One sided extendability and p-continuous analytic capacities

Using complex methods combined with Baire's Theorem we show that one-sided extendability, extendability and real analyticity are rare phenomena on various spaces of functions in the topological sense. These considerations led us to introduce the p-continuous analytic capacity and variants of it, $p \in \{ 0, 1, 2, \cdots \} \cup \{ \infty \}$, for compact or closed sets in $\mathbb{C}$. We use these capacities in order to characterize the removability of singularities of functions in the spaces $A^p$.


Introduction
In [2] it is proven that the set X of nowhere analytic functions in C ∞ ([0, 1]) contains a dense and G δ subset of C ∞ ([0, 1]). In [1] using Fourier methods it is shown that X is itself a dense and G δ subset of C ∞ ([0, 1]). Furthermore, combining the above methods with Borel's Theorem ( [7]) and a version of Michael's Selection Theorem ( [12]) the above result has been extended to C ∞ (γ), where γ is any analytic curve. In the case where γ is the unit circle T every function f ∈ C ∞ (T ) can be written as a sum f = g + w where g belongs to A ∞ (D) and is holomorphic in the open unit disc D and very smooth up to the boundary and w has similar properties in D c . Now, if we assume that f is extendable somewhere towards one side of T , say in D c , then because w is regular there, it follows that g ∈ A ∞ (D) is extendable. But the phenomenon of somewhere extendability has been proven to be a rare phenomenon in the Frechet space A ∞ (D) ( [8]). It follows that the phenomenon of one sided somewhere extendability is a rare phenomenon in C ∞ (T ) or more generally in C p (γ), p ∈ {∞} ∪ {0, 1, 2, · · · } for any analytic curve γ ( [1]). After the preprint [1] has been circulated, P. Gautier noticed that the previous result holds more generally for Jordan arcs without the assumption of analyticity of the curve. Indeed, applying complex methods, appearing in the last section of [1], we prove this result. It suffices to use the Oswood-Caratheodory Theorem combined with Montel's Theorem and the Poisson integral formula applied to the boundary values of bounded holomorphic functions in H ∞ (D). In fact, this complex method is most natural to our considerations of extendability, real-analyticity and one sided-extendability. The proofs are simplified and the results hold under much more general assumptions than the assumptions imposed by the Fourier method. This complex method is developed in the present paper. In section 4 we prove that extandability and real analyticity are rare phenomena in various spaces of functions on locally injective curves γ. For the real analyticity result we assume that γ is analytic and the result holds in any C k (γ), k ∈ {0, 1, 2, 3, · · · } ∪ {∞} endowed with its nature topology. For the other results the phenomena are proven to be rare in C k (γ), provided that the locally injective curve γ has smoothness at least of degree k. In section 5 initially we consider a finite set of disjoint curves γ 1 , γ 2 , · · · , γ n . Then in the case where γ 1 , γ 2 , · · · , γ n are disjoint Jordan curves in C bounding a domain Ω of finite connectivity we consider the spaces A p (Ω), p ∈ {0, 1, 2, 3, · · · } ∪ {∞} which by the maximum principle can be seen as function spaces on ∂Ω = γ * 1 ∪ · · · ∪ γ * n . In these spaces we show that the above phenomena of extendability or real analyticity are rare phenomena. For the real analyticity result we assume analyticity of ∂Ω, but for the extendability result we do not need to assume any smoothness of the boundary. In section 6 we consider the one sided extendability from a locally injective curve γ and we prove that this is a rare phenomenon in various spaces of functions. We construct a denumerable family G n of Jordan domains G containing in their boundary a non-trivial arc J of the image γ * of γ, such that each other domain Ω with similar properties contains some G n . We show that the phenomenon of extendability is rare for each domain G. Then by denumerable union (or intersection of the complements) we obtain our result with the aid of Baire's Category Theorem. We mention that the one-sided extendability of a function f : γ * → C is meant as the existence of a function F : G ∪ J → C which is hololomorphic on the Jordan domain G, continuous on G ∪ J and such that on the arc J of γ * we have F | J = f | J . Such notions of one-sided extendability have been considered in [3] and the references therein, but in the present article and [1] it is, as far as we know, the first time where the phenomenon is proven to be rare. At the end of section 6 we prove similar results on one-sided extendability on the space A p (Ω), where Ω is a finitely connected domain in C bounded by a finite set of disjoint Jordan curves γ 1 , γ 2 , · · · , γ n . Now, the extension F of a function f ∈ A p (Ω) has to coincide with f only on a non-trivial arc of the boundary of Ω, not on an open subset of Ω. Certainly if the continuous analytic capacity of ∂Ω is zero, the latter automatically happens, but not in general. In section 7 we consider Ω a domain in C, L a compact subset of Ω and we consider the phenomenon of extendability of a function f ∈ A p (Ω \ L) to a function F in A p (Ω). There is a dichotomy. Either for every f this is possible or generically for all f ∈ A p (Ω \ L) this fails. In order to characterize when each horn of the above dichotomy holds we are led to define the p-continuous analytic capacity a p (L) (p ∈ {0, 1, 2, 3, · · · }∪{∞}), where a 0 (L) is the known continuous analytic capacity a 0 (L) = a(L) ( [5]). The study of the above capacities and variants of it is done in section 3. For p = 1 the p-continuous analytic capacity a 1 is distinct from the continuous analytic capacity a 0 = a. In particular, if K 1/3 is the usual Cantor set lying on [0, 1] and L = K 1/3 × K 1/3 , then a 0 (L) > 0, but a 1 (L) = 0. This means that for any open set U containing L there exists a function in A(U \L) which is not holomorhic in U , but if the derivative of a function in A(U \ L) extends continuously on L, then the function is holomorphic in U . Generic versions of this fact imply that A 1 (U \ L) is of first category in A 0 (U \ L) = A(U \ L). If we replace the spaces A p with theÃ p spaces of Whitney type, then the extension on L is equivalent to the fact that the interior of L is void. Thus we can define the continuous analytic capacitiesã p (L) but they vanish if and only if the interior of L is empty. We prove what is needed in section 7. More detailed study of those capacities will be done in future papers; for instance we can investigate the semiadditivity of a p , whether the vanishing of a p on a compact set L is a local phenomenon and whether replacing the continuous analytic capacity a by the Ahlfors analytic capacity γ we can define capacities γ p satisfying the analogous properties. Certainly the spaces A p (Ω) will be replaced by H ∞ p (Ω), the space of holomorphic functions on Ω such that for every l ∈ N, l ≤ p the derivative f (l) of order l is bounded on Ω. We will also examine if a dichotomy result as in section 7 holds for the spaces H ∞ p (Ω) in the place of A p (Ω). All these in future papers. In section 2 some preliminary geometry of locally injective curves is presented; for instance a curve is real analytic if and only if real analyticity of a function on the curve is equivalent to holomorphic extendability of the function on discs centred on points of the curve. We also show that if the map γ defining the curve is a homeomorphism with non-vanishing derivative, then the spaces C k (γ), k ∈ {0, 1, 2, 3, · · · } ∪ {∞} are independent of the particular parametrization γ and depend only on the image γ * of γ. Thus, in some cases it makes sense to write C k (∂Ω) and prove generic results in these spaces, k ∈ {0, 1, 2, 3, · · · } ∪ {∞}. Finally, we mention that some of the results of section 5 are valid for analytic curves γ; that is, they hold when we use a conformal parametrization of γ. Naturally comes the question whether these results remain true if we change the parametrization of the curve; in particular what happens if we consider the parametrization with respect to the arc length s? Answering this question was the motivation of [9], [10] where it is proven that arc length is a global conformal parameter for any analytic curve. Thus, the results of section 5 remain true if we use the arc length parametrization. Finally, we mention that in the present paper we start with qualitative categorical results, which lead us to quantitative notions as the p-continuous analytic capacity a p and the p-analytic capacity γ p .

