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

28 Mar 2018

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.