Parameter dependence of solutions of the Cauchy–Riemann equation on weighted spaces of smooth functions

Abstract

Let $\varOmega$ be an open subset of $\mathbb {R}^{2}$ and E a complete complex locally convex Hausdorff space. The purpose of this paper is to find conditions on certain weighted Fréchet spaces $\mathcal {EV}(\varOmega )$ of smooth functions and on the space E to ensure that the vector-valued Cauchy–Riemann operator ${\overline{\partial }}:\mathcal {EV}(\varOmega ,E)\rightarrow \mathcal {EV}(\varOmega ,E)$ is surjective. This is done via splitting theory and positive results can be interpreted as parameter dependence of solutions of the Cauchy–Riemann operator.

1 Introduction

Let E be a linear space of functions on a set U and $P(\partial ):{\mathcal {F}}(\varOmega )\rightarrow {\mathcal {F}}(\varOmega )$ be a linear partial differential operator with constant coefficients which acts continuously on a locally convex Hausdorff space of (generalized) differentiable scalar-valued functions ${\mathcal {F}}(\varOmega )$ on an open set $\varOmega \subset \mathbb {R}^{n}$. We call the elements of U parameters and say that a family $(f_{\lambda })_{\lambda \in U}$ in ${\mathcal {F}}(\varOmega )$ depends on a parameter w.r.t. E if the map $\lambda \mapsto f_{\lambda }(x)$ is an element of E for every $x\in \varOmega$. The question of parameter dependence is whether for every family $(f_{\lambda })_{\lambda \in U}$ in ${\mathcal {F}}(\varOmega )$ depending on a parameter w.r.t. E there is a family $(u_{\lambda })_{\lambda \in U}$ in ${\mathcal {F}}(\varOmega )$ with the same kind of parameter dependence which solves the partial differential equation

$$\begin{aligned} P(\partial )u_{\lambda }=f_{\lambda },\quad \lambda \in U. \end{aligned}$$

In particular, it is the question of ${\mathcal {C}}^{k}$-smooth (holomorphic, distributional, etc.) parameter dependence if E is the space ${\mathcal {C}}^{k}(U)$ of k-times continuously partially differentiable functions on an open set $U\subset \mathbb {R}^{d}$ (the space ${\mathcal {O}}(U)$ of holomorphic functions on an open set $U\subset {\mathbb {C}}$, the space of distributions ${\mathcal {D}}(V)'$ on an open set $V\subset \mathbb {R}^{d}$ where $U={\mathcal {D}}(V)$, etc.).

The question of parameter dependence has been subject of extensive research varying in the choice of the spaces E, ${\mathcal {F}}(\varOmega )$ and the properties of the partial differential operator $P(\partial )$, e.g. being (hypo)elliptic, parabolic or hyperbolic. Even partial differential differential operators $P_{\lambda }(\partial )$ where the coefficients also depend ${\mathcal {C}}^{k}([0,1])$-smoothly [49], ${\mathcal {C}}^{\infty }$-smoothly [61], holomorphically [50, 61] or differentiable resp. real analytic [13] on the parameter $\lambda$ were considered. The case that the coefficients of the partial differential differential operator $P(x,\partial )$ are non-constant functions in $x\in \varOmega$ was treated for ${\mathcal {F}}(\varOmega )={\mathscr {A}}(\mathbb {R}^{n})$, the space of real analytic functions on $\mathbb {R}^{n}$, as well [3].

The answer to the question of ${\mathcal {C}}^{k}$-smooth (holomorphic, distributional, etc.) parameter dependence is obviously affirmative if $P(\partial )$ has a linear continuous right inverse. The problem to determine those $P(\partial )$ which have such a right inverse was posed by Schwartz in the early 1950s (see [21, p. 680]). In the case that ${\mathcal {F}}(\varOmega )$ is the space of ${\mathcal {C}}^{\infty }$-smooth functions or distributions on an open set $\varOmega \subset \mathbb {R}^{n}$ the problem was solved in [51, 52] and in the case of ultradifferentiable functions or ultradistributions in [53] by means of Phragmén-Lindelöf type conditions. The case that ${\mathcal {F}}(\varOmega )$ is a weighted space of ${\mathcal {C}}^{\infty }$-smooth functions on $\varOmega =\mathbb {R}^{n}$ or its dual was handled in [40], even for some $P(x,\partial )$ with smooth coefficients, the case of tempered distributions in [38] and of Fourier (ultra-)hyperfunctions in [44, 45]. For Hörmander’s spaces $B_{p,\kappa }^{loc}(\varOmega )$ as ${\mathcal {F}}(\varOmega )$ the problem was studied in [25].

The necessary condition of surjectivity of the partial differential operator $P(\partial )$ was studied in many papers, e.g. in [1, 23, 28, 48, 67] on ${\mathcal {C}}^{\infty }$-smooth functions and distributions, in [9, 26, 43] on real analytic functions, in [8, 14] on Gevrey classes, in [10, 12, 41, 42, 55] on ultradifferentiable functions of Roumieu type, in [22] on ultradistributions of Beurling type, in [7, 11] on ultradifferentiable functions and ultradistributions and in [47] on the multiplier space ${\mathcal {O}}_{M}$.