Preliminaries
In most of our results it is important what is the degree of smoothness of a curve and the relation of real analyticity of functions on a curve with the holomorphic extendability of them around the curve. That is why we present here some basic results concerning locally injective curves in C.
In this way C k (γ), k = 0, 1, 2, ... becomes a Banach space if I is compact , C k (γ), k = 0, 1, 2, ... a Frechet space if I is not compact and C ∞ (γ) a Frechet space. Therefore Baire's theorem is at our disposal. Definition 2.3. Let γ : I → C be a continuous and locally injective function, where I is an interval. We will say that the curve γ is analytic at t 0 ∈ I if there exist an open set t 0 ∈ V ⊆ C, a real number δ > 0 with (t 0 − δ, t 0 + δ) ∩ I ⊂ V and a holomorphic and injective function F : V → C such that F | (t 0 −δ,t 0 +δ) = γ| (t 0 −δ,t 0 +δ) . If γ is analytic at every t ∈ I, we will say that γ is an analytic curve.
Lemma 2.4. Let t 0 ∈ I and γ : I → C be a continuous and locally injective function, where I is an interval.
For every function f : I → C we suppose that 1) and 2) are equivalent: 1) There exists a power series of real variable ∞ n=0 a n (t − t 0 ) n , a n ∈ C with a positive radius of convergence r > 0 and there exists 0 < δ ≤ r such that with a positive radius of convergence s > 0 and 0 < ≤ s such that Then γ is analytic at t 0 .
The power series ∞ n=0 b n (z −γ(t 0 )) n has a positive radius of convergence and so there exists α > 0 such that γ(t) ∈ D(γ(t 0 ), α) for every t ∈ (t 0 − , t 0 + ) ∩ I and the function f : for every t ∈ (t 0 − η, t 0 + η) ∩ I and h is holomorphic and injective and the proof is complete.
Remark 2.5. The above proof shows that if γ in Lemma 2.4 belongs to C 1 (I), then the conclusion of the lemma is true even if we only assume that 1) ⇒ 2) is true.
The following lemma is the inverse of Lemma 2.4.
Lemma 2.6. Let t 0 ∈ I and γ : I → C be a continuous and locally injective function, which is analytic at t 0 , where I is an interval. Let also f : I → C.
Then the followings are equivalent: 1) There exists a power series of real variable ∞ n=0 a n (t − t 0 ) n , a n ∈ C with positive radius of convergence r > 0 and there exists a 0 < δ ≤ r such that 2) There exists a power series with complex variable with positive radius of convergence s > 0 and there exists > 0 such that Proof. Because γ is an analytic curve at t 0 , there exists an open disk D(t 0 , ε) ⊆ C, where ε > 0 and a holomorphic and injective function Γ : z ∈ D(t 0 , δ), which is well defined and holomorphic in D(t 0 , δ). We have that is a holomorphic function. We consider the function is an open set) which is a holomorphic function and Therefore, there exist b n ∈ C,n = 1, 2, 3, ... and δ > 0 such that for every z ∈ D(γ(t 0 ), δ) ⊆ Γ(D(t 0 , ε)) and thus (ii) ⇒ (i) We consider the function GoΓ : D(t 0 , a) → C is holomorphic. Therefore, there exist a n ∈ C,n = 1, 2, 3, ... such that and consequently Definition 2.7. Let γ : I → C be a locally injective curve and z 0 = γ(t 0 ), t 0 ∈ I, where I is an interval. A function f : γ * → C belongs to the class of nonholomorphically extendable at (t 0 , z 0 = γ(t 0 )) functions, if there are no open disk D(z 0 , r), r > 0 and η > 0 and a holomorphic function F : Otherwise we say that f is holomorphically extendable at (t 0 , z 0 = γ(t 0 )).
The following proposition associates the phenomenon of real-analyticity and that of holomorphically extendability. Proposition 2.9. Let γ : I → C be an analytic curve at t 0 , where I is an interval and t 0 ∈ I. A function f : γ * → C is real analytic at (t 0 , z 0 = γ(t 0 )) if and only if f is holomorphically extendable at (t 0 , z 0 = γ(t 0 )).
The following theorem is a consequence of Lemma 2.4 and Proposition 2.9.
Theorem 2.10. Let γ : I → C be a continuous and locally injective curve, where I is an interval and t 0 ∈ I. Then, γ is analytic at t 0 if and only if for every function f : γ * → C the following are equivalent: 1) f is real analytic at (t 0 , z 0 = γ(t 0 )) 2) f is holomorphically extendable at (t 0 , z 0 = γ(t 0 )) Now, we will examine a different kind of differentiability. Definition 2.11. Let γ : I → C be a continuous and injective curve, where I is an interval or the unit circle. We define the derivative of a function f : if the above limit exists and is a complex number.
Remark 2.12. If γ : I → C is a hemeomorphism from I to γ * , then we can if the above limit exists and is a complex number.
In this way C k (γ * ) becomes a Banach space if k < ∞ and I is compact. Otherwise, C k (γ * ) is a Frechet space.
Proof. The function f : γ * → C with f (γ(t)) = γ(t) for t ∈ I belongs to the class C k (γ * ) and therefore the function γ = f • γ : I → C belongs to the class C k (I).
Now, we will prove the inverse of the previous proposition: If γ ∈ C k (I), then C k (I) = C k (γ * ) • γ. In order to do that we need the following lemma, which will also be useful later.
Lemma 2.15. Let X be an interval I ⊂ R or the unit circle T , γ ∈ C k (X), k ∈ {1, 2, ...} ∪ {∞} and f ∈ C k (γ * ) and g = f • γ. Then g ∈ C k (X) and there exist polynomials where the derivatives of γ are with respect to the real variable t in the case X = I and with respect to the complex variable t, |t| = 1 in the case X = T .
