Relations of the Spaces Ap(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{A^p(\varOmega )}}$$\end{document} and Cp(∂Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{C^p(\partial \varOmega )}}$$\end{document}

Let Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega $$\end{document} be a Jordan domain in C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}$$\end{document}, J an open arc of ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \varOmega $$\end{document} and ϕ:D→Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi : D\rightarrow \varOmega $$\end{document} a Riemann map from the open unit disk D onto Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega $$\end{document}. Under certain assumptions on ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} we prove that if a holomorphic function f∈H(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in H(\varOmega )$$\end{document} extends continuously on Ω∪J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega \cup J$$\end{document} and p∈{1,2,⋯}∪{∞}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in \{1, 2, \dots \}\cup \{\infty \}$$\end{document}, then the following equivalence holds: the derivatives f(l),1≤l≤p,l∈N,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{(l)}, 1\le l\le p, l\in \mathbb {N},$$\end{document} extend continuously on Ω∪J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega \cup J$$\end{document} if and only if the function fJ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left. f\right| _{J}$$\end{document} has continuous derivatives on J with respect to the position of orders l,1≤l≤p,l∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l, 1\le l\le p, l\in \mathbb {N}$$\end{document}. Moreover, we show that for the relevant function spaces, the topology induced by the l-derivatives on Ω,0≤l≤p,l∈N,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega , 0\le l\le p, l\in \mathbb {N},$$\end{document} coincides with the topology induced by the same derivatives taken with respect to the position on J.


Introduction
In this paper we investigate the relationship between the continuous extendability of the derivatives of a function f ∈ A(Ω), for some Jordan domain Ω, and the differentiability of the map t → f (γ(t)) for some parametrization γ of ∂Ω. Here, A(Ω) is the collection of all complex functions holomorphic on Ω and continuous on Ω. Specifically, it is well known that the first p derivatives of a function f ∈ A(D), D being the unit disk in C, continuously extend over D if and only if the map t → f (e it ) is p times continuously differentiable [6]. We generalize this for functions that are holomorphic on the unit disk but now continuously extend over an open arc of T = ∂D and prove an analogous equivalence for functions defined on Jordan domains that have sufficiently smooth Riemann maps.
The spaces A p (Ω), p ∈ {0, 1, 2, . . .} ∪ {∞}, consist of all holomorphic functions f in Ω whose derivatives f (l) , l ∈ {0, 1, . . .}, l ≤ p, extend continuously on Ω. It is well known that for any f in the disc algebra A(D), f ∈ A p (D) if and only if the map t → f (e it ) is C p smooth. In other words, A p (D) = A(D) ∩ C p (T) both as sets and as topological spaces. Additionally, if f ∈ A(D) and g(t) = f (e it ), t ∈ R, the equation that relates the continuous extension of f on T and the derivative of g is as expected, i.e. dg dt (t) = ie it f (e it ). To prove this, one can use the Poisson representation, to recover the values of f in the disk from its boundary values, i.e. f (re it ) = (g * P r )(t) where P r denotes the Poisson kernel, differentiate both sides with respect to t and let r → 1 − . A detailed proof can be found in [6].
In this paper we prove analogous results for functions f ∈ A(D) whose derivatives continuously extend on an open arc of the unit circle but not necessarily the entire circle. Moreover, using Riemann's mapping theorem, we can drop our initial assumption f ∈ A(D) and instead assume that it only extends continuously over the specific arc we are interested in. Precisely, if f : D → C is holomorphic and continuously extends over an open arc J ⊆ T, then its first p derivatives continuously extend over that arc if and only if the This motivates a more general definition of the spaces A p .
In Sect. 3 we consider functions f holomorphic on a Jordan domain Ω and continuous on Ω ∪ J, for some open arc J of ∂Ω. We prove that for any p ∈ {1, 2, . . . } ∪ {∞}, the derivatives f (l) , 0 ≤ l ≤ p, continuously extend over Ω ∪ J if and only if the continuous extension of f on J is p times continuously differentiable on J with respect to the position [2]. To do this, we place a smoothness assumption for the Riemann map φ : D → Ω from the open unit disk D onto Ω. The condition is that (φ −1 ) has a continuous extension on Ω ∪ J and that (φ −1 ) (z) = 0 on Ω ∪ J.
Professor Paul Gauthier informed us that results in the same direction have already been proven in a paper of Steve Bell and Laszlo Lempert entitled "A C ∞ Schwarz Reflection principle in one and several complex variables" which appeared in J. Differential Geometry 32(1990)899-915. However, they do not cover our result Theorem 5. He also added that section 8 of a paper of Steven R. Bell and Steven G. Krantz entitled "Smoothness to the boundary of conformal maps", which appeared in Rocky Mountain Journal of Mathematics, vol. 17, no 1, winter 1987, shows that our result is the best possible in a certain sense.

