The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes

This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator on spaces $\mathcal{EV}(\Omega,E)$ of $\mathcal{C}^{\infty}$-smooth vector-valued functions whose growth on strips along the real axis with holes $K$ is induced by a family of continuous weights $\mathcal{V}$. Vector-valued means that these functions have values in a locally convex Hausdorff space $E$ over $\mathbb{C}$. We characterise the weights $\mathcal{V}$ which give a counterpart of the Grothendieck-K\"othe-Silva duality $\mathcal{O}(\mathbb{C}\setminus K)/\mathcal{O}(\mathbb{C})\cong\mathscr{A}(K)$ with non-empty compact $K\subset\mathbb{R}$ for weighted holomorphic functions. We use this duality to prove that the kernel $\operatorname{ker}\overline{\partial}$ of the Cauchy-Riemann operator $\overline{\partial}$ in $\mathcal{EV}(\Omega):=\mathcal{EV}(\Omega,\mathbb{C})$ has the property $(\Omega)$ of Vogt. Then an application of the splitting theory of Vogt for Fr\'{e}chet spaces and of Bonet and Doma\'nski for (PLS)-spaces in combination with some previous results on the surjectivity of the Cauchy-Riemann operator $\overline{\partial}\colon\mathcal{EV}(\Omega)\to\mathcal{EV}(\Omega)$ yields the surjectivity of the Cauchy-Riemann operator on $\mathcal{EV}(\Omega,E)$ if $E:=F_{b}'$ with some Fr\'{e}chet space $F$ satisfying the condition $(DN)$ or if $E$ is an ultrabornological (PLS)-space having the property $(PA)$. This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on $\mathcal{EV}(\Omega)$.


Introduction
The smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator ∂ ∶= (1 2)(∂ 1 + i∂ 2 ) on the space C ∞ (Ω) of smooth complex-valued functions on an open set Ω ⊂ R 2 is whether for every family (f λ ) λ∈U in C ∞ (Ω) depending smoothly (holomorphically, distributionally) on a parameter λ in an open set U ⊂ R d there is a family (u λ ) λ∈U in C ∞ (Ω) with the same kind of parameter dependence such that Here, smooth (holomorphic, distributional) parameter dependence of (f λ ) λ∈U means that the map λ ↦ f λ (x) is an element of C ∞ (U ) (of the space of holomorphic functions O(U ) on U ⊂ C open, the space of distributions D(V ) ′ for open V ⊂ R d where U = D(V )) for each x ∈ Ω.
The parameter dependence problem for a variety of partial differential operators on several spaces of (generalised) differentiable functions has been extensively studied, see e.g. [4,7,8,38,39,19] and the references and background in [3,27]. The answer to this problem for the Cauchy-Riemann operator is affirmative since the Cauchy-Riemann operator on the space C ∞ (Ω, E) of E-valued smooth functions is surjective if E = C ∞ (U ) (O(U ), D(V ) ′ ) by [9, Corollary 3.9, p. 1112] which is a consequence of the splitting theory of Bonet and Domański for PLS-spaces [3,4], the topological isomorphy of C ∞ (Ω, E) to Schwartz' ε-product C ∞ (Ω)εE and the fact that ∂∶ C ∞ (Ω) → C ∞ (Ω) is surjective on the nuclear Fréchet space C ∞ (Ω) (with its usual topology). More generally, the Cauchy-Riemann operator (1) is surjective if E is a Fréchet space by Grothendieck's classical theory of tensor products [14] or if E ∶= F ′ b where F is a Fréchet space satisfying the condition (DN ) by [38,Theorem 2.6,p. 174] or if E is an ultrabornological PLS-space having the property (P A) by [9, Corollary 3.9, p. 1112] since ker ∂ has the property (Ω) by [38,Proposition 2.5 (b), p. 173]. The first and the last result cover the case that E = C ∞ (U ) or O(U ) whereas the last covers the case E = D(V ) ′ as well. More examples of the second or third kind of such spaces E are arbitrary Fréchet-Schwartz spaces, the space S(R d ) ′ of tempered distributions, the space D(V ) ′ of distributions, the space D (w) (V ) ′ of ultradistributions of Beurling type and some more (see [4], [9, Corollary 4.8, p. 1116] and [27,Example 3,p. 7]).
In this paper we consider the Cauchy-Riemann operator on spaces EV(Ω, E) of weighted smooth E-valued functions where E is a locally convex Hausdorff space over C with a system of seminorms (p α ) α∈A generating its topology. These spaces consist of functions f ∈ C ∞ (Ω, E) fulfilling additional growth conditions induced by a family V ∶= (ν n ) n∈N of continuous functions ν n ∶ Ω → (0, ∞) on a sequence of open sets (Ω n ) n∈N with Ω = ⋃ n∈N Ω n given by the constraint f n,m,α ∶= sup x∈Ωn β∈N 2 0 , β ≤m for every n ∈ N, m ∈ N 0 and α ∈ A. The aim is to derive sufficient conditions on V and (Ω n ) n∈N such that where F is a Fréchet space satisfying the condition (DN ) or if E is an ultrabornological PLS-space having the property (P A).