However, if $P(\partial ):{\mathcal {C}}^{\infty }(\varOmega )\rightarrow {\mathcal {C}}^{\infty }(\varOmega )$, $\varOmega \subset \mathbb {R}^{n}$ open, is elliptic, then $P(\partial )$ has a linear right inverse (by means of a Hamel basis of ${\mathcal {C}}^{\infty }(\varOmega )$) and it has a continuous right inverse due to Michael’s selection theorem [56, Theorem 3.2”, p. 367] and [29, Satz 9.28, p. 217], but $P(\partial )$ has no linear continuous right inverse if $n\ge 2$ by a result of Grothendieck [62, Theorem C.1, p. 109]. Nevertheless, the question of parameter dependence w.r.t. E has a positive answer for several locally convex Hausdorff spaces E due to tensor product techniques. In this case the question of parameter dependence obviously has a positive answer if the topology of E is stronger than the topology of pointwise convergence on U and

$$\begin{aligned} P(\partial )^{E}:{\mathcal {C}}^{\infty }(\varOmega ,E) \rightarrow {\mathcal {C}}^{\infty }(\varOmega ,E) \end{aligned}$$

is surjective where ${\mathcal {C}}^{\infty }(\varOmega ,E)$ is the space of ${\mathcal {C}}^{\infty }$-smooth E-valued functions on $\varOmega$ and $P(\partial )^{E}$ the version of $P(\partial )$ for E-valued functions. From Grothendieck’s classical theory of tensor products [24] and the surjectivity of $P(\partial )$ it follows that $P(\partial )^{E}$ is also surjective if E is a Fréchet space. In general, Grothendieck’s theory of tensor products can be applied if $P(\partial )$ is surjective and ${\mathcal {F}}(\varOmega )$ a nuclear Fréchet space. However, the surjectivity of $P(\partial )^{E}$, $n\ge 2$, can even be extended beyond the class of Fréchet spaces E due to the splitting theory of Vogt for Fréchet spaces [64, 65] and of Bonet and Domański for PLS-spaces [4, 6] if, in addition, ${\text {ker}}P(\partial )$ has the property $(\varOmega )$ and E is the dual of a Fréchet space with the property (DN) or an ultrabornological PLS-space with the property (PA). The splitting theory of Bonet and Domański can also be applied if ${\mathcal {F}}(\varOmega )$ is a non-Fréchet PLS-space and for PLH-spaces ${\mathcal {F}}(\varOmega )$, e.g. ${\mathcal {D}}_{L^{2}}$ and $B_{2,\kappa }^{loc}(\varOmega )$ which are non-PLS-spaces, the splitting theory of Dierolf and Sieg [15, 16] is available. For applications we refer the reader to the already mentioned papers [4, 6, 15, 16, 64, 65] as well as [5, 18] where ${\mathcal {F}}(\varOmega )$ is the space of ultradistributions of Beurling type or of ultradifferentiable functions of Roumieu type and E, amongst others, the space of real analytic functions and to [30] where ${\mathcal {F}}(\varOmega )$ is the space of ${\mathcal {C}}^{\infty }$-smooth functions or distributions.

Notably, the preceding results imply that the inhomogeneous Cauchy–Riemann equation with a right-hand side $f\in {\mathcal {E}}(\varOmega ,E):={\mathcal {C}}^{\infty }(\varOmega ,E)$, where $\varOmega \subset \mathbb {R}^{2}$ is open and E a locally convex Hausdorff space over ${\mathbb {C}}$ whose topology is induced by a system of seminorms $(p_{\alpha })_{\alpha \in {\mathfrak {A}}}$, given by

$$\begin{aligned} {\overline{\partial }}^{E}u:=(1/2)(\partial _{1}^{E}+i\partial _{2}^{E})u=f \end{aligned}$$

(1)

has a solution $u\in {\mathcal {E}}(\varOmega ,E)$ if E is a Fréchet space or $E:=F_{b}'$ where F is a Fréchet space satisfying the condition (DN) or if E is an ultrabornological PLS-space having the property (PA). Among these spaces E are several spaces of distributions like ${\mathcal {D}}(V)'$, the space of tempered distributions, the space of ultradistributions of Beurling type etc. In the present paper we study this problem under the constraint that the right-hand side f fulfils additional growth conditions given by an increasing family of positive continuous functions ${\mathcal {V}}:=(\nu _{n})_{n\in \mathbb {N}}$ on an increasing sequence of open subsets $(\varOmega _{n})_{n\in \mathbb {N}}$ of $\varOmega$ with $\varOmega =\bigcup _{n\in \mathbb {N}}\varOmega _{n}$, namely,

$$\begin{aligned} |f|_{n,m,\alpha }:=\sup _{\begin{array}{c} x\in \varOmega _{n}\\ \beta \in \mathbb {N}^{2}_{0},\,|\beta |\le m \end{array}} p_{\alpha }\bigl ((\partial ^{\beta })^{E}f(x)\bigr )\nu _{n}(x)<\infty \end{aligned}$$

for every $n\in \mathbb {N}$, $m\in \mathbb {N}_{0}$ and $\alpha \in {\mathfrak {A}}$. Let us call the space of such functions $\mathcal {EV}(\varOmega ,E)$. Our interest is in conditions on ${\mathcal {V}}$ and $(\varOmega _{n})_{n\in \mathbb {N}}$ such that there is a solution $u\in \mathcal {EV}(\varOmega ,E)$ of (1), i.e. we search for conditions that guarantee the surjectivity of

$$\begin{aligned} {\overline{\partial }}^{E}:\mathcal {EV}(\varOmega ,E)\rightarrow \mathcal {EV}(\varOmega ,E). \end{aligned}$$

Using Grothendieck’s theory of tensor products, this was already done in [33] in the case that E is a Fréchet space. In the present paper we want to extend this result beyond the class of Fréchet spaces E. Concerning the sequence $(\varOmega _{n})_{n\in \mathbb {N}}$, we concentrate on the case that it is a sequence of strips along the real axis, i.e. $\varOmega _{n}:=\{z\in {\mathbb {C}}\;|\;|{{\,\mathrm{Im}\,}}(z)|<n\}$. The case that this sequence has holes along the real axis is treated in [35].