Therefore, the result holds also for i + 1 and the proof is complete.
Proposition 2.16. Let γ : I → C be a homeomorphism with γ (t) = 0, where I is an interval or the unit circle and k ∈ {1, 2, 3, · · · } ∪ {∞}. If Proof. By Lemma 2.15 we have that C k (I) ⊃ C k (γ * ) • γ. We will now prove by induction that for t 0 ∈ I. So, the function dg dz exists and is continuous and thus g ∈ C 1 (γ * ). If the result is true for k, we will prove that it is also true for k + 1. If for t 0 ∈ I. By the hypothesis of the induction, we have that f •γ −1 ∈ C k (γ * ).
It is also true that γ ∈ C k (I) . It follows that dg dz ∈ C k (γ * ) or equivalently g ∈ C k+1 (γ * ) and the proof is complete for k finite. The case k = ∞ follows from the previous result for all k finite.
Proof. At first, we will prove the proposition in the special case where X is a compact interval I ⊂ R or the unit circle T and k = ∞. In order to do so, we will find 0 < M, N < ∞, such that , for every f 1 , f 2 ∈ C k (γ) = C k (γ * ), where d 1 , d 2 are the metrics of C k (γ) and C k (γ * ), respectively.
Let f 1 , f 2 ∈ C k (γ * ) and In addition, Consequently, We also notice that We will prove by induction that it is easy to see that P i,i (z 1 , z 2 , ..., z i ) = z 1 i . Thus, and thus N 1 = 1 s 1 . If the result holds for every 1 ≤ j ≤ i < k, then we will prove that it also holds for i + 1 ≤ k. Using our induction hypothesis and equation (1) we find and the result holds also for i + 1. The induction is complete. Now, it is easy to see that It easily follows from the above that even in the case where X is any type of interval I ⊂ R and/or k = ∞ the respective topologies of the spaces C k (γ) and C k (γ * ) are the same. The basic open subsets of C k (γ) are defined by a compact subset of X, an l ∈ {0, 1, 2, ...}, l ≤ k, a function f ∈ C k (γ) and an ε > 0. But if we recall the definition of the topology of C k (γ * ) and use the above, we realize that this basic open subset of C k (γ) is also an open subset of C k (γ * ). Similarly, every basic open subset of C k (γ * ) is an open open subset of C k (γ). The proof is complete.
Combining Propositions 2.14, 2.16 and 2.17, we obtain the following theorem.
In addition the spaces C k (γ) and C k (γ * ) share the same topology.
Remark 2.19. One can prove a slightly stronger statement than the one in Theorem 2.18. We do not need to assume γ (t) = 0. Then C k (I) = C k (γ * )•γ if and only if γ ∈ C k (I) and γ (t) = 0 for all t ∈ I. Remark 2.20. With the definition of the derivative as in the Remark 2.12 we can define the spaces C k (E) for more general sets E ⊂ C but it may occur that the space C k (E) is not complete.
Theorem 2.18 shows that if γ is a homeomorphism and γ (t) = 0 for t ∈ I, then C k (γ) ≈ C k (γ * ). Therefore, we have the the following corollary: where I is an interval or the unit circle and k ∈ {1, 2, 3, · · · } ∪ {∞}. Then, the space C k (γ) is independent of the parametrization of γ and coincides with the space C k (γ * ).

Continuous analytic capacities
Next we present a few facts for the notion of continuous analytic capacity ( [5]) that we will need in sections 5 and 7 below. Section 7 leads us to generalise this notion and thus consider the p-continuous analytic capacity.  By Tietze's extension theorem the extensions in both previous definitions can be considered as extensions on the whole of C ∪ {∞} without increase of the original norm f ∞ .
the continuous analytic capacity of L.
We consider an arbitrary f ∈ A(Ω ∪ {∞}). Since f can be continuously extended over L, it belongs to C(U ) ∩ H(U \ L) and thus to H(U ). Therefore f is analytic in C and continuous in C ∪ {∞} and hence it is constant. Thus a(L) = 0. Now assume a(L) = 0 and we consider any f ∈ C(U ) ∩ H(U \ L). There exist two closed curves γ 1 and γ 2 in U so that γ 1 surrounds L and γ 2 surrounds γ 1 . We define the analytic functions Then the function g which equals φ 2 − f in the interior of γ 2 and φ 1 in the exterior of γ 1 is well defined and belongs to A(Ω ∪ {∞}). Therefore g is constant and thus f is analytic in the interior of γ 2 . Hence f ∈ H(U ).
Due to the local nature of the proof of the next theorem we shall state a few facts about the so-called Vitushkin's localization operator ( [4]).
Let U ⊂ C be open and f ∈ C(C) ∩ H(U ). Let also g ∈ C 1 (C) have compact support. We define the function The function G is continuous in Proof. One direction is immediate from Theorem 3.4 and Definition 3.5 and hence we assume that a(L) = 0. We consider an arbitrary open set U ⊂ C which intersects L and an arbitrary f ∈ C(U ) ∩ H(U \ L) and we shall prove that f extends analytically over U ∩ L. Now L may not be contained in U but since analyticity is a local property we shall employ Vitushkin's localization operator.
We also consider the restriction F of f in D(z 0 , 3δ) and we extend F so that it is continuous in C ∪ {∞}.
We define as in (2) the function Since a(L) = 0, we have a(D(z 0 , 2δ) ∩ L) = 0 and hence G is constant 0 in C. Therefore f is analytic in D(z 0 , δ).
Since z 0 ∈ U ∩ L is arbitrary we conclude that f ∈ H(U ).
Theorem 3.7. ( [5]) If L is a Jordan arc with locally finite length, then a(L) = 0. The same holds for any countable union of such curves. Therefore, line segments, circular arcs, analytic curves and boundaries of convex sets are all of zero continuous analytic capacity.
Let Ω be the complement in C of a compact set and on Ω, where the closure of Ω is taken in C.
If L is a closed subset of C then we define a p (L) = sup{a p (M ) : M compact subset of L}.
If L is a closed subset of C then we define It is obvious that a p (L) and a p (L) are decreasing function of p.
The following theorem corresponds to Theorem 3.4. The proof is a repetition of the proof of Theorem 3.4.
This is supposed to hold uniformly for z, w in compact subsets of U . The definition of the spaceÃ p (Ω ∪ {∞}) is analogous to the Definition 3.9 of A p (Ω ∪ {∞}).
Note that, since f ∈ H(U ), relation (3) is automatically true for z ∈ U and thus the "point" of the definition is when z ∈ ∂U .
If p is finite, then f ∈Ã p (U ) admits as a norm f Ãp (U ) the smallest M such that It is easy to see thatÃ p (U ) with this norm is complete. If p is infinite, then, using the norms for the finite cases in the standard way,Ã ∞ (U ) becomes a Frechet space.
