{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:5KKVHCRAUU773IK47HXGESOSRY","short_pith_number":"pith:5KKVHCRA","schema_version":"1.0","canonical_sha256":"ea95538a20a53ffda15cf9ee6249d28e161b59a7a715ca788f0626202f33776f","source":{"kind":"arxiv","id":"1312.4934","version":3},"attestation_state":"computed","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, "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1312.4934","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2013-12-17T20:47:06Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"5b3940849a385aecec8cfff9f6ebf00eae2778d98e6675285561f3055c45413e","abstract_canon_sha256":"eb40467cf3d24f9114a5edaa5b213294a8e3b6106037638c70167036a4ed4495"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:37:42.953466Z","signature_b64":"Yr9DvDOxhysI9MFDKZV1M/t48q1j0fqVTuBDRUDYqGWd8jRDxf8VXyQUZHwMI0w1YybVucaSv8W2HbuzmCWoCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ea95538a20a53ffda15cf9ee6249d28e161b59a7a715ca788f0626202f33776f","last_reissued_at":"2026-05-18T02:37:42.952953Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:37:42.952953Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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