Extendability Over an Open Arc of the Unit Circle
For 0 ≤ p ≤ +∞, A p (D) denotes the space of holomorphic functions on D whose derivatives of order l ∈ N, 0 ≤ l ≤ p, extend continuously over D. It is topologized via the semi-norms: Vol. 73 (2018) Relations of the Spaces A p (Ω) and C p (∂Ω) Page 3 of 13 86 The following theorem is well known. A detailed proof can be found in [6], We now generalize this on functions that are holomorphic on D and continuously extend over an open arc J of the unit circle. We prove that for any such function f and p ∈ {1, 2, . . .} ∪ {∞} the first p derivatives of f continuously extend over D∪J if and only if the map t → f (e it ) is p times continuously differentiable in I = {t ∈ [a, a + 2π] : e it ∈ J} for a suitable a ∈ R. Denote by A p (D, J) the space of holomorphic functions whose first p derivatives continuously extend over D ∪ J and let C p (J) be the class of functions f : J → C, such that the map t → f (e it ), t ∈ I, is p times continuously differentiable. The aim is to show the equality A p (D, J) = A(D, J) ∩ C p (J). For simplicity, we take J = {e it : 0 < t < 1} throughout this section. For any z = re iθ ∈ C we denote by P z (t) or P r (t) the Poisson kernel [1].

Proposition 1. If u : [0, 1] → R is a continuous function then:
is well defined in C\J and C ∞ harmonic.
Proof. To see that A(z) is well defined in C\J observe that: ) > 0 and: for all t ∈ [0, 2π]. Since the quantity on the right hand side is integrable we deduce that A(z) is indeed well defined in C\J.
In order to prove that A is C ∞ harmonic it suffices to show that g(z) = 1 2π since A is the real part of g according to (3). Note that for z = z 0 : and hence for z sufficiently close to z 0 : Therefore g is holomorphic and the proof is complete.
Because every continuous function u : and v(x) = 0 otherwise, one can expect that such a function when convolved with the Poisson kernel would retain the nice properties. More specifically, it will uniformly converge to 0 and to u(x) on the compact subsets of the respective open arcs. We prove this in Propositions 2 and 3.

Proposition 2. If A(z)
is as before, then for all θ ∈ (1, 2π) and l ∈ N: The convergence is uniform in the compact subsets of (1, 2π).
Vol. 73 (2018) Relations of the Spaces A p (Ω) and C p (∂Ω) Page 5 of 13 86 Proof. We start with l = 1. It suffices to prove that for all [θ 1 , and hence sup − −−− → 0 since the right hand side of (15) converges to 0 as r → 1 − . Note that no matter how many times we differentiate P r in respect to θ we will have a finite sum of fractions with numerator c(1 − r 2 ) k cos(θ) l sin(θ) m for c = 0, k, l, m ∈ N and denominator a power of (1 + r 2 − 2r cos(θ − t)). So for any l ≥ 2 the same arguments apply.
The convergence is uniform in the compact subsets of (0, 1).
Proof. By a theorem of Borel (see [5]), we can find a function q : R → R of class C ∞ (R) such that q l (0) = u l (0) and q l (1) = u l (1) for 0 ≤ l ≤ p, l ∈ N. Thus, the function: is of class C p (T). Define: It is well known that uniformly for all θ ∈ R, 0 ≤ l ≤ p, l ∈ N:

