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We construct two Dirichlet forms $\\ce_m$ and $\\ce_M$ such that $\\ce_m\\leq \\ce\\leq \\ce_M$. These forms are potentially the smallest and largest such Dirichlet forms. In particular $\\ce_m\\supseteq \\ce_M$, $(\\ce_M)_m=\\ce_m$ and $(\\ce_m)_M=\\ce_M$. We analyze the family of local, inner "},"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":"1602.01167","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-02-03T02:22:42Z","cross_cats_sorted":[],"title_canon_sha256":"6b221d6ba660e39aabde79cf1f4132088715a9d51710039a2e4045afe62af630","abstract_canon_sha256":"37a71d865076010375ebf2ecd4b1d2ded8eab28104906ee3bbe35286bf43f0ab"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:23.426295Z","signature_b64":"tfrhF1VJifoPNMiw1Ta0ZhBum4TDhSAIqJVPxyd+BpS5CXWPX8NROpxF0CUykDJeyT2fXVQFewgvtfkmOVKtCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a633223f95abedf0e625adec4e253dadd3f590cf0f8553956fa338204091e545","last_reissued_at":"2026-05-18T01:21:23.425765Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:23.425765Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On extensions of local Dirichlet forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Derek W. Robinson","submitted_at":"2016-02-03T02:22:42Z","abstract_excerpt":"Let $\\ce$ be a Dirichlet form on $L_2(X\\,;\\mu)$ where $(X,\\mu)$ is locally compact $\\sigma$-compact measure space. Assume $\\ce$ is inner regular, i.e.\\ regular in restriction to functions of compact support, and local in the sense that\n  $\\ce(\\varphi,\\psi)=0$ for all $\\varphi, \\psi\\in D(\\ce)$ with $\\varphi\\,\\psi=0$. We construct two Dirichlet forms $\\ce_m$ and $\\ce_M$ such that $\\ce_m\\leq \\ce\\leq \\ce_M$. These forms are potentially the smallest and largest such Dirichlet forms. In particular $\\ce_m\\supseteq \\ce_M$, $(\\ce_M)_m=\\ce_m$ and $(\\ce_m)_M=\\ce_M$. We analyze the family of local, inner "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.01167","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":"1602.01167","created_at":"2026-05-18T01:21:23.425853+00:00"},{"alias_kind":"arxiv_version","alias_value":"1602.01167v1","created_at":"2026-05-18T01:21:23.425853+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.01167","created_at":"2026-05-18T01:21:23.425853+00:00"},{"alias_kind":"pith_short_12","alias_value":"UYZSEP4VVPW7","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_16","alias_value":"UYZSEP4VVPW7BZRF","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_8","alias_value":"UYZSEP4V","created_at":"2026-05-18T12:30:46.583412+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/UYZSEP4VVPW7BZRFVXWE4JJ5VX","json":"https://pith.science/pith/UYZSEP4VVPW7BZRFVXWE4JJ5VX.json","graph_json":"https://pith.science/api/pith-number/UYZSEP4VVPW7BZRFVXWE4JJ5VX/graph.json","events_json":"https://pith.science/api/pith-number/UYZSEP4VVPW7BZRFVXWE4JJ5VX/events.json","paper":"https://pith.science/paper/UYZSEP4V"},"agent_actions":{"view_html":"https://pith.science/pith/UYZSEP4VVPW7BZRFVXWE4JJ5VX","download_json":"https://pith.science/pith/UYZSEP4VVPW7BZRFVXWE4JJ5VX.json","view_paper":"https://pith.science/paper/UYZSEP4V","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1602.01167&json=true","fetch_graph":"https://pith.science/api/pith-number/UYZSEP4VVPW7BZRFVXWE4JJ5VX/graph.json","fetch_events":"https://pith.science/api/pith-number/UYZSEP4VVPW7BZRFVXWE4JJ5VX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UYZSEP4VVPW7BZRFVXWE4JJ5VX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UYZSEP4VVPW7BZRFVXWE4JJ5VX/action/storage_attestation","attest_author":"https://pith.science/pith/UYZSEP4VVPW7BZRFVXWE4JJ5VX/action/author_attestation","sign_citation":"https://pith.science/pith/UYZSEP4VVPW7BZRFVXWE4JJ5VX/action/citation_signature","submit_replication":"https://pith.science/pith/UYZSEP4VVPW7BZRFVXWE4JJ5VX/action/replication_record"}},"created_at":"2026-05-18T01:21:23.425853+00:00","updated_at":"2026-05-18T01:21:23.425853+00:00"}