In [28,24] this was done in the case that E is a Fréchet space using conditions on V and (Ω n ) n∈N which guarantee that EV(Ω) is a nuclear Fréchet space, EV(Ω, E) is topological isomorphic to EV(Ω)εE for complete E and ∂∶ EV(Ω) → EV(Ω) is surjective. By proving that ker ∂ has property (Ω) under some additional assumptions on V this was extended in [27] to E ∶= F ′ b where F is a Fréchet space satisfying the condition (DN ) or ultrabornological PLS-spaces E with (P A) in the case that the Ω n are strips along the real axis, i.e. Ω n ∶= {z ∈ C Im(z) < n} for n ∈ N (see [27,Corollary 17,p. 21]). In particular, these conditions are satisfied if ν n (z) ∶= exp(a n Re(z) γ ), z ∈ C, for some 0 < γ ≤ 1 and a n ↗ 0 by [27,Corollary 18,p. 21]. In the present paper we consider the case that the Ω n are strips along the real axis with holes around non-empty compact sets K ⊂ [−∞, ∞] and we are confronted with the task of deriving sufficient conditions on V such that ker ∂ has (Ω). The corresponding spaces EV(Ω, E) and their subspaces of holomorphic functions are of interest because they are the basic spaces for the theory of vector-valued Fourier hyperfunctions, see e.g. [15,16,18,20,22,29,30].
Let us summarise the content of our paper. In Section 2 we recall necessary definitions and preliminaries which are needed in the subsequent sections. Section 3 is dedicated to a counterpart for weighted holomorphic functions of the Silva-Köthe-Grothendieck duality where K ⊂ R is a non-empty compact set and A (K) the space of germs of real analytic functions on K (see Theorem 3.10, Corollary 3.11, Corollary 3.13). In Section 4 we use this duality to characterise the weights V such that the kernel ker ∂ satisfies property (Ω) in the case that (Ω n ) n∈N is a sequence of strips along the real axis with holes around a non-empty compact set K ⊂ [−∞, ∞] (see Theorem 4.3,Corollary 4.4). The preceding conditions on V are used in Section 5 to obtain the surjectivity of the Cauchy-Riemann operator on EV(Ω, E) in the case that (Ω n ) n∈N is a sequence of strips along the real axis with holes around K for E ∶= F ′ b where F is a Fréchet space satisfying the condition (DN ) or an ultrabornological PLS-space E having the property (P A) (see Theorem 5.1). Especially, these conditions hold if ν n (z) ∶= exp(a n Re(z) γ ), z ∈ C, for some 0 < γ ≤ 1 and a n ↗ 0 (see Corollary 5.2).

Notation and Preliminaries
The notation and preliminaries are essentially the same as in [25,28,27,Section 2]. We denote by R ∶= R ∪ {±∞} the two-point compactifaction of R and set C ∶= R + iR. We define the distance of two subsets Moreover, we denote by B r (x) ∶= {w ∈ R 2 w − x < r} the Euclidean ball around x ∈ R 2 with radius r > 0 and identify R 2 and C as (normed) vector spaces. We denote the complement of a subset M ⊂ R 2 by M C ∶= R 2 ∖ M , the closure of M by M and the boundary of M by ∂M . For a function f ∶ M → C and K ⊂ M we denote by f K the restriction of f to K and by the sup-norm on K. By C(Ω) we denote the space of continuous C-valued functions on a set Ω ⊂ R 2 and by L 1 (Ω) the space of (equivalence classes of) C-valued Lebesgue integrable functions on a measurable set Ω ⊂ R 2 . By E we always denote a non-trivial locally convex Hausdorff space over the field C equipped with a directed fundamental system of seminorms (p α ) α∈A . If E = C, then we set (p α ) α∈A ∶= { ⋅ }. We recall that for a disk D ⊂ E, i.e. a bounded, absolutely convex set, the vector space E D ∶= ⋃ n∈N nD becomes a normed space if it is equipped with gauge functional of D as a norm (see [17, p. 151]). The space E is called locally complete if E D is a Banach space for every closed disk D ⊂ E (see [17,10.2.1 Proposition,p. 197]). Further, we denote by L(F, E) the space of continuous linear maps from a locally convex Hausdorff space F to E and sometimes use the notation ⟨T, f ⟩ ∶= T (f ), f ∈ F , for T ∈ L(F, E). If E = C, we write F ′ ∶= L(F, C) for the dual space of F . If F and E are (linearly topologically) isomorphic, we write F ≅ E. We denote by L b (F, E) the space L(F, E) equipped with the locally convex topology of uniform convergence on the bounded subsets of F .