u(t)P z (t)dt and B(z)
Note that G(z) = A(z) + B(z) for all |z| < 1 and hence for 0 ≤ l ≤ p, l ∈ N: From (18) Proposition 2 we have that d l G dθ l (re iθ ) → u (l) (θ) and d l B dθ l (re iθ ) → 0 as r → 1 − , uniformly in the compact subsets of (0, 1). As a result, while the convergence is uniform in the compact subsets of (0, 1).
Remark 1. By linearity, Propositions 1, 2, 3 and Lemma 1 hold for complex functions f = u + iv where u = Ref and v = Imf, since we can apply them to the real and imaginary part separately.
We now adapt the proof of Theorem 1 for functions whose derivatives only extend over an open arc of the unit circle.
The following are equivalent: f (l) continuously extends on D ∪ J for all 0 ≤ l ≤ p, l ∈ N if and only if g is p times continuously differentiable in (0, 1). In that case, for all t ∈ (0, 1) Proof. We prove it by induction on p. For p = 1, let t 0 ∈ (0, 1) and t 1 < t 0 < t 2 such that [t 1 , t 2 ] ⊂ (0, 1). Assuming that f (l) continuously extends over D ∪ J for all 0 ≤ l ≤ p, l ∈ N, let f r (t) = f (re it ), h(t) = ie it f (e it ) and: for all t ∈ (0, 1) and 0 < r < 1.
and hence from a well-known theorem f r → hdt + c, for a some c ∈ C, as r → 1, while the convergence is uniform in [t 1 , t 2 ]. Additionally, f r → g, as r → 1, uniformly in [t 1 , t 2 ] and therefore g (t) = h(t) = ie it f (e it ), t ∈ (t 1 , t 2 ). However, t 0 was arbitrary thus, g ∈ C 1 ((0, 1)) and (22) holds. For the converse let g ∈ C 1 ((0, 1)). We now use the Poisson representation: Vol. 73 (2018) Relations of the Spaces A p (Ω) and C p (∂Ω) Page 7 of 13 86 for |z| < 1. Define: for all 0 < r < 1 and t ∈ R. Since [t 1 , t 2 ] ⊂ (0, 1), Propositions 2 and 3 imply that: uniformly for t ∈ [t 1 , t 2 ]. Since t 0 was arbitrarily chosen in (0, 1) we deduce that f extends continuously on D ∪ J. To complete the induction, let us assume that the theorem holds for some p ≥ 1. If f (l) continuously extends on D ∪ J for all 0 ≤ l ≤ p + 1, l ∈ N it follows that (f ) (l) continuously extends on D ∪ J for all 0 ≤ l ≤ p, l ∈ N. By the induction hypothesis the map t → f (e it ) belongs in the class C p ((0, 1)) and since, by the case of p = 1, g (t) = ie it f (e it ) we have g ∈ C p ((0, 1)) and hence g ∈ C p+1 ((0, 1)). For the converse, if g ∈ C p+1 ((0, 1)) it follows from (22) that g (t) = ie it f (e it ) ∈ C p ((0, 1)) and therefore, the map t → f (e it ) is of class C p ((0, 1)). By the induction hypothesis, (f ) (l) continuously extends on D ∪ J, for all 0 ≤ l ≤ p, l ∈ N and hence f (l) continuously extends on D ∪ J, for all 0 ≤ l ≤ p + 1, l ∈ N. The case of p = ∞ follows easily.
Using Riemann's mapping theorem [1] we can drop the assumption of continuity over D. Indeed, we can just have f continuously extend over the open arc we are dealing with, i.e. f ∈ A(D, J).  ∈ (0, 1). The following are equivalent: f (l) continuously extends over D ∪ J, for all 0 ≤ l ≤ p, l ∈ N, if and only if g is p times continuously differentiable in (0, 1). In that case: for all t ∈ (0, 1). Proof. We prove it by induction on p. For p = 1, the only if part is proven like Theorem 2; we also obtain (30). For the converse, let t 0 ∈ (0, 1), t 1 < t 0 < t 2 such that [t 1 , t 2 ] ⊆ (0, 1) and J = {e it : t 1 < t < t 2 }. Set V = {|z| < 1, z = 0 : z |z| ∈ J }. It is easily verified that V is simply connected and hence there is a conformal map φ : D → V which extends to homeomorphism over the closures φ : D → V , by the Osgood-Carathéodory theorem [4]. Since  1 , t 2 )) with non-vanishing derivative (see [1] pp. 233-235). Since g ∈ C 1 ((t 1 , t 2 )) and t → φ(e it ) is in C ∞ ((0, 1)) we have that the map t → f (φ(e it )) is in C 1 ((0, 1)). By Theorem 2 the derivative (f • φ) continuously extends on D ∪ J and hence f continuously extends in

Jordan Domains
. Since any two Riemann maps differ by an automorphism of the unit disk it is easily verified that the spaces C p φ (J) do not depend on the chosen Riemann map. Next, we consider differentiability on J with respect to the position [2]. Definition 1. Let J be a Jordan arc and f : J → C. We define the derivative of f on z 0 ∈ J by: df dz Vol. 73 (2018) Relations of the Spaces A p (Ω) and C p (∂Ω) Page 9 of 13 86 if this limit exists and is a complex number.
In order to go one step further , we consider the derivative df dz on J of Definition 1 and we take its derivative on J with respect to the position.

Definition 2.
A function f : J → C belongs to the class C 1 (J) if df dz (z) exists and is continuous for z ∈ J. Inductively, suppose that d p−1 f dz p−1 is well defined on J for some p = 2, . . . , +∞, we say that f is of class exists and is continuous on J.
Let us first prove a straightforward fact concerning the parametrization of an arc J of ∂Ω induced by a Riemann map φ.
On the right hand side of (34) f denotes the continuous extension of f (z) from Ω to Ω ∪ J.
Before proceeding to the proof, let us note that the choice of the Riemann map is irrelevant. Suppose that Φ, Ψ are two Riemann maps and J an open arc of ∂Ω such that Φ −1 is of class A n (Ω, J), n ∈ {1, 2, . . .}∪{∞}, and (Φ −1 ) (z) = 0 for all z ∈ Ω ∪J. Then, Ψ −1 is also of class A n (Ω, J) and (Ψ −1 ) is non zero in Ω ∪J. To see this, observe that Φ −1 •Ψ is an automorphism of the unit disk and hence there are a ∈ D and c ∈ T such that Ψ = Φ•φ a , where φ a (z) = c z−a 1−az , z ∈ D. Note that φ a is holomorphic in D(0, 1 a ) ⊃ D with non vanishing derivative