There is a fundamental result of Whitney ( [13]) saying that if f ∈Ã p (U ), then f can be extended in C in such a way that the extended f belongs to C p (C) and that the partial derivatives of f of order ≤ p in C are extensions of the original partial derivatives of f in U .
Definition 3.15. Let L be a compact subset of C and p ∈ {0, 1, 2, · · · }∪{∞}. Let also Ω = C \ L. For p = ∞, we denotẽ In the case p = ∞ the norm f Ãp (U ) is replaced by the distance of f from 0 in the metric space structure ofÃ ∞ (U ). Obviously,ã 0 (L) = a(L). If L is a closed subset of C then we definẽ It turns out that, in the case p ≥ 1, there is a simple topological characterization of the compact sets L withã p (L) = 0. Conversely, let L have empty interior and let f belong toÃ 1 (Ω ∪ {∞}). Then f is analytic in Ω and at every z ∈ L we have Thus f is analytic at z (with derivative equal to f (z)) and hence analytic in all of C. Since f is continuous at ∞, it is a constant. ThereforeÃ 1 (Ω ∪ {∞}) contains only the constant functions andã 1 (L) = 0.
Proof. Due to the last theorem, it is enough to find a compact L with empty interior and withã 0 (L) = a(L) > 0. This set L is a Cantor type set. We consider a sequence (a n ) with 0 < a n < 1 2 for all n = 1, 2, 3, . . . and construct a sequence (L n ) of decreasing compact sets as follows. L 0 is the unit square [0, 1] × [0, 1] and L 1 is the union of the four squares at the four corners of L 0 with sidelength equal to a 1 . We then continue inductively. If L n is the union of 4 n squares each of sidelength equal to l n = a 1 · · · a n , then each of these squares produces four squares at its four corners each of sidelength equal to a 1 · · · a n a n+1 . The union of these new squares is L n+1 . We denote I n,k , k = 1, . . . , 4 n , the squares whose union is L n . Finally, we define It is clear that L is a totally disconnected compact set. The area of L n equals |L n | = 4 n (a 1 · · · a n ) 2 = (2a 1 · · · 2a n ) 2 .

Now we assume that
Under this condition we find that the area of L equals Then it is well known ( [5]) that the function f is not identically equal to 0 and henceã 0 (L) = a(L) > 0.
Remark 3.18. The latter part of the above proof shows that if a compact set L (not necessarily of Cantor type) has strictly positive area, then a(L) > 0, which is a well known fact ( [5]).
The problem of the characterization of the compact sets L with a p (L) = 0 seems to be more complicated.
We will show that there exists a compact set L such that a 0 (L) and a 1 (L) are essentially different; that is a 0 (L) > 0 and a 1 (L) = 0.
Theorem 3.19. There is a compact subset L of C such that a 1 (L) = 0 < a 0 (L).
Proof. We consider the same Cantor type set L which appeared in the proof of the previous theorem. We keep the same notation.
Let z 0 ∈ Ω. Then there is n 0 such that z 0 / ∈ L n for all n ≥ n 0 . By Cauchy's formula, for every n ≥ n 0 we have where γ n,k is the boundary curve of the square I n,k . Let z n,k be any point of Ω inside I n,k (for example, the center of the square). It is geometrically obvious that for every z ∈ γ n,k there is a path (consisting of at most two line segments) γ with length l(γ) ≤ 2l n joining z and z n,k and contained in Ω (with the only exception of its endpoint z).
where n → 0 uniformly for z ∈ γ n,k and for k = 1, . . . , 4 n . Therefore This holds for all n ≥ n 0 and hence f (z 0 ) = 0 for all z 0 ∈ Ω. We proved that the only element f of A 1 (Ω ∪ {∞}) with f (∞) = 0 is the zero function and thus a 1 (L) = 0.
We will now see a different proof of the above theorem. The proof is longer from the previous one, but it provides us a more general result.
Proof. First, we observe that the area of L is 0, as where each L n is the union of 4 n squares of area 9 −n . It is known ( [6], [14]) that there exists a function g continuous on S 2 and holomorphic off L, such that g (∞) = lim z→∞ z(g(z) − g(∞)) = 0, which implies that a 0 (L) > 0. We will prove that a 1 (L) = 0 or equivalently that every function in where χ ε is the characteristic function of the square S ε with center at 0 and sides parallel to the axes with length ε. It is easy to see from the continuity of f that the convolutions f * ϕ ε belong to C 1 (C) and converge uniformly on on Ω. Let (a, b) ∈ Ω and h ∈ R. Then It is easy to see that for almost every (x, y) ∈ S ε and every h ∈ R the segment ] is a subset of Ω. Thus, from Mean value theorem for almost every (x, y) ∈ S ε and every h ∈ R there exists q ∈ R, such that as h converges to 0. By Dominated convergence theorem (5) holds true. Similarly, Since the Cauchy-Riemann equations are satisfied for f almost everywhere, we have that ∂u ∂x * ϕ ε = ∂v ∂y * ϕ ε and ∂u ∂y * ϕ ε = − ∂v ∂x * ϕ ε .
Thus, the Cauchy-Riemann equations are satisfied for every f * ϕ ε on Ω. Since the interior of L is void, the set Ω is dense in C. From the continuity of the partial derivatives of every f * ϕ ε on C, the Cauchy-Riemann equations are satisfied for every f * ϕ ε on C, which implies that every f * ϕ ε is holomorphic on C. Finally, the functions f * ϕ ε converge uniformly on D(0, 2) to f , as ε → 0, which combined with Weierstrass theorem implies that f is holomorphic on D(0, 2) and therefore f is holomorphic on C.
Remark 3.21. The above proof also shows that if L is a compact subset of C with zero area and if almost for every line ε which is parallel to the x-axis and almost for every line ε which is parallel to the y-axis, ε ∩ L = ∅, then a 1 (L) = 0. In fact, it suffices that these intersections are finite for a dense set of ε parallel to the x-axis and for a dense set of ε parallel to the y-axis.
Let L ⊂ C be a closed set without isolated points. We denote by C(L) the set of continuous functions f : L → C. This space endowed with the topology of uniform convergence on the compact subsets of L is a complete metric space and thus Baire's theorem is at our disposal. Then the function h(z) = f (z) + δ 2z , z ∈ L belongs to A(M, z 0 , r) and therefore is holomorphic on D(z 0 , r). But then the function δ 2z will be holomorphic on D(z 0 , r), which is absurd. Thus, the interior of A(M, z 0 , r) is void.
2)If D(z 0 , r) is not contained in L, then there exists w ∈ D(z 0 , r)\L. Let (f n ) n≥1 be a sequence in A(M, z 0 , r) where f n converges uniformly on compact subsets of L to a function f defined on L. Then, for n = 1, 2, . . . there are holomorphic functions F n : D(t 0 , r) → C bounded by M such that F n | D(z 0 ,r)∩L = f n | D(z 0 ,r)∩L . By Montel's theorem, there exists a subsequence (F kn ) of (F n ) which converges uniformly to a function F on the compact subsets of D(z 0 , r) which is holomorphic and bounded by M on D(z 0 , r).
