract_canon_sha256":"30ae95c8c15f5249b2da2e8db45827a45e5c63b689efcd7a579f874c72f289eb"},"schema_version":"1.0"},"canonical_sha256":"8ac27517c49d452f05f71bda708491b1c2d9afd5949a93637eee5737b9b5c702","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:00.663613Z","signature_b64":"Gw4Q7nJ/lOY3AlmitmqX9wDb0oe+K1k9rk5+CiX9KMPfBkkL+oRiatavKdag9rb9vf7eqOkSH4A68IMAP8aaBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ac27517c49d452f05f71bda708491b1c2d9afd5949a93637eee5737b9b5c702","last_reissued_at":"2026-05-18T01:19:00.663136Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:00.663136Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1308.3607","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":"2026-05-18T01:19:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+SnvHcQR2EKdvlROAHlp9iVt3cdmg31bBa1lttBU/loAxPQtyjwYah8hZMmMImKf2v7UFqNTo27DYkcF5OwmCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T07:30:05.248437Z"},"content_sha256":"88e72f5290562aee14253007ca7d31923670f1e5acbe053df8ea60948161097c","schema_version":"1.0","event_id":"sha256:88e72f5290562aee14253007ca7d31923670f1e5acbe053df8ea60948161097c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:RLBHKF6ETVCS6BPXDPNHBBERWH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Harmonic functions on metric measure spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.AP","authors_text":"Bobo Hua, Chao Xia, Martin Kell","submitted_at":"2013-08-16T11:31:50Z","abstract_excerpt":"In this paper, we study harmonic functions on metric measure spaces with Riemannian Ricci curvature bounded from below, which were introduced by Ambrosio-Gigli-Savar\\'e. We prove a Cheng-Yau type local gradient estimate for harmonic functions on these spaces. Furthermore, we derive various optimal dimension estimates for spaces of polynomial growth harmonic functions on metric measure spaces with nonnegative Riemannian Ricci curvature."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.3607","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":"2026-05-18T01:19:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/vpTE1EH8kvb95LDokfGOhDnHsDLjaCYrJWgXz5ZsGMmX6du+OjKArF7rOvMZpETs3zSeJBppZXyyoqdveeuCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T07:30:05.249140Z"},"content_sha256":"d1ed00e73b2305c2d9d01c6a5a7d6fbb3b3afd38c498cf090bdec9910908b5bf","schema_version":"1.0","event_id":"sha256:d1ed00e73b2305c2d9d01c6a5a7d6fbb3b3afd38c498cf090bdec9910908b5bf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RLBHKF6ETVCS6BPXDPNHBBERWH/bundle.json","state_url":"https://pith.science/pith/RLBHKF6ETVCS6BPXDPNHBBERWH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RLBHKF6ETVCS6BPXDPNHBBERWH/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":"2026-06-11T07:30:05Z","links":{"resolver":"https://pith.science/pith/RLBHKF6ETVCS6BPXDPNHBBERWH","bundle":"https://pith.science/pith/RLBHKF6ETVCS6BPXDPNHBBERWH/bundle.json","state":"https://pith.science/pith/RLBHKF6ETVCS6BPXDPNHBBERWH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RLBHKF6ETVCS6BPXDPNHBBERWH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:RLBHKF6ETVCS6BPXDPNHBBERWH","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":"30ae95c8c15f5249b2da2e8db45827a45e5c63b689efcd7a579f874c72f289eb","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-08-16T11:31:50Z","title_canon_sha256":"9794a5af6eb9f64c8e8c3178589669414096195dc75626fbabeac6899e06d85e"},"schema_version":"1.0","source":{"id":"1308.3607","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.3607","created_at":"2026-05-18T01:19:00Z"},{"alias_kind":"arxiv_version","alias_value":"1308.3607v2","created_at":"2026-05-18T01:19:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.3607","created_at":"2026-05-18T01:19:00Z"},{"alias_kind":"pith_short_12","alias_value":"RLBHKF6ETVCS","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"RLBHKF6ETVCS6BPX","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"RLBHKF6E","created_at":"2026-05-18T12:27:59Z"}],"graph_snapshots":[{"event_id":"sha256:d1ed00e73b2305c2d9d01c6a5a7d6fbb3b3afd38c498cf090bdec9910908b5bf","target":"graph","created_at":"2026-05-18T01:19:00Z","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 harmonic functions on metric measure spaces with Riemannian Ricci curvature bounded from below, which were introduced by Ambrosio-Gigli-Savar\\'e. We prove a Cheng-Yau type local gradient estimate for harmonic functions on these spaces. Furthermore, we derive various optimal dimension estimates for spaces of polynomial growth harmonic functions on metric measure spaces with nonnegative Riemannian Ricci curvature.","authors_text":"Bobo Hua, Chao Xia, Martin Kell","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-08-16T11:31:50Z","title":"Harmonic functions on metric measure spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.3607","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":"sha256:88e72f5290562aee14253007ca7d31923670f1e5acbe053df8ea60948161097c","target":"record","created_at":"2026-05-18T01:19:00Z","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":"30ae95c8c15f5249b2da2e8db45827a45e5c63b689efcd7a579f874c72f289eb","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-08-16T11:31:50Z","title_canon_sha256":"9794a5af6eb9f64c8e8c3178589669414096195dc75626fbabeac6899e06d85e"},"schema_version":"1.0","source":{"id":"1308.3607","kind":"arxiv","version":2}},"canonical_sha256":"8ac27517c49d452f05f71bda708491b1c2d9afd5949a93637eee5737b9b5c702","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8ac27517c49d452f05f71bda708491b1c2d9afd5949a93637eee5737b9b5c702","first_computed_at":"2026-05-18T01:19:00.663136Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:00.663136Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Gw4Q7nJ/lOY3AlmitmqX9wDb0oe+K1k9rk5+CiX9KMPfBkkL+oRiatavKdag9rb9vf7eqOkSH4A68IMAP8aaBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:00.663613Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.3607","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:88e72f5290562aee14253007ca7d31923670f1e5acbe053df8ea60948161097c","sha256:d1ed00e73b2305c2d9d01c6a5a7d6fbb3b3afd38c498cf090bdec9910908b5bf"],"state_sha256":"c3e5c4cd2591e8636391dbd431601413ad697828a9b428ce6aabc5974dd4318f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yuP0LLxxBV6f+p2iBWM6ylFFzksdiwzjFrYTUp0H6+B0AKD5xhIkvlSq0tzCJ/fxJIYqQQdK5y190SVUy6ZmDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T07:30:05.253198Z","bundle_sha256":"92543c131d7711a2d3ceffa423c6b906941dd677097f34f591697fa18e6899de"}}