We recall the following well-known definitions concerning continuous partial differentiability of vector-valued functions (c.f. [23, p. 237 for the nth unit vector e n ∈ R 2 the limit For k ∈ N a function f is said to be k-times continuously partially differentiable (f is C k ) if f is C 1 and all its first partial derivatives are Due to the vector-valued version of Schwarz' theorem (∂ β ) E f is independent of the order of the partial derivatives on the right-hand side, we call β ∶= β 1 +β 2 the order of differentiation and write exists in E for every z 0 ∈ Ω. As before we define derivatives of higher order recursively, i.e. for n ∈ N 0 we set and Ω = ⋃ n∈N Ω n . Let V ∶= (ν n ) n∈N be a countable family of positive continuous functions ν n ∶ Ω → (0, ∞) such that ν n ≤ ν n+1 for all n ∈ N. We call V a directed family of continuous weights on Ω and set for n ∈ N a) The subscript α in the notation of the seminorms is omitted in the C-valued case.

Duality
We recall the well-known topological Silva-Köthe-Grothendieck isomorphy where E is a quasi-complete locally convex Hausdorff space, ∅ ≠ K ⊂ R is compact, O(C ∖ K, E) is equipped with the topology of uniform convergence on compact subsets of C ∖ K, the quotient space with the induced quotient topology and A (K) is the space of germs of real analytic functions on K with its inductive limit topology (see e.g. [ where the closure and the complement are taken in C. The definition of S 1 (K) is motivated by 3.1. Definition. Let K ⊂ R be a compact set, V ∶= (ν n ) n∈N a directed family of continuous weights on C and E a locally convex Hausdorff space. Using Definition 2.1, we set with Ω n ∶= S n (K) for all n ∈ N and f K,n,m,α ∶= f n,m,α , f ∈ EV(C ∖ K, E), and f K,n,α ∶= f n,α , f ∈ OV(C ∖ K, E), for n ∈ N, m ∈ N 0 and α ∈ A. We omit the index K in f K,n,α,m and f K,n,α if no confusion seems to be likely.
The spaces OV(C ∖ K, E) play the counterpart of O(C ∖ K, E) for our version of the isomorphy (2). Next, we introduce some conditions which guarantee the existence of a counterpart of A (K) for our purpose.
Condition. Let K ⊂ R be a non-empty compact set and V ∶= (ν n ) n∈N a directed family of continuous weights on C. For every n ∈ N let (qV ∞ ) there be I 1 (n) > n such that for every ε > 0 there is a compact set Q ⊂ Condition (qV ∞ ) means that the quotient νn ν I 1 (n) vanishes at infinity whereas (qL 1 ) means that the quotient νn be open and f ∈ O(Ω). For z ∈ Ω and n ∈ N 0 we denote the point evaluation of the nth complex derivative at z by δ 3.2. Proposition. Let K ⊂ R be a non-empty compact set and V ∶= (ν n ) n∈N a directed family of continuous weights on C. For n ∈ N let f (z) ν n (z) −1 and the spectral maps for n, k ∈ N, n ≤ k, be given by the restrictions π n,k ∶ Oν −1 n (U n (K)) → Oν −1 k (U k (K)), π n,k (f ) ∶= f U k (K) . If V fulfils (qV ∞ ), then a) the inductive limit exists and is a DFS-space. b) the span of the set of point evaluations of complex derivatives {δ (n) x0 if K has no isolated points. Proof. a)(i) First, we prove that the normed space Oν −1 n (U n (K)) is a Banach space. Let (f k ) k∈N be a Cauchy sequence in Oν −1 n (U n (K)). Let ε > 0 and M ⊂ U n (K) be compact. Then there exists N ∈ N such that for all k, m ≥ N Thus (f k ) k∈N is also a Cauchy sequence in the Fréchet space O(U n (K))∩C(U n (K)) equipped with the topology induced by the system of seminorms ⋅ M with compact M ⊂ U n (K). Therefore it converges to f ∈ O(U n (K)) ∩ C(U n (K)). Since every Cauchy sequence is bounded, there exists C(n) ≥ 0 with f k (z) ν n (z) −1 ≤ C(n) for all z ∈ U n (K) and k ∈ N, implying f ∈ Oν −1 n (U n (K)) by pointwise convergence. Using the pointwise convergence again, we get for all z ∈ U n (K) and k ≥ N , n ≤ m, are injective by virtue of the identity theorem and the definition of sets U n (K). Thus the considered spectrum is an embedding spectrum. ( Thus B n is bounded in O(U n (K)) w.r.t. the system of seminorms generated by ⋅ M for compact M ⊂ U n (K). As this space is a Fréchet-Montel space, B n is relatively compact and hence relatively sequentially compact in O(U n (K)).
