{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:GIFYCP4YMAIFTDGHYVM2JC6RMP","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":"2dadcf88c6271d220459d7ea3b4a123558fd2575d5e8613235ac07df903f0c9c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-05-21T22:08:11Z","title_canon_sha256":"3a6e30d610a56e61f63c262b5a39eb00c1c9b84359a5ea90fb64f11bc65ef42d"},"schema_version":"1.0","source":{"id":"1005.4090","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.4090","created_at":"2026-05-18T02:58:02Z"},{"alias_kind":"arxiv_version","alias_value":"1005.4090v1","created_at":"2026-05-18T02:58:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.4090","created_at":"2026-05-18T02:58:02Z"},{"alias_kind":"pith_short_12","alias_value":"GIFYCP4YMAIF","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"GIFYCP4YMAIFTDGH","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"GIFYCP4Y","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:58ed498120bdc71044b2094537df6ae7c72c374105352cb4ba84091caff9dd9f","target":"graph","created_at":"2026-05-18T02:58:02Z","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":"We prove a superposition principle for Riesz potentials of nonnegative continuous functions on Lie groups of Heisenberg type. More precisely, we show that the Riesz potential $$ R_\\alpha(\\rho)(g) = \\int_{\\G} N(g^{-1} g')^{\\alpha-Q} \\rho(g') dg', \\qquad 0<\\alpha<Q, $$ of a nonnegative function $\\rho\\in C_0(\\G)$ on a group $\\G$ of Heisenberg type is necessarily either $p$-subharmonic or $p$-superharmonic, depending on $p$ and $\\alpha$. Here $N$ denotes the non-isotropic homogeneous norm on such groups, as introduced by Kaplan. This result extends to a wide class of nonabelian stratified Lie grou","authors_text":"Jeremy Tyson, Nicola Garofalo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-05-21T22:08:11Z","title":"Riesz potentials and p-superharmonic functions in Lie groups of Heisenberg type"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.4090","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:dc7b3c750476b356ee9ce54fb96909ad33b62c14939dfc75d1adc8d57e70ca6b","target":"record","created_at":"2026-05-18T02:58:02Z","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":"2dadcf88c6271d220459d7ea3b4a123558fd2575d5e8613235ac07df903f0c9c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-05-21T22:08:11Z","title_canon_sha256":"3a6e30d610a56e61f63c262b5a39eb00c1c9b84359a5ea90fb64f11bc65ef42d"},"schema_version":"1.0","source":{"id":"1005.4090","kind":"arxiv","version":1}},"canonical_sha256":"320b813f986010598cc7c559a48bd163e40a099fd5c72788cba462037e25fa24","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"320b813f986010598cc7c559a48bd163e40a099fd5c72788cba462037e25fa24","first_computed_at":"2026-05-18T02:58:02.141128Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:02.141128Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0BMQUrhouCPfn56dD5Izldh8PeT5AuipAepkNq0L2Dd4jU79t6u7Dq7IkdFnYV4lgKkhe+f88n0EvN0VhVhdAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:02.141678Z","signed_message":"canonical_sha256_bytes"},"source_id":"1005.4090","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dc7b3c750476b356ee9ce54fb96909ad33b62c14939dfc75d1adc8d57e70ca6b","sha256:58ed498120bdc71044b2094537df6ae7c72c374105352cb4ba84091caff9dd9f"],"state_sha256":"2fa56d7fcd9cc477f9e9602c785996f2f40c4d58f3d1722a2c1ba1e572d9132a"}