Let us briefly outline the content of our paper. In Sect. 2 we summarise the necessary definitions and preliminaries which are needed in the subsequent sections. In Sect. 3 we recall the definitions of the topological invariants $(\varOmega )$, (DN) and (PA) and provide some examples of spaces E having these invariants. Then we prove our main result on the surjectivity of Cauchy–Riemann operator on $\mathcal {EV}(\varOmega ,E)$ which depends on these invariants (see Theorem 5). To apply our main result, the kernel ${\text {ker}}{\overline{\partial }}$ needs to have $(\varOmega )$ and in Sect. 4 we provide sufficient conditions on the weights and the sequence $(\varOmega _{n})_{n\in \mathbb {N}}$ which guarantee $(\varOmega )$ (see Theorem 10 and Corollary 13). We close this section with a special case of our main theorem where $(\varOmega _{n})_{n\in \mathbb {N}}$ is a sequence of strips along the real axis (see Corollary 17) and for example $\nu _{n}(z):=\exp (a_{n}|{{\,\mathrm{Re}\,}}(z)|^{\gamma })$ for some $0<\gamma \le 1$ and $a_{n}\nearrow 0$ (see Corollary 18).

2 Notation and preliminaries

The notation and preliminaries are essentially the same as in [33, 36, Sect. 2]. We define the distance of two subsets $M_{0}, M_{1} \subset \mathbb {R}^{2}$ w.r.t. a norm $\Vert \cdot \Vert$ on $\mathbb {R}^{2}$ via

$$\begin{aligned} \mathrm {d}^{\Vert \cdot \Vert }(M_{0},M_{1}) :={\left\{ \begin{array}{ll} \inf _{x\in M_{0},\,y\in M_{1}}\Vert x-y\Vert , &{} M_{0},\,M_{1} \ne \emptyset , \\ \infty , &{} M_{0}= \emptyset \;\text {or}\; M_{1}=\emptyset . \end{array}\right. } \end{aligned}$$

Moreover, we denote by $\Vert \cdot \Vert _{\infty }$ the sup-norm, by $|\cdot |$ the Euclidean norm on $\mathbb {R}^{2}$, by $\mathbb {B}_{r}(x):=\{w\in \mathbb {R}^{2}\;|\;|w-x|<r\}$ the Euclidean ball around $x\in \mathbb {R}^{2}$ with radius $r>0$ and identify $\mathbb {R}^{2}$ and ${\mathbb {C}}$ as (normed) vector spaces. We denote the complement of a subset $M\subset \mathbb {R}^{2}$ by $M^{C}:= \mathbb {R}^{2}{\setminus } M$, the closure of M by ${\overline{M}}$ and the boundary of M by $\partial M$. For a function $f:M\rightarrow {\mathbb {C}}$ and $K\subset M$ we denote by $f_{\mid K}$ the restriction of f to K and by

$$\begin{aligned} \Vert f\Vert _{K}:=\sup _{x\in K}|f(x)| \end{aligned}$$

the sup-norm on K. By $L^{1}(\varOmega )$ we denote the space of (equivalence classes of) ${\mathbb {C}}$-valued Lebesgue integrable functions on a measurable set $\varOmega \subset \mathbb {R}^{2}$ and by $L^{q}(\varOmega )$, $q\in \mathbb {N}$, the space of functions f such that $f^{q}\in L^{1}(\varOmega )$. If $(a_{n})_{n\in \mathbb {N}}$ is a strictly increasing real sequence, we write $a_{n}\nearrow 0$ resp. $a_{n}\nearrow \infty$ if $a_{n}<0$ for all $n\in \mathbb {N}$ and $\lim _{n\rightarrow \infty }a_{n}=0$ resp. $a_{n}\ge 0$ for all $n\in \mathbb {N}$ and $\lim _{n\rightarrow \infty }a_{n}=\infty$.

By E we always denote a non-trivial locally convex Hausdorff space over the field ${\mathbb {C}}$ equipped with a directed fundamental system of seminorms $(p_{\alpha })_{\alpha \in {\mathfrak {A}}}$. If $E={\mathbb {C}}$, then we set $(p_{\alpha })_{\alpha \in {\mathfrak {A}}}:=\{|\cdot |\}$. Further, we denote by L(F, E) the space of continuous linear maps from a locally convex Hausdorff space F to E and sometimes write $\langle T,f\rangle :=T(f)$, $f\in F$, for $T\in L(F,E)$. If $E={\mathbb {C}}$, we write $F':=L(F,{\mathbb {C}})$ for the dual space of F. If F and E are (linearly topologically) isomorphic, we write $F\cong E$. We denote by $L_{t}(F,E)$ the space L(F, E) equipped with the locally convex topology of uniform convergence on the finite subsets of F if $t=\sigma$, on the precompact subsets of F if $t=\gamma$, on the absolutely convex, compact subsets of F if $t=\kappa$ and on the bounded subsets of F if $t=b$.