(iv) What remains to be shown is that for all n ∈ N there exists m > n such that π n,m is a compact map. Because the considered spaces are Banach spaces, it suffices to prove the existence of m > n such that (π n,m (f k )) k∈N has a convergent subsequence in Oν −1 m (U m (K)) for every sequence (f k ) k∈N in B n . According to (qV ∞ ), we choose m ∶= I 1 (n) > n. Let ε > 0. Then there is a compact set Q ⊂ U m (K) In addition, we set and therefore Hence the subsequence (π n,m (f k l )) l∈N converges in Oν −1 m (U m (K)), proving the compactness of π n,m .
It follows from (i) − (iv) and [31,Proposition 25.20,p. 304] that the inductive limit x0 is linear and for k ∈ N and f ∈ Oν −1 k (U k (K)) we derive from Cauchy's inequality that We consider the polar set of F , i.e.
is valid for any n ∈ N 0 . Thus f is identical to zero on a neighbourhood of x 0 (by Taylor series expansion) since f is holomorphic near x 0 ∈ U n (K). Due to the assumptions every component of U n (K) contains a point x 0 ∈ K ∩ R so f is identical to zero on U n (K) by the identity theorem and continuity, yielding to y = 0.
. Due to the assumptions every component Z of U n (K) contains a point x 0 ∈ K ∩ R and every point in Z ∩K ∩R is an accumulation point of Z ∩K ∩R. So f is identical to zero on U n (K) by the identity theorem.
The parts (iii)-(iv) of the proof of Proposition 3.2 a) are just slight modifications of [2, Theorem (b), p. 67-68] which cannot be directly applied due to the closure U n (K) being involved. In the case ν n (z) −1 ∶= exp((1 n) Re(z) ), z ∈ C, for all n ∈ N the spaces OV −1 ind (K) play an essential in the theory of Fourier hyperfunctions and it is already mentioned in [20, p. 469] resp. proved in [18,1.11 Satz,p. 11] and [22, 3.5 Theorem, p. 17] that they are DFS-spaces.
a) The set U t (K) has finitely many components. b) Let K ≠ ∅ and Z be a component of U t (K). We define a ∶= min(Z ∩ K) If Z ∩ R is bounded from below and unbounded from above and a exists, there exists 0 < R ≤ 1 t such that for all 0 < r ≤ R: is bounded from below and unbounded from above and a does not exist, then Z = (t, ∞) + i(−1 t, 1 t). If Z ∩ R is bounded from above and unbounded from below and b does not exist, then Proof. a) We only consider the case Since all Z j are pairwise disjoint, this implies that J has to be finite. The others cases follow analogously. b)(i) Since Z ∩ K is closed in R and therefore compact, a and b exist. Hence then a exists and analogously to (i) there exists (v) This follows directly from the definition of U t (K) and as Z is a component of U t (K).

3.5.
Definition. Let n ∈ N, K ⊂ R be a non-empty compact set and (Z j ) j∈J denote the components of U n (K). A component Z j of U n (K) fulfils one of the cases of Remark 3.4 b) and so for a = a j , b = b j (in the cases (i)-(iii)), for 0 < r j < R j = R (in the cases (i)-(iv)) resp. 0 < r j < 1 n =∶ R j (in the case (v)) we define where Z j fulfils (v) in the last two cases. By Remark 3.4 a) there is w.l.o.g. k ∈ N with U n (K) = ⋃ k j=1 Z j . We set r ∶= (r j ) 1≤j≤k and the path where γ j is the path along the boundary of V rj (Z j ) in C in the positive sense (counterclockwise). 3.6. Proposition. Let K ⊂ R be a non-empty compact set and V ∶= (ν n ) n∈N a directed family of continuous weights on C which fulfils (qV ∞ ) and (qL 1 ). Let n ∈ N, γ K,n,r be the path from Definition 3.5 and E a locally convex Hausdorff space. If then a) F ⋅ ϕ is Pettis-integrable along γ K,n,r for all F ∈ OV(C ∖ K, E) and ϕ ∈ Oν −1 n (U n (K)). b) there are m ∈ N and C > 0 such that for all α ∈ A, F ∈ OV(C ∖ K, E) and ϕ ∈ Oν −1 n (U n (K)) which gives ∫ γ K,n,r F (ζ)ϕ(ζ)dζ = e K,n,r . First, let V rj (Z j ) be bounded for some 1 ≤ j ≤ k. Then there is a parametrisation γ j ∶ [0, 1] → C which has a continuously differentiable extensionγ j on (−1, 2). As is well-defined and