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The behaviour of the corresponding evolution semigroup $S_t$ can be described in terms of a function $(A,B) \\mapsto d(A ;B)\\in[0,\\infty]$ over pairs of measurable subsets of $\\Ri^d$. Then \\[ |(\\phi_A,S_t\\phi_B)|\\leq e^{-d(A;B)^2(4t)^{-1}}\\|\\phi_A\\|_2\\|\\phi_B\\|_2 \\] for all $t>0$ and all $\\phi_A\\in L_2(A)$, $\\phi_B\\in L_2(B)$. Moreover $S_tL_2(A)\\subseteq L_2(A)$ for all $t>0$ if"},"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":"math/0601349","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2006-01-14T01:06:51Z","cross_cats_sorted":[],"title_canon_sha256":"0ef6bd408e4d6949d12594d1790901376ffadbd09fb818e00332c5f0c12e33f5","abstract_canon_sha256":"b5069d097514d71df7737247952a37ae3368ac6b998956b62eefd53a7beaaa78"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:03:22.976334Z","signature_b64":"Q/TRHV48BbvG4+ujyRds5aJZVHHKcK+adqtjy1YX/SjiTiZqItggruzGgydm4EYfQYuQA9RzES6d4wYoZRJYDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a498bf836a23fe83774d7aced4056d554d8b6fa2b2937aa743dacf9e8bf72f7","last_reissued_at":"2026-05-18T03:03:22.975739Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:03:22.975739Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dirichlet forms and degenerate elliptic operators","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Adam Sikora, A.F.M. ter Elst, Derek W. Robinson, Yueping Zhu","submitted_at":"2006-01-14T01:06:51Z","abstract_excerpt":"It is shown that the theory of real symmetric second-order elliptic operators in divergence form on $\\Ri^d$ can be formulated in terms of a regular strongly local Dirichlet form irregardless of the order of degeneracy. The behaviour of the corresponding evolution semigroup $S_t$ can be described in terms of a function $(A,B) \\mapsto d(A ;B)\\in[0,\\infty]$ over pairs of measurable subsets of $\\Ri^d$. Then \\[ |(\\phi_A,S_t\\phi_B)|\\leq e^{-d(A;B)^2(4t)^{-1}}\\|\\phi_A\\|_2\\|\\phi_B\\|_2 \\] for all $t>0$ and all $\\phi_A\\in L_2(A)$, $\\phi_B\\in L_2(B)$. 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