{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:JRJPOR2NUX2KAVN4YMXZEL6IXL","short_pith_number":"pith:JRJPOR2N","schema_version":"1.0","canonical_sha256":"4c52f7474da5f4a055bcc32f922fc8baf851a9d398eb50a6ba6ad1468f2f00b5","source":{"kind":"arxiv","id":"1601.03919","version":1},"attestation_state":"computed","paper":{"title":"Harmonic functions on metric measure spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CA"],"primary_cat":"math.MG","authors_text":"Micha{\\l} Gaczkowski, Przemys{\\l}aw G\\'orka, Tomasz Adamowicz","submitted_at":"2016-01-15T13:50:08Z","abstract_excerpt":"We introduce and study strongly and weakly harmonic functions on metric measure spaces defined via the mean value property holding for all and, respectively, for some radii of balls at every point of the underlying domain. Among properties of such functions we investigate various types of Harnack estimates on balls and compact sets, weak and strong maximum principles, comparison principles, the H\\\"older and the Lipshitz estimates and some differentiability properties. The latter one is based on the notion of a weak upper gradient. The Dirichlet problem for functions satisfying the mean value p"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1601.03919","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-01-15T13:50:08Z","cross_cats_sorted":["math.AP","math.CA"],"title_canon_sha256":"9794a5af6eb9f64c8e8c3178589669414096195dc75626fbabeac6899e06d85e","abstract_canon_sha256":"91dff8dda6d200dc94014d048637a05283d52a637858d7710aaa3ec71c0dee2d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:49.684636Z","signature_b64":"OlOaGvqIDamr29zhoyeqUxA07ZxNEAXxI18+ndORDJCWfoYLiGwlLgxrJJ9FbQ8azLwwcFQPGCQ7bkGTWEKGCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4c52f7474da5f4a055bcc32f922fc8baf851a9d398eb50a6ba6ad1468f2f00b5","last_reissued_at":"2026-05-18T01:22:49.683991Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:49.683991Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Harmonic functions on metric measure spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CA"],"primary_cat":"math.MG","authors_text":"Micha{\\l} Gaczkowski, Przemys{\\l}aw G\\'orka, Tomasz Adamowicz","submitted_at":"2016-01-15T13:50:08Z","abstract_excerpt":"We introduce and study strongly and weakly harmonic functions on metric measure spaces defined via the mean value property holding for all and, respectively, for some radii of balls at every point of the underlying domain. Among properties of such functions we investigate various types of Harnack estimates on balls and compact sets, weak and strong maximum principles, comparison principles, the H\\\"older and the Lipshitz estimates and some differentiability properties. The latter one is based on the notion of a weak upper gradient. The Dirichlet problem for functions satisfying the mean value p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03919","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1601.03919","created_at":"2026-05-18T01:22:49.684085+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.03919v1","created_at":"2026-05-18T01:22:49.684085+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.03919","created_at":"2026-05-18T01:22:49.684085+00:00"},{"alias_kind":"pith_short_12","alias_value":"JRJPOR2NUX2K","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_16","alias_value":"JRJPOR2NUX2KAVN4","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_8","alias_value":"JRJPOR2N","created_at":"2026-05-18T12:30:25.849896+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JRJPOR2NUX2KAVN4YMXZEL6IXL","json":"https://pith.science/pith/JRJPOR2NUX2KAVN4YMXZEL6IXL.json","graph_json":"https://pith.science/api/pith-number/JRJPOR2NUX2KAVN4YMXZEL6IXL/graph.json","events_json":"https://pith.science/api/pith-number/JRJPOR2NUX2KAVN4YMXZEL6IXL/events.json","paper":"https://pith.science/paper/JRJPOR2N"},"agent_actions":{"view_html":"https://pith.science/pith/JRJPOR2NUX2KAVN4YMXZEL6IXL","download_json":"https://pith.science/pith/JRJPOR2NUX2KAVN4YMXZEL6IXL.json","view_paper":"https://pith.science/paper/JRJPOR2N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.03919&json=true","fetch_graph":"https://pith.science/api/pith-number/JRJPOR2NUX2KAVN4YMXZEL6IXL/graph.json","fetch_events":"https://pith.science/api/pith-number/JRJPOR2NUX2KAVN4YMXZEL6IXL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JRJPOR2NUX2KAVN4YMXZEL6IXL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JRJPOR2NUX2KAVN4YMXZEL6IXL/action/storage_attestation","attest_author":"https://pith.science/pith/JRJPOR2NUX2KAVN4YMXZEL6IXL/action/author_attestation","sign_citation":"https://pith.science/pith/JRJPOR2NUX2KAVN4YMXZEL6IXL/action/citation_signature","submit_replication":"https://pith.science/pith/JRJPOR2NUX2KAVN4YMXZEL6IXL/action/replication_record"}},"created_at":"2026-05-18T01:22:49.684085+00:00","updated_at":"2026-05-18T01:22:49.684085+00:00"}