linear. We estimate where we used [31, Proposition 22.14, p. 256] in the first and second to last equation to get from p α to sup e ′ ∈B ○ α and back. If K ⊂ R, then all V rj (Z j ), 1 ≤ j ≤ k, are bounded and we deduce our statement with e K,n,r ∶= ∑ k j=1 e j , m ∶= max 1≤j≤k m j and Second, let us consider the case Next, we prove that (e k,q ) q>1 r k is a Cauchy sequence in E. We choose M ∶= max(m k , I 2 (n)) with I 2 (n) from condition (qL 1 ). For q, p ∈ N, q > p > 1 r k > n, we obtain dt F M,α ϕ n and observe that ( ∫ q 0 νn(t) ν I 2 (n) (t) dt) q is a Cauchy sequence in C by condition (qL 1 ). Therefore (e k,q ) q>1 r k is a Cauchy sequence in E, has a limit e k in the sequentially complete space E and We fix p ∈ N, p > 1 r k > n, and conclude that Consequently, our statement holds also in the case ∞ ∈ K, −∞ ∈ K and in the remaining cases it follows analogously. c) We note that for all e ′ ∈ E ′ . Thus statement c) follows from Cauchy's integral theorem and the Hahn-Banach theorem if K ⊂ R. Now, let us consider the case ∞ ∈ K, −∞ ∈ K. We denote by γ k resp.γ k the part of γ K,n,r resp. γ K,n,r in the unbounded component of U n (K). It suffices to show that We choose I 1 (n) from condition (qV ∞ ). Let ε > 0 and w.l.o.g. r k <r k . Then there is a compact set We choose q ∈ R such that q > 1 r k and q ∈ U I1(n) (K) ∖ Q and define the path γ + 0,q ∶ [r k ,r k ] → C, γ + 0,q (t) ∶= q + it. We deduce that for m k ∈ N, (1 m k ) < min(r k , 1 I 1 (n)), and every e ′ ∈ E ′ where we used condition (qL 1 ) for the second equality. In the same way we obtain with Hence we get (5) by Cauchy's integral theorem and the Hahn-Banach theorem as well. The remaining cases follow similarly. d) The proof is similar to c). Let F ∈ OV(C, E). Again, it suffices to prove that This follows from Cauchy's integral theorem and the Hahn-Banach theorem if K ⊂ R. Again, we only consider the case ∞ ∈ K, −∞ ∈ K and only need to show that where γ k is the part of γ K,n,r in the unbounded component of U n (K). Let ε > 0 and choose q as in c). Then we have with for every e ′ ∈ E ′ by (qV ∞ ) and (qL 1 ). Cauchy's integral theorem and the Hahn-Banach theorem imply our statement.
An essential role in the proof of O(C ∖ K, E) O(C, E) ≅ L b (A (K), E) for nonempty compact K ⊂ R and quasi-complete E (see (2)) plays the fundamental solution z ↦ 1 (πz) of the Cauchy-Riemann operator. By the identity theorem we can consider OV(C, E) as a subspace of OV(C ∖ K, E) and we equip the quotient space OV(C ∖ K, E) OV(C, E) with the induced locally convex quotient topology (which may not be Hausdorff, see Remark 3.12). We want to prove the isomorphy , E) for non-empty compact K ⊂ R under some assumptions on K, the weights V and the space E. Since we have to deal with functions having some growth given by the weights V, we have to use a fundamental solution z ↦ g(z) (πz), where g is an entire function with g(0) = 1, of the Cauchy-Riemann operator which is suitable for our growth conditions. Condition (CT). Let K ⊂ R be a non-empty compact set and V ∶= (ν n ) n∈N a directed family of continuous weights on C. Let there be g K ∈ O(C), g K (0) = 1, such that with G K (z) ∶= g K (z − ⋅), z ∈ C ∖ K, it holds that 1) for all z 0 ∈ C ∖ K there is n ∈ N such that G K (B 1 n (z 0 )) ⊂ Oν −1 n (U n (K)) and 2) for all n ∈ N there is J 2 (n) > n such that 3) for all n ∈ N there is J 3 (n) > n such that 5) for all n ∈ N and z ∈ C ∖ K sup ζ∈Sn(K) (CT ) stands for Cauchy transformation which is the name of the inverse of the isomorphism we are searching for.

3.7.
Remark. a) Since g K is an entire function, the estimates in the conditions (CT.2) and (CT. We will see that the conditions (qV ∞ ), (qL 1 ) and (CT ) hold with g K (z) ∶= exp(−z 2 ), z ∈ C, for ν n (z) ∶= exp(a n Re(z) γ ), z ∈ C, where 0 < γ ≤ 1 and (a n ) n∈N is increasing without change of sign.