The so-called $\varepsilon$-product of Schwartz is defined by

$$\begin{aligned} F\varepsilon E:=L_{e}(F_{\kappa }',E) \end{aligned}$$

(2)

where $L(F_{\kappa }',E)$ is equipped with the topology of uniform convergence on equicontinuous subsets of $F'$. This definition of the $\varepsilon$-product coincides with the original one by Schwartz [59, Chap. I, Sect. 1, Définition, p. 18].

We recall the following well-known definitions concerning continuous partial differentiability of vector-valued functions (c.f. [34, p. 237]). A function $f:\varOmega \rightarrow E$ on an open set $\varOmega \subset \mathbb {R}^{2}$ to E is called continuously partially differentiable (f is ${\mathcal {C}}^{1}$) if for the n-th unit vector $e_{n}\in \mathbb {R}^{2}$ the limit

$$\begin{aligned} (\partial ^{e_{n}})^{E}f(x):=(\partial _{n})^{E}f(x) :=\lim _{\begin{array}{c} h\rightarrow 0\\ h\in \mathbb {R}, h\ne 0 \end{array}}\frac{f(x+he_{n})-f(x)}{h} \end{aligned}$$

exists in E for every $x\in \varOmega$ and $(\partial ^{e_{n}})^{E}f$ is continuous on $\varOmega$ ($(\partial ^{e_{n}})^{E}f$ is ${\mathcal {C}}^{0}$) for every $n\in \{1,2\}$. For $k\in \mathbb {N}$ a function f is said to be k-times continuously partially differentiable (f is ${\mathcal {C}}^{k}$) if f is ${\mathcal {C}}^{1}$ and all its first partial derivatives are ${\mathcal {C}}^{k-1}$. A function f is called infinitely continuously partially differentiable (f is ${\mathcal {C}}^{\infty }$) if f is ${\mathcal {C}}^{k}$ for every $k\in \mathbb {N}$. The linear space of all functions $f:\varOmega \rightarrow E$ which are ${\mathcal {C}}^{\infty }$ is denoted by ${\mathcal {C}}^{\infty }(\varOmega ,E)$. Let $f\in {\mathcal {C}}^{\infty }(\varOmega ,E)$. For $\beta =(\beta _{n})\in \mathbb {N}_{0}^{2}$ we set $(\partial ^{\beta _{n}})^{E}f:=f$ if $\beta _{n}=0$, and

$$\begin{aligned} (\partial ^{\beta _{n}})^{E}f :=\underbrace{(\partial ^{e_{n}})^{E}\cdots (\partial ^{e_{n}})^{E}}_{\beta _{n} \text {-times}}f \end{aligned}$$

if $\beta _{n}\ne 0$ as well as

$$\begin{aligned} (\partial ^{\beta })^{E}f :=(\partial ^{\beta _{1}})^{E}(\partial ^{\beta _{2}})^{E}f. \end{aligned}$$

Due to the vector-valued version of Schwarz’ theorem $(\partial ^{\beta })^{E}f$ is independent of the order of the partial derivatives on the right-hand side, we call $|\beta |:=\beta _{1}+\beta _{2}$ the order of differentiation and write $\partial ^{\beta }f:=(\partial ^{\beta })^{{\mathbb {C}}}f$.

A function $f:\varOmega \rightarrow E$ on an open set $\varOmega \subset \mathbb {C}$ to E is called holomorphic if the limit

$$\begin{aligned} \biggl (\frac{\partial }{\partial z}\biggr )^{E}f(z_{0}) :=\lim _{\begin{array}{c} h\rightarrow 0\\ h\in \mathbb {C}, h\ne 0 \end{array}}\frac{f(z_{0}+h)-f(z_{0})}{h} \end{aligned}$$

exists in E for every $z_{0}\in \varOmega$ and the space of such functions is denoted by ${\mathcal {O}}(\varOmega ,E)$. The exact definition of the spaces from the introduction is as follows.

Definition 1

[34, Definition 3.2, p. 238] Let $\varOmega \subset \mathbb {R}^{2}$ be open and $(\varOmega _{n})_{n\in \mathbb {N}}$ a family of non-empty open sets such that $\varOmega _{n}\subset \varOmega _{n+1}$ and $\varOmega =\bigcup _{n\in \mathbb {N}} \varOmega _{n}$. Let ${\mathcal {V}}:=(\nu _{n})_{n\in \mathbb {N}}$ be a countable family of positive continuous functions $\nu _{n}:\varOmega \rightarrow (0,\infty )$ such that $\nu _{n}\le \nu _{n+1}$ for all $n\in \mathbb {N}$. We call ${\mathcal {V}}$ a directed family of continuous weights on $\varOmega$ and set for $n\in \mathbb {N}$

