{"paper":{"title":"Functional Inequalities for Stable-Like Dirichlet Forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Feng-Yu Wang, Jian Wang","submitted_at":"2012-05-21T07:40:01Z","abstract_excerpt":"Let $V\\in C^2(\\R^d)$ such that $\\mu_V(\\d x):= \\e^{-V(x)}\\,\\d x$ is a probability measure, and let $\\aa\\in (0,2)$. Explicit criteria are presented for the $\\aa$-stable-like Dirichlet form $$\\E_{\\aa,V}(f,f):= \\int_{\\R^d\\times\\R^d} \\ff{|f(x)-f(y)|^2}{|x-y|^{d+\\alpha}}\\,\\d y\\,\\e^{-V(x)}\\,\\d x$$ to satisfy Poincar\\'e-type (i.e., Poincar\\'e, weak Poincar\\'e and super Poincar\\'e) inequalities. As applications, sharp functional inequalities are derived for the Dirichlet form with $V$ having some typical growths. Finally, the main result of \\cite{MRS} on the Poincar\\'e inequality is strengthened"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.4508","kind":"arxiv","version":3},"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"}