3.8. Proposition. Let K ⊂ R be a non-empty compact set and V ∶= (ν n ) n∈N a directed family of continuous weights on C which fulfils (qV ∞ ) and (qL 1 ), γ K,n,r the path from Definition 3.5 and E a locally convex Hausdorff space. If (i) K ⊂ R and E is locally complete, or (ii) E is sequentially complete, then the map E) and ϕ ∈ Oν −1 n (U n (K)), n ∈ N, is well-defined, linear and continuous. For all non-empty compact sets K 1 ⊂ K it holds that on OV −1 ind (K). Proof. In the following we omit the index K in H K if no confusion seems to be likely. Let f = [F ] ∈ OV(C ∖ K, E) OV(C E) and ϕ ∈ OV −1 ind (K). Then there is n ∈ N such that ϕ ∈ Oν −1 n (U n (K)). Due to Proposition 3.6 a) and d) H(f )(ϕ) ∈ E and H(f ) is independent of the representative F of f . From Proposition 3.6 c) follows that H(f ) is well-defined on OV −1 ind (K), i.e. for all k ∈ N, k ≥ n, and ϕ ∈ Oν −1 n (U n (K)) it holds that H(f )(ϕ) = H(f )(ϕ U k (K) ) = H(f )(π n,k (ϕ)).
For every n ∈ N there are m ∈ N and C > 0 such that for all f = [F ] ∈ OV(C ∖ K, E) OV(C E), ϕ ∈ Oν −1 n (U n (K)) and α ∈ A p α (H(f )(ϕ)) ≤ C F m,α ϕ n by Proposition 3.6 b), which implies that H(f ) ∈ L(Oν −1 n (U n (K)), E) for every n ∈ N. We deduce that H(f ) ∈ L(OV −1 ind (K), E) by [10, 3.6 Satz, p. 117]. Let denote the quotient map. We equip the quotient space with its usual quotient topology generated by the system of quotient seminorms given by for l ∈ N and α ∈ A. Then the quotient space, equipped with these seminorms, is a locally convex space (but maybe not Hausdorff). Since (7) holds for every representative F of f , we obtain for every f ∈ OV(C ∖ K, E) OV(C, E), ϕ ∈ Oν −1 n (U n (K)), n ∈ N, and α ∈ A that Now, let M ⊂ OV −1 ind (K) be a bounded set. Since the sequence (B n ) n∈N of closed unit balls B n of Oν −1 n (U n (K)) is a fundamental system of bounded sets in OV −1 ind (K) by [31, Proposition 25.19, p. 303], there exist n ∈ N and λ > 0 with M ⊂ λB n . We derive from (8) proving the continuity of H. Moreover, let K 1 ⊂ R be compact and K 1 ⊂ K. We observe that for every F ∈ OV(C ∖ K 1 , E) and ϕ ∈ Oν −1 n (U n (K)), n ∈ N, it holds that by (qV ∞ ) and (qL 1 ) using Cauchy's integral theorem and the Hahn-Banach theorem as in Proposition 3.6 c) and d). This yields to on OV −1 ind (K). Now, we take a closer look at the potential inverse of H K .
3.9. Proposition. Let K ⊂ R be a non-empty compact set and V ∶= (ν n ) n∈N a directed family of continuous weights on C which fulfils (qV ∞ ), (CT.1) and (CT.2), and E be a locally convex Hausdorff space. Then the map , is well-defined, linear and continuous.

3.10.
Theorem. Let K ⊂ R be a non-empty compact set and V ∶= (ν n ) n∈N a directed family of continuous weights on C which fulfils (qV ∞ ), (qL 1 ) and (CT ), and E a locally convex Hausdorff space. If (i) K ⊂ R and E is locally complete, or (ii) K ∩ {±∞} has no isolated points in K and E is sequentially complete, then the map Proof. As before we omit the index K of H K , Θ K and G K if it is not necessary. As a consequence of Proposition 3.8 and Proposition 3.9 the maps H and Θ are linear and continuous. First, we prove that Θ ○ H = id on OV(C ∖ K, E) OV(C, E), which implies the injectivity of H. Let p ∈ N, p ≥ 2. We choose n ∈ N such that d ⋅ (S p (K), U n (K)) > 0. We define the path Γ p ∶= Γ − − Γ + with Further, we choose m ∈ N such that 1 m < min 1≤j≤k r j < 1 n and m > p where r = (r j ) 1≤j≤k is from the path γ K,n,r in the definition of H. Due to this choice Γ ± and γ K,n,r are within S m (K).
