{"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 $\\phi_A\\in L_2(A)$, $\\phi_B\\in L_2(B)$. Moreover $S_tL_2(A)\\subseteq L_2(A)$ for all $t>0$ if"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0601349","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"}