{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:QQCMIOMIGHUHV7IZFAS4QHPQWL","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":"d52643593053066d15ae295ef9feca882e4ed874e8af8841e37cb8bf6898550c","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-09-05T14:55:05Z","title_canon_sha256":"4178ab127e68e61624f9e1006134672dc1c3667f9886d3f4d7c5203305ee73c7"},"schema_version":"1.0","source":{"id":"1309.1366","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.1366","created_at":"2026-05-18T02:17:12Z"},{"alias_kind":"arxiv_version","alias_value":"1309.1366v2","created_at":"2026-05-18T02:17:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.1366","created_at":"2026-05-18T02:17:12Z"},{"alias_kind":"pith_short_12","alias_value":"QQCMIOMIGHUH","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"QQCMIOMIGHUHV7IZ","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"QQCMIOMI","created_at":"2026-05-18T12:27:57Z"}],"graph_snapshots":[{"event_id":"sha256:d8a1fa61adf65c9613d74693b73764f5f87229e0ab9bd172b9032dd8a206f93a","target":"graph","created_at":"2026-05-18T02:17:12Z","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 $(M, \\rho,\\mu)$ be an RD-space satisfying the non-collapsing condition. In this paper, the authors introduce Besov-type spaces $B_{p,q}^{s,\\tau}(M)$ and Triebel--Lizorkin-type spaces $F_{p,q}^{s,\\tau}(M)$ associated to a non-negative self-adjoint operator $L$ whose heat kernels satisfy some Gaussian upper bound estimate, H\\\"older continuity, and the stochastic completeness property. Characterizations of these spaces via Peetre maximal functions and heat kernels are established for full range of indices. Also, frame characterizations of these spaces are given. When $L$ is the Laplacian oper","authors_text":"Dachun Yang, Liguang Liu, Wen Yuan","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-09-05T14:55:05Z","title":"Besov-Type and Triebel--Lizorkin-Type Spaces Associated with Heat Kernels"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.1366","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:d2734b030cbe9b73201f5d524ce1d8469fad73b1df6409cf6612dc46d12567db","target":"record","created_at":"2026-05-18T02:17:12Z","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":"d52643593053066d15ae295ef9feca882e4ed874e8af8841e37cb8bf6898550c","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-09-05T14:55:05Z","title_canon_sha256":"4178ab127e68e61624f9e1006134672dc1c3667f9886d3f4d7c5203305ee73c7"},"schema_version":"1.0","source":{"id":"1309.1366","kind":"arxiv","version":2}},"canonical_sha256":"8404c4398831e87afd192825c81df0b2c837d0894f62a2d64f9ad4cb990b53fb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8404c4398831e87afd192825c81df0b2c837d0894f62a2d64f9ad4cb990b53fb","first_computed_at":"2026-05-18T02:17:12.515356Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:17:12.515356Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4za3ryuDpELKjYBoVhzpljnWSGvqWMI4BFq0QqTslsSgPxwocZLXFP6N4HfkYxHI4Z/k5PeGfq0647p+NLZJDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:17:12.516071Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.1366","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d2734b030cbe9b73201f5d524ce1d8469fad73b1df6409cf6612dc46d12567db","sha256:d8a1fa61adf65c9613d74693b73764f5f87229e0ab9bd172b9032dd8a206f93a"],"state_sha256":"97884d683eb73b4c1127699d99460d7eb677f16d8e45796c06c1da2218181f06"}