Because F kn → f at D(z 0 , r) ∩ L we have that F | D(z 0 ,r)∩L = f | D(z 0 ,r)∩L and so f ∈ A(M, z 0 , r). Therefore, A(M, z 0 , r) is a closed subset of C (L).
If A(M, z 0 , r) has not empty interior, then there exists a function f in the interior of A(M, z 0 , r), a compact set K ⊂ L and δ > 0 such that We choose 0 < a < δ inf z∈K |z − w|. We notice that this is possible because inf z∈K |z−w| > 0, since w ∈ L and K ⊂ L. The function h(z) = f (z)+ a 2(z − w) for z ∈ L belongs to A(M, z 0 , r) and therefore it has a holomorphic and bounded extension H on D(z 0 , r), such that H| D(z 0 ,r)∩L = h| D(z 0 ,r)∩L . However, there exists a holomorphic function F : D(z 0 , r) → C which coincides with f on D(z 0 , r) ∩ L. By analytic continuation H(z) = F (z) + a 2(z − w) for z ∈ D(z 0 , r) \ {z 0 }, since they are equal on L ∩ (D(z 0 , r) \ {z 0 }), which contains infinitely many points close to z 0 , all of them being non isolated. As a result H is not bounded at D(z 0 , r) which is a contradiction. Thus, A(M, z 0 , r) has empty interior.   The proof of the above results can be used to prove similar results at some special cases. Let γ : I → C be a a continuous and locally injective curve, where I is an interval in R of any type. The symbol γ * will be used instead of γ(I). It is obvious that γ * has no isolated points. We also recall Definition 2.2 of C k (γ). Definition 4.6. Let γ : I → C be a locally injective curve and z 0 = γ(t 0 ), t 0 ∈ I, where I is an interval. A function f : γ * → C belongs to the class of nonholomorphically extendable at (t 0 , z 0 = γ(t 0 )) functions, if there are no open disk D(z 0 , r), r > 0 and η > 0 and a holomorphic function F : D(z 0 , r) → C, such that γ((t 0 − η, t 0 + η) ∩ I) ⊂ D(z 0 , r) and F (γ(t)) = f (γ(t)) for all t ∈ (t 0 − η, t 0 + η) ∩ I. Otherwise we say that f is holomorphically extendable at (t 0 , z 0 = γ(t 0 )). Theorem 4.7. Let k, l ∈ {0, 1, 2, ...}∪{∞} such that l ≤ k. Let also γ : I → C be a locally injective function belonging to C k (I), and t 0 ∈ I, where I is an interval. The class of non-holomorphically extendable at (t 0 , z 0 = γ(t 0 )) functions belonging to C l (γ) is a dense and G δ subset of C l (γ).
Theorem 4.9. Let k, l ∈ {0, 1, 2, ...} ∪ {∞} such that l ≤ k. Let also γ : I → C be a locally injective function belonging to C k (I), where I is an interval. The class of nowhere holomorphically extendable functions belonging to C l (γ) is a dense and G δ subset of C l (γ).
Proof. Let t n , n = 1, 2, 3, ... be a dense in I sequence of points of I. Then the class of nowhere holomorphically extendable functions belonging to C k (γ) coincides with the intersection over every n = 1, 2, 3, ... of the classes of non-holomorphically extendable at (t n , z n = γ(t n )) functions belonging to C l (γ). Since the classes of non-holomorphically extendable at (t n , z n = γ(t n )) functions belonging to C l (γ) are dense and G δ subsets of C l (γ) according to Theorem 4.7, it follows that the class of nowhere holomorphically extendable functions belonging to C l (γ) is a dense and G δ subset of C l (γ) from Baire's theorem.
Now,using results of non-extendability, we will prove results for real analyticity. Proposition 2.9 and Theorem 4.9 immediately prove the following theorems.
Theorem 4.10. Let γ : I → C be an analytic curve, where I is an interval and t 0 ∈ I. For k = 0, 1, 2, . . . or k = ∞ the class of functions f ∈ C k (γ) which are not real analytic at (t 0 , z 0 = γ(t 0 )) is a dense and G δ subset of C k (γ).
Theorem 4.11. Let γ : I → C be an analytic curve, where I is an interval. For k = 0, 1, 2, . . . or k = ∞ the class of functions f ∈ C k (γ) which are nowhere real analytic is a dense and G δ subset of C k (γ).  Proof. The map S : C k (γ) → C k (δ) defined by S(g) = g • Φ −1 , g ∈ C k (γ) is an isometry onto. Also a function g ∈ C k (γ) is nowhere analytic if and only if S(g) is nowhere analytic. Theorem 4.11 combined with the above facts yields the result.
Corollary 4.14. Let J ⊂ R be an interval and γ(t) = t, t ∈ J or J = R and γ(t) = e it , t ∈ R. Let X denote the image of γ, that is X = J or X is the unit circle, respectively. Let Φ : X → C be a homeomorphism of X on Φ(X) ⊂ C and δ = Φ • γ. Then the set of functions f ∈ C k (δ), k ∈ {0, 1, 2, ...} ∪ {∞} which are nowhere analytic is a G δ and dense subset of C k (δ).
Proof. The curve γ is an analytic curve defined on an interval. The result follows from Proposition 4.13.
Remark 4.15. According to Corollary 4.14 for any Jordan curve δ or Jordan arc δ with a suitable parametrization generically on C k (δ), k ∈ {0, 1, 2, ...} ∪ {∞} every function is nowhere analytic. In fact this holds for all parametrizations of δ * and the spaces C k (δ) are the same for all parametrizations so that δ is a homeomorphism between the unit circle T or [0, 1] and δ * (see preliminaries).

Extendability of functions on domains of finite connectivity
We start this section with the following general fact.
where A i,k are dense and open subsets of X i for i = 1, ..., n, k = 1, 2, ..., then k are open and dense subsets of X 1 , ..., X n . Baire's theorem completes the proof.
We can consider the above space as the class of functions f , which are defined on the disjoint union γ * 1 ∪...∪γ * n of the locally injective curves γ 1 , ..., γ n , where f | γ i belongs to C p i (γ i ).  Proof. Let A i be the class of nowhere holomorphically extendable functions belonging to C p i (γ i ) for i = 1, ..., n. Then the set A 1 × ... × A n coincides with the class of nowhere holomorphically extendable functions belonging to C p 1 ,...,pn (γ 1 , ..., γ n ). It follows from Theorem 4.9 that the sets A 1 , ..., A n are dense and G δ subsets of C p 1 (γ 1 ), ..., C pn (γ n ), respectively, which combined with Proposition 5.1 implies that the class A 1 × ... × A n is a dense and G δ subset of C p 1 ,...,pn (γ 1 , ..., γ n ).  The proof of the following theorem is similar to the proof of Theorem 5.5.