• (a)
  $$\begin{aligned} {\mathcal {E}}\nu _{n}(\varOmega _{n}, E):= \left\{ f \in {\mathcal {C}}^{\infty }(\varOmega _{n}, E)\; | \; \forall \;\alpha \in {\mathfrak {A}},\,m \in \mathbb {N}_{0}^{2}:\; |f|_{n,m,\alpha } < \infty \right\} \end{aligned}$$

  and

  $$\begin{aligned} \mathcal {EV}(\varOmega , E):= \left\{ f\in {\mathcal {C}}^{\infty }(\varOmega , E)\; | \;\forall \; n \in \mathbb {N}: \; f_{\mid \varOmega _{n}}\in {\mathcal {E}}\nu _{n}(\varOmega _{n}, E)\right\} \end{aligned}$$

  where

  $$\begin{aligned} |f|_{n,m,\alpha }:=\sup _{\begin{array}{c} x \in \varOmega _{n}\\ \beta \in \mathbb {N}_{0}^{2}, \, |\beta | \le m \end{array}} p_{\alpha }\bigl ((\partial ^{\beta })^{E}f(x)\bigr )\nu _{n}(x). \end{aligned}$$
• (b)
  $$\begin{aligned} {\mathcal {E}}\nu _{n,{\overline{\partial }}}(\varOmega _{n},E):= \left\{ f \in {\mathcal {E}}\nu _{n}(\varOmega _{n}, E)\; | \; f\in {\text {ker}}{\overline{\partial }}^{E} \right\} \end{aligned}$$

  and

  $$\begin{aligned} \mathcal {EV}_{{\overline{\partial }}}(\varOmega , E):=\{f\in \mathcal {EV}(\varOmega , E)\;|\; f\in {\text {ker}}{\overline{\partial }}^{E}\}. \end{aligned}$$
• (c)
  $$\begin{aligned} {\mathcal {O}}\nu _{n}(\varOmega _{n},E):= \left\{ f \in {\mathcal {O}}(\varOmega _{n}, E)\; | \; \forall \;\alpha \in {\mathfrak {A}}:\;|f|_{n,\alpha } < \infty \right\} \end{aligned}$$

  and

  $$\begin{aligned} \mathcal {OV}(\varOmega , E):=\{ f\in {\mathcal {O}}(\varOmega , E)\; | \;\forall \; n \in \mathbb {N}: \; f_{\mid \varOmega _{n}}\in {\mathcal {O}}\nu _{n}(\varOmega _{n}, E)\} \end{aligned}$$

  where

  $$\begin{aligned} |f|_{n,\alpha }:=\sup _{\begin{array}{c} x \in \varOmega _{n} \end{array}} p_{\alpha }(f(x))\nu _{n}(x). \end{aligned}$$

The subscript $\alpha$ in the notation of the seminorms is omitted in the ${\mathbb {C}}$-valued case. The letter E is omitted in the case $E={\mathbb {C}}$ as well, e.g. we write ${\mathcal {E}}\nu _{n}(\varOmega _{n}):={\mathcal {E}}\nu _{n}(\varOmega _{n},{\mathbb {C}})$ and $\mathcal {EV}(\varOmega ):=\mathcal {EV}(\varOmega ,{\mathbb {C}})$ .

A projective limit F of a sequence of locally convex Hausdorff spaces $(F_{n})_{n\in \mathbb {N}}$ is called weakly reduced if for every $n\in \mathbb {N}$ there is $m\in \mathbb {N}$ such that $\pi _{n}(F)$ is dense in $F_{m}$ w.r.t. the topology of $F_{n}$ where $\pi _{n}:F\rightarrow F_{n}$ is the canonical projection. The spaces $\mathcal {FV}(\varOmega ,E)$, ${\mathcal {F}}={\mathcal {E}}$, ${\mathcal {O}}$, are projective limits, namely, we have

$$\begin{aligned} \mathcal {FV}(\varOmega , E)\cong \lim _{\begin{array}{c} \longleftarrow \\ n\in \mathbb {N} \end{array}}{\mathcal {F}}\nu _{n}(\varOmega _{n}, E) \end{aligned}$$

where the spectral maps are given by the restrictions

$$\begin{aligned} \pi _{k,n}:{\mathcal {F}}\nu _{k}(\varOmega _{k}, E)\rightarrow {\mathcal {F}}\nu _{n}(\varOmega _{n}, E),\; f\mapsto f_{\mid \varOmega _{n}},\;k\ge n. \end{aligned}$$

3 Main result

In this section we prove our main result that the surjectivity of the vector-valued Cauchy–Riemann operator on $\mathcal {EV}(\varOmega ,E)$ is inherited from the surjectivity on $\mathcal {EV}(\varOmega )$ if the kernel $\mathcal {EV}_{{\overline{\partial }}}(\varOmega )$ in the scalar-valued case has $(\varOmega )$, and $E:=F_{b}'$ where F is a Fréchet space satisfying the condition (DN) or E is an ultrabornological PLS-space having the property (PA). Therefore we recall the definitions of the topological invariants $(\varOmega )$, (DN) and (PA) and give some examples.

A Fréchet space F with an increasing fundamental system of seminorms satisfies $(\varOmega )$ if

$$\begin{aligned} \forall \; p\in \mathbb {N}\; \exists \; q\in \mathbb {N}\;\forall \; k\in \mathbb {N}\;\exists \; n\in \mathbb {N},\,C>0\;\forall \; r>0:\; U_{q}\subset Cr^n U_k + \frac{1}{r} U_p \end{aligned}$$

(3)

where (see [54, Chap. 29, Definition, p. 367]).

