{"paper":{"title":"Mean Lipschitz conditions on Bergman space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CV","authors_text":"A. G. Siskakis, G. Stylogiannis, P. Galanopoulos","submitted_at":"2013-12-17T20:47:06Z","abstract_excerpt":"For $f$ analytic on the unit disc let $r_t(f)(z)=f(e^{it}z)$ and $f_r(z)=f(rz)$, rotations and dilations respectively. We show that for $f$ in the Bergman space $A^p$ and $0<\\alpha\\leq 1$ the following are equivalent. \\begin{itemize} \\item[(i)] $\\n{r_t(f)-f}_{A^p}=\\og(|t|^{\\alpha}), \\quad t\\to 0$, \\item[(ii)] $\\n{(f')_r}_{A^p} =\\og\\left (1-r)^{\\alpha-1}\\right ), \\quad r\\to 1^{-}$, \\item[(iii)] $\\n{f_r-f}_{A^p}=\\og((1-r)^{\\alpha}),\\quad r\\to 1^{-}$. \\end{itemize}\n  The Hardy space analogues of these conditions are known to be equivalent by results of Hardy and Littlewood and of E. Storozhenko, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.4934","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"}