From now on, we will consider that p 1 = p 2 = · · · = p n . As we did for the spaces C p 1 ,...,pn , we will prove analogue generic results in the space where Ω is a planar domain bounded by the disjoint Jordan curves γ 1 , ..., γ n , n ∈ {1, 2, ...}. More specifically, we will define the following spaces: Remark 5.9. In particular cases it is true that A p (Ω) is included in C p (∂Ω) as a closed subset. We will not examine now under which more general sufficient conditions this remains true. Proof. Let M > 0, r > 0, z 1 ∈ D(z 0 , r) ∩ Ω and d > 0 such that D(z 1 , d) ⊂ D(z 0 , r) ∩ Ω. Let also A(p, Ω, z 0 , r, z 1 , d, M ) be the set of A p (Ω) functions f for which there exists a holomorphic function F on D(z 0 , r), such that |F (z)| ≤ M for every z ∈ D(z 0 , r) and F | D(z 1 ,d) = f | D(z 1 ,d) . We will first show that the class A(p, Ω, z 0 , r, z 1 , d, M ) is a closed subset of A p (Ω) with empty interior. Let (f n ) n≥1 be a sequence in A(p, Ω, z 0 , r, z 1 , d, M ) converging in the topology of A p (Ω) to a function f of A p (Ω). This implies that f n converges uniformly on Ω to f. Then, for n = 1, 2, . . . there are holomorphic functions F n : D(z 0 , r) → C bounded by M such that F n | D(z 1 ,d) = f n | D(z 1 ,d) . By Montel's theorem, there exists a subsequence (F kn ) of (F n ) which converges uniformly on the compact subsets of D(z 0 , r) to a function F which is holomorphic and bounded by M on D(z 0 , r).
In addition, if A(p, Ω, z 0 , r, z 1 , d, M ) has not empty interior, then there exist f ∈ A(p, Ω, z 0 , r, z 1 , d, M ), l ∈ {0, 1, 2, ...} and > 0 such that We choose w ∈ D(z 0 , r)\Ω and 0 < δ < min{inf  Proof. Let z l , l = 1, ... be a dense sequence of ∂Ω. If A(z l ) is the class of nonholomorphically extendable at z l functions of A p (Ω) in the sense of Riemann surfaces, then, from Theorem 5.12, the set A(z l ) is a dense and G δ subset of A p (Ω). The set ∞ l=1 A(z l ) coincides with the class of nowhere holomorphically extendable functions of A p (Ω) in the sense of Riemann surfaces and from Baire's theorem is a dense and G δ subset of A p (Ω).
Remark 5.15. In [8] it has been also proved that the class of nowhere holomorphically extendable functions of A ∞ (Ω) in the sense of Riemann surfaces is a dense and G δ subset of A ∞ (Ω). The method in [8] comes from the theory of Universal Taylor Series and is different from the method in the present paper. Now we will examine a different kind of extendability.
Definition 5.16. Let Ω be a bounded domain in C defined by a finite number of disjoint Jordan curves γ 1 , · · · , γ n . Let also z 0 ∈ ∂Ω. A continuous function f : Ω → C belongs to the class of non-holomorphically extendable at z 0 functions if there exist no pair of an open disk D(z 0 , r), r > 0 and a holomorphic function F : D(z 0 , r) → C such that F (z) = f (z) for every z ∈ D(z 0 , r)∩∂Ω. Otherwise we will say that f is holomorphically extendable at z 0 .
Remark 5.17. If γ i : This holds true because of the following observations: 1)For a constant η > 0 we can find r > 0 such that D(z 0 , r) ∩ γ * i ⊆ γ i ((t 0 − η, t 0 + η) ∩ I i ). This follows from the fact that the disjoint compact sets γ i [I i \ (t 0 − η, t 0 + η)] and {z 0 } have a strictly positive distance. 2)For a constant r > 0 we can find η > 0 such that γ i ((t 0 − η, t 0 + η) ∩ I i ) ⊂ D(z 0 , r), because of the continuity of the map γ i . Proof. Let M > 0, r > 0 and A(p, Ω, z 0 , r, M ) be the class of functions f ∈ A p (Ω) for which there exist a holomorphic function F : D(z 0 , r) → C such that F | D(z 0 ,r)∩∂Ω = f | D(z 0 ,r)∩∂Ω and |F (z)| ≤ M for z ∈ D(z 0 , r). We will show that this class is a closed subset of A p (Ω) with empty interior.
Similarly to the proof of Lemma 4.1 the class A(p, Ω, z 0 , r, M ) is a closed subset of A p (Ω).
If A(p, Ω, z 0 , r, M ) has not empty interior, then there exist a function f in the interior of A(p, Ω, z 0 , r, M ), a number b ∈ {0, 1, 2, ...} and δ > 0 such that We choose w ∈ D(z 0 , r)\Ω and 0 < a < δ min{inf z∈Ω |z−w|, inf z∈Ω |z−w| 2 , ..., w| b+1 }. This is possible because w ∈ Ω. Then, similarly to the proof of The-orem 4.7 we are led to a contradiction. Thus A(p, Ω, z 0 , r, M ) has empty interior. The set coincides with the class of non-holomorphically extendable at z 0 functions of A p (Ω) and Baire's theorem implies that this set is a dense and G δ subset of A p (Ω). Proof. The proof is similar to the proof of Theorem 4.9, taking into account the statement of Theorem 5.18.
Remark 5.21. If the continuous analytic capacity of the boundary of Ω is zero, then Definition 5.11 implies Definition 5.16. Indeed, let Ω be a bounded domain in C defined by a finite number of disjoint Jordan curves, such that the continuous analytic capacity of ∂Ω is zero, and let z 0 ∈ ∂Ω. Let f be a continuous function in Ω which does not belong to the class of Definition 5.16; That is there exist a pair of an open disk D(z 0 , r), r > 0 and a holomorphic function F : D(z 0 , r) → C such that F (z) = f (z) for every z ∈ D(z 0 , r) ∩ ∂Ω. We consider the function G : D(z 0 , r) → C Then G is continuous on D(z 0 , r) and holomorphic on D(z 0 , r) \ ∂Ω. But the continuous analytic capacity of ∂Ω is zero. From Theorem 3.6, G is holomorphic on D(z 0 , r) and, since the set D(z 0 , r) ∩ Ω is an open subset of C, there exist z 1 ∈ D(z 0 , r) ∩ Ω and d > 0 such that D(z 1 , d) ⊂ D(z 0 , r) ∩ Ω.
Obviously G coincides with f on D(z 1 , d) and thus f does not belong to the class of Definition 5.11. Now, as in section 4, we will associate the phenomenon of non-extendability with that of real analyticity on the spaces A p (Ω).