A Fréchet space satisfies (DN) by [54, Chap. 29, Definition, p. 359] if

A PLS-space is a projective limit $X=\lim \limits _{\begin{array}{c} \longleftarrow \\ N\in \mathbb {N} \end{array}}X_{N}$, where the $X_{N}$ given by inductive limits are DFS-spaces (which are also called LS-spaces), and it satisfies (PA) if

where denotes the dual norm of and $i^{M}_{N}$, $i^{K}_{N}$ the linking maps (see [6, Sect. 4, Eq. (24), p. 577]).

Due to [63, 1.4 Lemma, p. 110] and [6, Proposition 4.2, p. 577] we have the following relation between the properties (DN) and (PA).

Remark 2

Let F be a Fréchet-Schwartz space. Then F satisfies (DN) if and only if the DFS-space $E:=F_{b}'$ satisfies (PA).

Let us summarise some examples of ultrabornological PLS-spaces satisfying (PA) and spaces of the form $E:=F_{b}'$ where F is a Fréchet space satisfying (DN). The majority of them is already contained in [6, 19] and [64].

Example 3

(a) The following spaces are ultrabornological PLS-spaces with property (PA) and also strong duals of a Fréchet space satisfying (DN):

• the strong dual of a power series space of inifinite type $\varLambda _{\infty }(\alpha )_{b}'$,

• the strong dual of any space of holomorphic functions ${\mathcal {O}}(U)_{b}'$ where U is a Stein manifold with the strong Liouville property (for instance, for $U={\mathbb {C}}^{d}$),

• the space of germs of holomorphic functions ${\mathcal {O}}(K)$ where K is a completely pluripolar compact subset of a Stein manifold (for instance K consists of one point),

• the space of tempered distributions ${\mathcal {S}}(\mathbb {R}^{d})_{b}'$ and the space of Fourier ultra-hyperfunctions ${\mathcal {P}}'_{**}$ (with the strong topology),

• the weighted distribution spaces $(K\{pM\})_{b}'$ of Gelfand and Shilov if the weight M satisfies

  $$\begin{aligned} \sup _{|y|\le 1}M(x+y)\le C\inf _{|y|\le 1}M(x+y),\quad x\in \mathbb {R}^{d}, \end{aligned}$$

• ${\mathcal {D}}(K)_{b}'$ for any compact set $K\subset \mathbb {R}^{d}$ with non-empty interior,

• ${\mathcal {C}}^{\infty }({\overline{U}})_{b}'$ for any non-empty open bounded set $U\subset \mathbb {R}^{d}$ with ${\mathcal {C}}^{1}$-boundary.

(b) The following spaces are ultrabornological PLS-spaces with property (PA):

• an arbitrary Fréchet-Schwartz space,

• a PLS-type power series space $\varLambda _{r,s}(\alpha ,\beta )$ whenever $s=\infty$ or $\varLambda _{r,s}(\alpha ,\beta )$ is a Fréchet space,

• the spaces of distributions ${\mathcal {D}}(U)_{b}'$ and ultradistributions of Beurling type ${\mathcal {D}}_{(\omega )}(U)_{b}'$ for any open set $U\subset \mathbb {R}^{d}$,

• the kernel of any linear partial differential operator with constant coefficients in ${\mathcal {D}}(U)_{b}'$ or in ${\mathcal {D}}_{(\omega )}(U)_{b}'$ when $U\subset \mathbb {R}^{d}$ is open and convex,

• the space $L_{b}(X,Y)$ where X has (DN), Y has $(\varOmega )$ and both are nuclear Fréchet spaces. In particular, $L_{b}(\varLambda _{\infty }(\alpha ),\varLambda _{\infty }(\beta ))$ if both spaces are nuclear.

(c) The following spaces are strong duals of a Fréchet space satisfying (DN):

• the strong dual $F_{b}'$ of any Banach space F,

• the strong dual $\lambda ^{2}(A)_{b}'$ of the Köthe space $\lambda ^{2}(A)$ with a Köthe matrix $A=(a_{j,k})_{j,k\in \mathbb {N}_{0}}$ satisfying

  $$\begin{aligned} \exists \;p\in \mathbb {N}_{0}\;\forall \;k\in \mathbb {N}_{0}\;\exists \;n\in \mathbb {N}_{0},C>0:\;a_{j,k}^{2}\le Ca_{j,p}a_{j,n}. \end{aligned}$$

Proof

The statement for the spaces in (a) and (b) follows from [19, Corollary 4.8, p. 1116], [54, Proposition 31.12, p. 401], [54, Proposition 31.16, p. 402] and Remark 2. The first part of statement (c) is obvious since Banach spaces clearly satisfy the property (DN). The second part on the Köthe space $\lambda ^{2}(A)$ follows from [29, Satz 12.11 a), p. 305]. $\square$

Since we will use the $\varepsilon$-product $\mathcal {EV}(\varOmega )\varepsilon E$ to pass the surjectivity from ${\overline{\partial }}$ to ${\overline{\partial }}^{E}$, we remark the following which is not hard to prove (see [31, Sect. 39]).

Proposition 4

• (a) Let X be a semi-reflexive locally convex Hausdorff space and Y a Fréchet space. Then $L_{b}(X_{b}',Y_{b}')\cong L_{b}(Y,(X_{b}')_{b}')$ via taking adjoints.

• (b) Let X be a Montel space and E a locally convex Hausdorff space. Then $L_{b}(X_{b}',E)\cong X\varepsilon E$ where the topological isomorphism is the identity map.