is holomorphic on C ∖ {z} with values in E and like in Proposition 3.6 a) and b) we deduce that it is Pettis-integrable along γ u and Γ ± [s0,s1] with [s 0 , s 1 ] ⊂ R using [5, Proposition 2, p. 354] and the Mackey-Arens theorem. Then we have by (CT.5) for all e ′ ∈ E ′ . Hence we derive from Cauchy's integral formula that for all e ′ ∈ E ′ and z ∈ S p (K). Thus we have for all z ∈ S p (K). By (the proof) of Proposition 3.9 the functiong(z, for all z ∈ S p (K). But the right-hand side W p of (9), as a function in z, is weakly holomorphic on S p (∅) = {z ∈ C Im(z) < p}, which follows from and differentiation under the integral sign. The weak holomorphy and the local completeness of E imply that W p is holomorphic on S p (∅) by [12,Corollary 2,p. 404]. Thus W is extended by W p to a function in O(C, E) and the extensions for each p ∈ N coincide because of the identity theorem. We denote this extension by W as well.
For l ∈ N we choose p ∶= J 4 (l) ≥ 2 and m ∶=J 4 (p) > p from condition (CT.4). Then we have for z = x + iy ∈ S 1 l ⊂ S p (∅) yielding to W ∅,l,α = sup z∈S l (∅) p α (W (z))ν l (z) ≤ max W K,l,α , sup Hence W ∈ OV(C, E) and thus E), which implies the surjectivity of H. Due to the Hahn-Banach theorem this is equivalent to the condition that all T ∈ L(OV −1 ind (K), E), ϕ ∈ Oν −1 n (U n (K)), n ∈ N, and e ′ ∈ E ′ , it suffices to show the result for E = C.
As the span of the set of point evaluations of complex derivatives {δ (n) x0 for all ϕ ∈ Oν −1 k (U k (K)), k ∈ N. Let us take a closer look at the integral on the right-hand side of (10). Let m ∈ N, m ≥ 2. Theng(z, ⋅) = g K (z−⋅) z−⋅ ∈ O(B 1 m (x 0 )) for every z ∈ S m ({x 0 }). We fix the notationg z (ζ) ∶=g(z, ζ) for z ∈ S m ({x 0 }) and ζ ∈ B 1 m (x 0 ). We set l ∶= J 3 (m) > m with J 3 (m) from condition (CT.3). Then we get by Cauchy's inequality z (x 0 ) ≤ n!l n max x0 ,g(z, ⋅)⟩) ∈ OV(C ∖ {x 0 }). This means that the path of the integral on the right-hand side of (10) can be deformed using Cauchy's integral theorem in combination with condition (qV ∞ ) (like in Proposition 3.6 a) and b)) and we get with s ∶= min j r j > 0 for r = (r j ) 1 2πi for all ϕ ∈ Oν −1 k (U k (K)). Since g ∈ O(C), g(0) = 1, there is a sequence (a j ) j∈N0 in C such that a 0 = g(0) = 1 and Thus the Laurent series ofg(z, with an entire function h(⋅, x 0 ). By Cauchy's integral theorem and Cauchy's integral formula for derivatives we have for all ϕ ∈ Oν −1 k (U k (K)), k ∈ N. If K ∩ {±∞} has isolated points in K, e.g. K = {+∞}, then we cannot apply the preceding theorem directly since a counterpart for Proposition 3.2 b) is missing. However, we can make use of the relation (6) if OV −1 ind (R) is dense in OV −1 ind (K). 3.11. Corollary. Let K ⊂ R be a non-empty compact set and V ∶= (ν n ) n∈N a directed family of continuous weights on C which fulfils (qV ∞ ) and (qL 1 ) for K and R as well as (CT.1) and (CT.2) for K, and (CT ) for R with g K = g R . If E is a sequentially complete locally convex Hausdorff space and OV −1 Proof. H K and Θ K are well-defined, linear and continuous maps by Proposition 3.8 and Proposition 3.9. H R is a topological isomorphism with inverse Θ R by Theorem 3.10 (ii). The embedding of OV −1 ind (R) into OV −1 ind (K) is continuous and dense, hence defines the embedding of L(OV −1 ind (K), E) into L(OV −1 ind (R), E) (the density of the first embedding implies the injectivity of the latter one) and we have since g R = g K . Furthermore, it follows from (6) that by Theorem 3.10. Thus H K is bijective and Θ K its inverse. 3.13. Corollary. Let E be a locally convex Hausdorff space, K ⊂ R a non-empty compact set, (a n ) n∈N strictly increasing, a n < 0 for all n ∈ N or a n ≥ 0 for all n ∈ N and V ∶= (exp(a n µ)) n∈N where for some 0 < γ ≤ 1. If (i) K ⊂ R and E is locally complete, or (ii) K ∩ {±∞} has no isolated points in K and E is sequentially complete, or (iii) K is arbitrary, a n < 0 for all n ∈ N, lim n→∞ a n = 0, γ = 1 and E sequentially complete, then the map Proof. We only need to prove that the conditions of Theorem 3.10 in (i)-(ii) resp. Corollary 3.11 in (iii) are fulfilled. For (iii), i.e. K ⊂ R is a non-empty compact set, a n < 0 for all n ∈ N, lim n→∞ a n = 0 and γ = 1, we remark that (qV ∞ ): The choices I 1 (n) ∶= 2n and guarantee that this condition is fulfilled.

