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Relations of the spaces Ap (Ω) and C p (∂Ω)

Accepted version
Peer-reviewed

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Article

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Authors

Georgakopoulos, N 
Mastrantonis, V 
Nestoridis, V 

Abstract

Let Ω be a Jordan domain in C, J an open arc of ∂Ω and φ : D → Ω a Riemann map from the open unit disk D onto Ω. Under certain assumptions on φ we prove that if a holomorphic function f ∈ H(Ω) extends continuously on Ω ∪ J and p ∈ {1, 2, . . . } ∪ {∞}, then the following equivalence holds: the derivatives f (l) , 1 ≤ l ≤ p, l ∈ N, extend continuously on Ω ∪ J if and only if the function f|J has continuous derivatives on J with respect to the position of orders l, 1 ≤ l ≤ p, l ∈ N. Moreover, we show that for the relevant function spaces, the topology induced by the l−derivatives on Ω, 0 ≤ l ≤ p, l ∈ N, coincides with the topology induced by the same derivatives taken with respect to the position on J.

Description

This is the author accepted manuscript. It is currently under an indefinite embargo pending publication by Springer.

Keywords

Riemann map, Poisson Kernel, Jordan curve, Smoothness on the boundary

Journal Title

Results in Mathematics

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Publisher

Springer

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