{"paper":{"title":"Kernel estimate and capacity in Dirichlet type spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.CV","authors_text":"K. Kellay, O. El-Fallah, Y. Elmadani","submitted_at":"2014-11-04T20:39:12Z","abstract_excerpt":"Let $\\mu$ be a positive finite measure on the unit circle. The Dirichlet type space $\\mathcal{D}(\\mu)$, associated to $\\mu$, consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against the Poisson integral of $\\mu$. First, we give an estimate of the norm of the reproducing kernel $k^\\mu$ of $\\mathcal{D}(\\mu)$. Next, we study the notion of $\\mu$-capacity associated to $\\mathcal{D}(\\mu)$, in the sense of Beurling--Deny. Namely, we give an estimate of $\\mu$-capacity of arcs in terms of the norm of $k^\\mu$. We also provide a new condition on clo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.1036","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"}