At this point we can observe that if Ω is a bounded domain in C defined by disjoint Jordan curves γ 1 , ..., γ n and f ∈ A p (Ω), then the analogous of Proposition 2.9 under the above assumptions holds true, since nothing essential changes in its proof. So, we have the following proposition: Remark 5.26. We recall that for an analytic Jordan curve γ defined on [0, 1] there exist 0 < r < 1 < R and a holomorphic injective function Φ : D(0, r, R) → C, such that γ(t) = Φ(e it ), where D(0, r, R) = {z ∈ C : r < |z| < R}. This yields a natural parametrization of the curve γ * ; the parameter t is called a conformal parameter for the curve γ * . Theorem 5.25 holds if each of the Jordan curves γ 1 , ..., γ n is parametrized by such a conformal parameter t. Naturally one asks if the same result holds for other parametrizations; for instance does Theorem 5.25 remains true if each γ 1 , ..., γ n is parametrized by arc length? This was the motivation of [9] and [10], where it is proved that arc length is a global conformal parameter for any analytic curve. Thus, Theorem 5.25 remains also true if arc length is used as a parametrization for each analytic curve γ i .

One sided extendability
In this section, we consider one sided extensions from a locally injective curve γ. For instance, if γ * is homeomorphic to [0, 1], one can find an open disc D and an open arc J of γ * separating D to two components D + and D − . Those are Jordan domains containing in their boundaries a subarc J of γ. We will show that generically in C k (γ) every function h cannot be extended to a function F : Ω ∪ J → C continuous on Ω ∪ J and holomorphic in Ω; where Ω = D + or Ω = D − . That is the one sided extendability is a rare phenomenon in C k (γ), provided that γ is of class at least C k . In order to prove this fact we need the following lemmas, which are well known in algebraic topology. We include their elementary proofs for the purpose of completeness. . Also, the sets V 1 , V 2 are closed in the relative topology of δ * \{δ(t s )} ⊃ W \{δ(t s )}. It follows that the sets W ∩ V 1 , W ∩ V 2 are closed in the relative topology of W \ {δ(t s )}. Consequently, the set W \ {δ(t s )} is not connected, which is absurd, since it is an open disk without one of its interior points. Thus, the interior of δ * in C is void. Proposition 6.2. Let I ⊂ R be an interval. Let also γ : I → C be a continuous and locally injective curve. Then the interior of γ * in C is void . coincides with γ * and Baire's theorem implies that the interior of γ * is void in C.
Proposition 6.2 implies the following: Corollary 6.3. Let γ : I → C be a continuous and locally injective curve on the interval I ⊂ R. Let also Ω be a Jordan domain, such that ∂Ω contains an arc of γ * , γ([t 1 , t 2 ]), t 1 < t 2 , t 1 , t 2 ∈ I. Then the set Ω \ γ * is not empty.
Let γ : I → C be a continuous and locally injective curve defined on the interval I ⊂ R. Naturally one asks if a Jordan domain as in Corollary 6.3 exists. Now, we will construct denumerably many such Jordan domains, such that every Ω, as in Corollary 6.3, contains one of these domains.
We distinguish two cases according to whether the segments [P, γ(r)], [Q, γ( r)] intersect or not. If the segments [P, γ(r)], [Q, γ( r)] intersect at a point w, then the union of the segments [w, γ(r)], [w, γ( r)] and γ[ r, r] is the image of a Jordan curve, the interior of which is one of the desired Jordan domains. If the segments [P, γ(r)], [Q, γ( r)] do not intersect, then we consider a simple polygonal line (that is without self intersections) in C \ γ([s 1 , s 2 ]), which connects P, Q, the vertices of which belong to Q + iQ. This is possible, since the set C\γ([s 1 , s 2 ]) is a domain ( [11]). We consider one G n is a bounded holomorphic function on D, we know that for every n, the radial limits of G n exist almost everywhere on the unit circle and so we can consider the respective functions g n defined almost everywhere on the unit circle which are extensions of the previous g n . These g n are also bounded by M . Let ε > 0 be a positive number. For the Poisson kernel P r , 0 ≤ r < 1 and for every n = 0, 1, 2 . . . , it holds that G n (re iθ ) = 1 2π π −π P r (t)g n (θ − t)dt.
If Ω is a Jordan domain, such that ∂Ω contains an arc of γ * , γ([t 1 , t 2 ]), t 1 < t 0 < t 2 , t 1 , t 2 ∈ I, then A(k, Ω, M ) denotes the set of functions f ∈ C k (γ) for which there exists a continuous and bounded by M function F : Ω ∪ γ((t 1 , t 2 )), which is holomorphic on Ω and F | γ((t 1 ,t 2 )) = f | γ((t 1 ,t 2 )) . Let G n , n = 1, 2, ... be the denumerably many Jordan domains constructed above, such that s 1 < t 0 < s 2 where s 1 , s 2 ∈ Q ∩ I are as in the construction of G n and t 0 is the fixed real number in the statement of Theorem 6.6 . From Proposition 6.4, the sets A(k, G n , M ) are closed subsets of C k (γ) with empty interior. We will prove that the class of non one sided holomorphically extendable at (t 0 , γ(t 0 )) functions of C k (γ) coincides with the set (C k (γ) \ A(k, G n , M )), and thus, Baire's theorem will imply that the above set is a dense and G δ subset of C k (γ).
Obviously, the set contains the class of non one sided holomorphically extendable at (t 0 , γ(t 0 )) functions of C k (γ). Reversely, let Ω be a Jordan domain whose boundary contains an arc γ([t 1 , t 2 ]), t 1 < t 0 < t 2 , t 1 , t 2 ∈ I. Let also f ∈ C k (γ), for which there exists a continuous function F : Ω ∪ γ((t 1 , t 2 )), which is holomorphic on Ω and F | γ((t 1 ,t 2 )) = f | γ((t 1 ,t 2 )) . From the construction of the above Jordan domains G n , we can find P, Q ∈ Ω ∩ (Q + iQ) and t 1 < r < t 0 < r < t 2 , such that the segments [P, γ(r)], [Q, γ( r)] intersect ∂Ω only at γ(r), γ( r), respectively. The only modifications needed in order to do that are the following: we fix t 1 < t 3 < t 0 < t 4 < t 2 , t 3 , t 4 ∈ Q and consider a number smaller than the half of the distance of γ(t 3 ) from the compact set ∂Ω \ γ(t 1 , t 0 ) and a number smaller than the half of the distance of γ(t 4 ) from the compact set ∂Ω \ γ(t 0 , t 2 ). We continue as in the construction, with the only possible difference that the rational numbers t 3 and t 4 will replace s 1 , s 2 in the construction of the G n after Corollary 6.3 and, if a simple polygonal line, which connects P, Q and with vertices in Q + iQ, is needed, we consider it also in the domain Ω. This Jordan domain is one of the denumerable Jordan domains, G n 0 , constructed above and G n 0 is contained in Ω ∪ γ([ r, r]). It easily follows that F | Gn 0 is bounded by some number M = 1, 2, 3, ... and thus f belongs to A(k, G n 0 , M ). Therefore, the class of non one sided holomorphically extendable at (t 0 , γ(t 0 )) functions of C k (γ) is a subset of the set which combined with the above completes the proof.
