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Izv. RAN. Ser. Mat., 2018, Volume 82, Issue 1, Pages 34–64 (Mi izv8484)  

Morera-type theorems in the hyperbolic disc

V. V. Volchkov, Vit.V.Volchkov

Donetsk National University

Abstract: Let $G$ be the group of conformal automorphisms of the unit disc $\mathbb{D}=ż\in\mathbb{C}\colon |z|<1\}$. We study the problem of the holomorphicity of functions $f$ on $\mathbb{D}$ satisfying the equation
$$ \int_{\gamma_{\varrho}} f(g (z))  dz=0 \quad \forall   g\in G, $$
where $\gamma_{\varrho}=ż\in\mathbb{C}\colon |z|=\varrho\}$ and $\rho\in (0,1)$ is fixed. We find exact conditions for holomorphicity in terms of the boundary behaviour of such functions. A by-product of our work is a new proof of the Berenstein–Pascuas two-radii theorem.

Keywords: holomorphicity, conformal automorphism, boundary behaviour.

DOI: https://doi.org/10.4213/im8484

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English version:
Izvestiya: Mathematics, 2018, 82:1, 31–60

Bibliographic databases:

UDC: 517.444
MSC: 30A05, 43A80, 43A90, 44A15, 44A35, 45Q05
Received: 05.12.2015
Revised: 18.09.2016

Citation: V. V. Volchkov, Vit.V.Volchkov, “Morera-type theorems in the hyperbolic disc”, Izv. RAN. Ser. Mat., 82:1 (2018), 34–64; Izv. Math., 82:1 (2018), 31–60

Citation in format AMSBIB
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  • Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya Izvestiya: Mathematics
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