Theorem 5

Let $\mathcal {EV}(\varOmega )$ be a Schwartz space and $\mathcal {EV}_{{\overline{\partial }}}(\varOmega )$ a nuclear subspace satisfying property $(\varOmega )$. Assume that the scalar-valued operator ${\overline{\partial }}:\mathcal {EV}(\varOmega )\rightarrow \mathcal {EV}(\varOmega )$ is surjective. Moreover, if

• (a) $E:=F_{b}'$ where F is a Fréchet space over ${\mathbb {C}}$ satisfying (DN), or

• (b) E is an ultrabornological PLS-space over ${\mathbb {C}}$ satisfying (PA),

then

$$\begin{aligned} {\overline{\partial }}^{E}:\mathcal {EV}(\varOmega ,E)\rightarrow \mathcal {EV}(\varOmega ,E) \end{aligned}$$

is surjective.

Proof

Throughout this proof we use the notation $X'':=(X_{b}')_{b}'$ for a locally convex Hausdorff space X. In both cases, (a) and (b), the space E is a complete locally convex Hausdorff space. The space $\mathcal {EV}(\varOmega )$ is a Fréchet space by [34, Proposition 3.7, p. 240] and so its closed subspace $\mathcal {EV}_{{\overline{\partial }}}(\varOmega )$ as well. Further, $\mathcal {EV}(\varOmega )$ is a Schwartz space and $\mathcal {EV}_{{\overline{\partial }}}(\varOmega )$ nuclear, thus both spaces are reflexive. As the Fréchet-Schwartz space $\mathcal {EV}(\varOmega )$ is a Montel space,

$$\begin{aligned} S:\mathcal {EV}(\varOmega )\varepsilon E\rightarrow \mathcal {EV}(\varOmega ,E),\; u\longmapsto [z\mapsto u(\delta _{z})], \end{aligned}$$

is a topological isomorphism by [36, 3.21 Example b), p. 14] where $\delta _{z}$ is the point-evaluation at $z\in \varOmega$. We denote by ${\mathcal {J}}:E\rightarrow E'^{*}$ the canonical injection in the algebraic dual $E'^{*}$ of the topological dual $E'$ and for $f\in \mathcal {EV}(\varOmega ,E)$ we set

$$\begin{aligned} R_{f}^{t}:\mathcal {EV}(\varOmega )'\rightarrow E'^{\star },\; y\longmapsto \bigl [e'\mapsto y(e'\circ f) \bigr ]. \end{aligned}$$

Then the map $f\mapsto {\mathcal {J}}^{-1}\circ R_{f}^{t}$ is the inverse of S by [36, 3.17 Theorem, p. 12]. The sequence

$$\begin{aligned} 0\rightarrow \mathcal {EV}_{{\overline{\partial }}}(\varOmega )\overset{i}{\rightarrow }\mathcal {EV}(\varOmega ) \overset{{\overline{\partial }}}{\rightarrow }\mathcal {EV}(\varOmega )\rightarrow 0, \end{aligned}$$

(4)

where i means the inclusion, is a topologically exact sequence of Fréchet spaces because ${\overline{\partial }}$ is surjective by assumption. Let us denote by $J_{0}:\mathcal {EV}_{{\overline{\partial }}}(\varOmega )\rightarrow \mathcal {EV}_{{\overline{\partial }}}(\varOmega )''$ and $J_{1}:\mathcal {EV}(\varOmega )\rightarrow \mathcal {EV}(\varOmega )''$ the canonical embeddings which are topological isomorphisms since $\mathcal {EV}_{{\overline{\partial }}}(\varOmega )$ and $\mathcal {EV}(\varOmega )$ are reflexive. Then the exactness of (4) implies that

$$\begin{aligned} 0\rightarrow \mathcal {EV}_{{\overline{\partial }}}(\varOmega )''\overset{i_{0}}{\rightarrow } \mathcal {EV}(\varOmega )'' \overset{{\overline{\partial }}_{1}}{\rightarrow }\mathcal {EV}(\varOmega )''\rightarrow 0, \end{aligned}$$

(5)

where $i_{0}:=J_{0}\circ i\circ J^{-1}_{0}$ and ${\overline{\partial }}_{1}:=J_{1}\circ {\overline{\partial }}\circ J^{-1}_{1}$, is an exact topological sequence. Topological as the (strong) bidual of a Fréchet space is again a Fréchet space by [54, Corollary 25.10, p. 298].

(a) Let $E:=F_{b}'$ where F is a Fréchet space with (DN). Then ${\text {Ext}}^{1}(F,\mathcal {EV}_{{\overline{\partial }}}(\varOmega )'')=0$ by [65, 5.1 Theorem, p. 186] since $\mathcal {EV}_{{\overline{\partial }}}(\varOmega )$ satisfies $(\varOmega )$ and therefore $\mathcal {EV}_{{\overline{\partial }}}(\varOmega )''$ as well. Combined with the exactness of (5) this implies that the sequence

$$\begin{aligned} 0\rightarrow L(F,\mathcal {EV}_{{\overline{\partial }}}(\varOmega )'')\overset{i^{*}_{0}}{\rightarrow }L(F,\mathcal {EV}(\varOmega )'') \overset{{\overline{\partial }}^{*}_{1}}{\rightarrow }L(F,\mathcal {EV}(\varOmega )'')\rightarrow 0 \end{aligned}$$

is exact by [57, Proposition 2.1, p. 13-14] where $i^{*}_{0}(B):=i_{0}\circ B$ and ${\overline{\partial }}^{*}_{1}(D):={\overline{\partial }}_{1}\circ D$ for $B\in L(F,\mathcal {EV}_{{\overline{\partial }}}(\varOmega )'')$ and $D\in L(F,\mathcal {EV}(\varOmega )'')$. In particular, we obtain that

$$\begin{aligned} {\overline{\partial }}^{*}_{1}:L(F,\mathcal {EV}(\varOmega )'')\rightarrow L(F,\mathcal {EV}(\varOmega )'') \end{aligned}$$

