{"paper":{"title":"Dirichlet forms and critical exponents on fractals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Ka-Sing Lau, Qingsong Gu","submitted_at":"2017-03-21T05:47:18Z","abstract_excerpt":"Let $B^{\\sigma}_{2, \\infty}$ denote the Besov space defined on a compact set $K \\subset {\\Bbb R}^d$ which is equipped with an $\\alpha$-regular measure $\\mu$. The {\\it critical exponent} $\\sigma^*$ is the supremum of the $\\sigma$ such that $B^{\\sigma}_{2, \\infty} \\cap C(K)$ is dense in $C(K)$. It is well-known that for many standard self-similar sets $K$, $B^{\\sigma^*}_{2, \\infty}$ are the domain of some local regular Dirichlet forms. In this paper, we explore new situations that the underlying fractal sets admit inhomogeneous resistance scalings, which yield two types of critical exponents. We"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.07061","kind":"arxiv","version":2},"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"}