Proof. Let t n ∈ I • , n = 1, 2, ..., where I • is the interior of I in R, be a dense sequence in I. Then the class of nowhere one sided holomorphically extendable functions of C k (γ) coincides with the intersection of the classes of non one sided holomorphically extendable at (t n , γ(t n )) functions of C k (γ), which is from Theorem 6.6 and Baire's theorem a dense and G δ subset of C k (γ).
Proof. Similar to the proof of Theorem 5.5.
First, we will prove that the set A(p, G) is a closed subset of A p (G). Let (f n ) n≥1 be a sequence in A(p, G) converging in the topology of A p (G) to a function f ∈ A p (G). This implies that f n converges uniformly on G to f . By the maximum principle, the extensions F n of f n form a uniformly Cauchy sequence on Ω. Thus, the limit F of F n on Ω is an extension of f . Therefore, f ∈ A(p, G) and A(p, G) is a closed subset of A p (G). Now, we will prove that the set A(p, G) has empty interior in A p (G). If A(p, G) has not empty in A p (G), then there exists a function f in the interior of A(p, G) and there exist l ∈ {0, 1, ...}, l ≤ p and d > 0, such that It is easy to see that the function f 0 is not identically equal to zero. Thus, m = max{sup z∈G |f 0 (j) (z)|, j = 0, 1, ..., l} > 0. Then, the function g(z) = f (z) + d 2m f 0 (z), z ∈ G belongs to A(p, G). Since the functions f, g belong to A(p, G), there are functions F, H ∈ A p (Ω), such that F | G = f , H| G = g.
Then, the function 2m d (H(z) − F (z)) belongs to A p (Ω) and is equal to f 0 in G, which contradicts our hypothesis. Thus, the set A(p, G) has empty interior in A p (G).
For p < ∞ and Ω unbounded A p (Ω) is a Frechet space, while if Ω is bounded it is a Banach space. Definition 7.6. Let L be a compact subset of C and U be an open subset of C, such that L ⊆ U . Let also z 0 ∈ ∂L. A function f ∈ H(U \ L) is extendable at z 0 if there exists r > 0 and F ∈ H(D(z 0 , r)) such that F | (U \L)∩D(z 0 ,r) = f | (U \L)∩D(z 0 ,r) . Otherwise, we say that f is not extendable at z 0 .
Below, we will use the above definition of extendability. Proof. We will first prove that the set E M,p,U,L,z 0 ,r is a closed subset of A p (U \L). Let (f n ) n≥1 be a sequence in E M,p,U,L,z 0 ,r converging in the topology of A p (U \ L) to a function f ∈ A p (U \ L). This implies that f n converges uniformly on U \ L to f and that there exists a sequence (F n ) n≥1 in H(D(z 0 , r)) such that F n | (U \L)∩D(z 0 ,r) = f n | (U \L)∩D(z 0 ,r) and F ∞ ≤ M for every n ≥ 1. By Montel's theorem, there exists a subsequence of (F n ), (F kn ), which converges uniformly on the compact subsets of D(z 0 , r) to a function F which is holomorphic and bounded by M on D(z 0 , r). Since F kn converges to f on (U \L)∩D(z 0 , r), the functions f and F are equal on (U \L)∩D(z 0 , r). Thus, f belongs to E M,p,U,L,z 0 ,r and E M,p,U,L,z 0 ,r is a closed subset of A p (U \ L).
If there exists f 0 ∈ A p (U \ L) which is not extendable at z 0 , the interior of E M,p,U,L,z 0 ,r is void in A p (U \ L), the proof of which is similar to the proof of Theorem 7.1.
Here, we have one more dichotomy, which is a local version of the first one.  L) and Baire's Theorem shows that A p (U \L)\E p,U,L,z 0 , which coincides with the set of non extendable functions of A p (U \ L) at z 0 , is a dense and G δ subset of A p (U \ L). Now, we will compare two notions: local extendability and existence of a holomorphic extension. At first, we will examine the case of a compact set L with empty interior. Proof. If there exists a holomorphic extension F of f on U , then obviously f is extendable at every z 0 ∈ ∂L = L.
Conversely, if f is extendable at every z 0 ∈ L, then for every z 0 ∈ L there exist a positive real number r z 0 and a holomorphic function F z 0 on D(z 0 , r z 0 ) such that D(z 0 , r z 0 ) ⊆ U and F z 0 | (U \L)∩D(z 0 ,rz 0 ) = f | (U \L)∩D(z 0 ,rz 0 ) . Let z 1 , z 2 ∈ L such that V = D(z 1 , r z 1 ) ∩ D(z 2 , r z 2 ) = ∅. Since L • = ∅, V \ L is a non-empty, open set. Thus, F z 1 , F z 2 are holomorphic on the domain V and coincide with f on V \ L. By analytic continuation, F z 1 = F z 2 on V . So, the function F defined on U such that F (z) = F z (z) for every z ∈ L and F (z) = f (z) for every z ∈ U \ L is a holomorphic extension of f on U .
Obviously, if f ∈ A p (U \ L), then F ∈ A p (U ).
Remark 7.10. If L • = ∅ the equivalence at Proposition 7.9 is not true. Indeed, if w ∈ L • = ∅, then the holomorphic function f (z) = 1 z−w for z ∈ U \L can not be extended to a holomorphic function on U , but it is extendable at every z 0 ∈ ∂L.
We again consider a compact set L ⊆ C and an open set U ⊆ C, such that L ⊆ U and a p ∈ {0, 1, 2, · · · } ∪ {∞}. Now, we want to find a similar connection between a p (L) and a p (L ∩ D(z 0 , r)); that is, is the condition a p (L) = 0 equivalent to the condition a p (L ∩ D(z 0 , r)) = 0 for all z 0 ∈ L?
If we suppose that L • = ∅, then there exist z 0 and r > 0 such that D(z 0 , r) ⊆ L. Thus, a p (L) and a p (L ∩ D(z 0 , r)) are strictly positive.
So, we do not need to assume that L • = ∅, since it follows from both the conditions a p (L) = 0 and a p (L ∩ D(z 0 , r)) = 0 for every z 0 ∈ L and for some r = r z 0 > 0. Also, the first condition obviously implies the second one.
Probably Theorem 3.6 holds even for p ≥ 1. Specifically, if a p (L) = 0 and V is an open set, then every function g ∈ A p (V \ L) belongs to A p (V ). This leads us to believe that the above conditions are in fact equivalent. However, this will be examined in future papers.