(Ω) for OV-spaces on strips with holes
In this section we derive sufficient conditions such that OV(C ∖ K) satisfies (Ω) for a non-empty compact set K ⊂ R. The basic idea is to prove that, under suitable conditions, the strong dual OV −1 ind (K) ′ b satisifies (Ω), then we use the duality OV(C ∖ K) OV(C) ≅ OV −1 ind (K) ′ b from the preceding section to obtain (Ω) for OV(C ∖ K) if OV(C) satisfies (Ω). Let us recall that a Fréchet space F with an increasing fundamental system of seminorms ( ⋅ k ) k∈N satisfies (Ω) by [31,Chap. 29,Definition,p. 367 We start with the following helpful observation concerning the inductive limit OV −1 ind (K), namely, that the choice of the sequence (1 n) n∈N for the neighbourhoods U n (K) = U 1 (1 n) (K) is irrelevant. 4.1. Remark. Let K ⊂ R be a non-empty compact set, V ∶= (ν n ) n∈N a directed family of continuous weights on C, (c n ) n∈N a strictly decreasing sequence in R with c n ≤ 1 for all n ∈ N and lim n→∞ c n = 0. For n ∈ N let and the spectral maps for n, k ∈ N, n ≤ k, be given by the restrictions Oν −1 n (U 1 cn (K)).
We recall an equivalent description of the property (Ω). By [31,Lemma 29.13,p. 369] a Fréchet space F with an increasing fundamental system of seminorms ( ⋅ k ) k∈N satisfies (Ω) if and only if holds where y * k ∶= sup{ y(x) x k ≤ 1} ∈ R ∪ {∞} is the dual norm. We introduce the following condition which we need for an application of Hadamard's Three Circles Theorem.
Condition (H3CT). Let K ⊂ R be a non-empty compact set with K ∩ {±∞} ≠ ∅ and V ∶= (ν n ) n∈N a directed family of continuous weights on C. Let there be a strictly decreasing sequence (c n ) n∈N in R with c n ≤ 1 for all n ∈ N and lim n→∞ c n = 0 such that We note that 0 < θ < 1 and state the following improvement of [22, 5.21 Lemma, p. 88].
Proof. a) Let p, q, k ∈ N, p < q < k, and f ∈ Oν −1 p (U 1 cp (K)). Considering the components of U 1 cp (K) we have to distinguish three different cases.
What remains to be shown is that because then we are done with C ∶= exp( a q (c p + c q ) γ ). If a n < 0, then θ = ln(c p c q ) ln(c p c k ) = (1 a p ) − (1 a q ) (1 a p ) − (1 a k ) = a k (a q − a p ) a q (a k − a p ) and (16) is equivalent to 0 ≥ a 2 k (a q − a p ) a q (a k − a p ) which holds if and only if 0 ≤ a 2 k (a q − a p ) + a q (a k − a p ) − a k (a q − a p ) a p − a 2 q (a k − a p ) = a 2 k (a q − a p ) + (a k − a q )a 2 p − a 2 q (a k − a q + a q − a p ) = (a 2 k − a 2 q )(a q − a p ) + (a k − a q )(a 2 p − a 2 q ) = (a k − a q )(a k + a q )(a q − a p ) − (a k − a q )(a q − a p )(a p + a q ) as a q (a k − a p ) < 0. Since a k − a q > 0 and a q − a p > 0, this is equivalent to 0 ≤ (a k + a q ) − (a p + a q ) = a k − a p , which is true. If a n ≥ 0, then θ = ln(c p c q ) ln(c p c k ) = a q − a p a k − a p and (16) is equivalent to which holds, as a k − a p > 0, if and only if 0 ≥ (a q − a p )a k + a k − a p − (a q − a p ) a p − (a k − a p )a q = a q a k − a p a k + a k a p − a q a p − a k a q + a p a q = 0.

Surjectivity of the Cauchy-Riemann operator
In our last section we prove our main result on the surjectivity of the Cauchy-Riemann operator on EV(C ∖ K, E) for non-empty compact K ⊂ R. This is done by using the results obtained so far and splitting theory. We recall that a Fréchet space (F, ( ⋅ k ) k∈N ) satisfies (DN ) by [31,Chap. 29, Definition, p. 359] if ∃ p ∈ N ∀ k ∈ N ∃ n ∈ N, C > 0 ∀ x ∈ F ∶ x 2 k ≤ C x p x n . A PLS-space is a projective limit X = lim