TY - JOUR
TI - Relations of the Spaces Ap(Ω) and Cp(∂Ω)
AU - Georgakopoulos, N.
AU - Mastrantonis, V.
AU - Nestoridis, V.
JO - Results in Mathematics
PY - 2018
VL - 73
TODO - 2
SP - null
PB - Birkhauser Verlag AG
SN - 1422-6383, 1420-9012
TODO - 10.1007/s00025-018-0842-5
TODO - null
TODO - 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. © 2018, The Author(s).
ER -