{"paper":{"title":"Superposition operators, Hardy spaces, and Dirichlet type spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Daniel Girela, Mar\\'ia Auxiliadora M\\'arquez, Petros Galanopoulos","submitted_at":"2016-11-16T13:16:45Z","abstract_excerpt":"For $0<p<\\infty $ and $\\alpha >-1$ the space of Dirichlet type $\\mathcal D^p_\\alpha $ consists of those functions $f$ which are analytic in the unit disc $\\mathbb D$ and satisfy $\\int_{\\mathbb D}(1-| z| )^\\alpha| f^\\prime (z)| ^p\\,dA(z)<\\infty $. The space $\\Dp$ is the closest one to the Hardy space $H^p$ among all the $\\mathcal D^p_\\alpha $. Our main object in this paper is studying similarities and differences between the spaces $H^p$ and $\\Dp$ ($0<p<\\infty $) regarding superposition operators. Namely, for $0<p<\\infty $ and $0<s<\\infty $, we characterize the entire functions $\\varphi $ such "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.05265","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"}