Abstract
Let \(\varOmega \) be a Jordan domain in \(\mathbb {C}\), J an open arc of \(\partial \varOmega \) and \(\phi : D\rightarrow \varOmega \) a Riemann map from the open unit disk D onto \(\varOmega \). Under certain assumptions on \(\phi \) we prove that if a holomorphic function \(f\in H(\varOmega )\) extends continuously on \(\varOmega \cup J\) and \(p\in \{1, 2, \dots \}\cup \{\infty \}\), then the following equivalence holds: the derivatives \(f^{(l)}, 1\le l\le p, l\in \mathbb {N},\) extend continuously on \(\varOmega \cup J\) if and only if the function \(\left. f\right| _{J}\) has continuous derivatives on J with respect to the position of orders \(l, 1\le l\le p, l\in \mathbb {N}\). Moreover, we show that for the relevant function spaces, the topology induced by the l-derivatives on \(\varOmega , 0\le l\le p, l\in \mathbb {N},\) coincides with the topology induced by the same derivatives taken with respect to the position on J.
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Georgakopoulos, N., Mastrantonis, V. & Nestoridis, V. Relations of the Spaces \({\varvec{A^p(\varOmega )}}\) and \({\varvec{C^p(\partial \varOmega )}}\). Results Math 73, 86 (2018). https://doi.org/10.1007/s00025-018-0842-5
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DOI: https://doi.org/10.1007/s00025-018-0842-5