(6)

is surjective. Via $E=F_{b}'$ and Proposition 4 ($X=\mathcal {EV}(\varOmega )$ and $Y=F$) we have the topological isomorphism

$$\begin{aligned} \psi :=S\circ {^{t}(\cdot )}:L(F,\mathcal {EV}(\varOmega )'')\rightarrow \mathcal {EV}(\varOmega ,E),\; \psi (u)=\bigl (S\circ {^{t}(\cdot )}\bigr )(u)=\bigl [z\mapsto {^{t}u}(\delta _{z})\bigr ], \end{aligned}$$

and the inverse

$$\begin{aligned} \psi ^{-1}(f) =(S\circ {^{t}(\cdot )})^{-1}(f) =({^{t}(\cdot )}\circ S^{-1})(f) ={^{t}({\mathcal {J}}^{-1}\circ R_{f}^{t})},\quad f\in \mathcal {EV}(\varOmega ,E). \end{aligned}$$

Let $g\in \mathcal {EV}(\varOmega ,E)$. Then $\psi ^{-1}(g)\in L(F,\mathcal {EV}(\varOmega )'')$ and by the surjectivity of (6) there is $u\in L(F,\mathcal {EV}(\varOmega )'')$ such that ${\overline{\partial }}^{*}_{1}u=\psi ^{-1}(g)$. So we get $\psi (u)\in \mathcal {EV}(\varOmega ,E)$. Next, we show that ${\overline{\partial }}^{E}\psi (u)=g$ is valid. Let $x\in F$, $z\in \varOmega$ and $h\in \mathbb {R}$, $h\ne 0$, and $e_{k}$ denote the kth unit vector in $\mathbb {R}^{2}$. From

$$\begin{aligned} \bigl (\frac{\delta _{z+he_{k}}-\delta _{z}}{h}\bigr )(f)=\frac{f(z+he_{k})-f(z)}{h}\underset{h\rightarrow 0}{\rightarrow }\partial ^{e_{k}}f(z), \end{aligned}$$

for every $f\in \mathcal {EV}(\varOmega )$ it follows that $\frac{\delta _{z+he_{k}}-\delta _{z}}{h}$ converges to $\delta _{z}\circ \partial ^{e_{k}}$ in $\mathcal {EV}(\varOmega )_{\sigma }'$. Since the Fréchet–Schwartz space $\mathcal {EV}(\varOmega )$ is in particular a Montel space, we deduce that $\frac{\delta _{z+he_{k}}-\delta _{z}}{h}$ converges to $\delta _{z}\circ \partial ^{e_{k}}$ in $\mathcal {EV}(\varOmega )_{\gamma }'=\mathcal {EV}(\varOmega )_{b}'$ by the Banach–Steinhaus theorem. Let $B\subset F$ be bounded. As ${^{t}u}\in L(\mathcal {EV}(\varOmega )_{b}',F_{b}')$, there are a bounded set $B_{0}\subset \mathcal {EV}(\varOmega )$ and $C>0$ such that

$$\begin{aligned}&\sup _{x\in B}\bigl |\bigl (\frac{{^{t}u}(\delta _{z+he_{k}})-{^{t}u}(\delta _{z})}{h}\bigr )(x) -{^{t}u}\bigl (\delta _{z}\circ \partial ^{e_{k}}\bigr )(x)\bigr |\\&=\sup _{x\in B}\bigl |{^{t}u}\bigl (\frac{\delta _{z+he_{k}}-\delta _{z}}{h}-\delta _{z}\circ \partial ^{e_{k}}\bigr )(x)\bigr | \le C\sup _{f\in B_{0}}\bigl |\bigl (\frac{\delta _{z+he_{k}}-\delta _{z}}{h}-\delta _{z}\circ \partial ^{e_{k}}\bigr )(f)\bigr | \underset{h\rightarrow 0}{\rightarrow }0, \end{aligned}$$

yielding to $(\partial ^{e_{k}})^{E}(\psi (u))(z)={^{t}u}(\delta _{z}\circ \partial ^{e_{k}})$. This implies ${\overline{\partial }}^{E}(\psi (u))(z)={^{t}u}(\delta _{z}\circ {\overline{\partial }})$. So for all $x\in F$ and $z\in \varOmega$ we have

$$\begin{aligned} {\overline{\partial }}^{E}(\psi (u))(z)(x)&={^{t}u}(\delta _{z}\circ {\overline{\partial }})(x) =u(x)(\delta _{z}\circ {\overline{\partial }}) =\langle \delta _{z}\circ {\overline{\partial }},J^{-1}_{1}(u(x))\rangle \\&=\langle \delta _{z},{\overline{\partial }}J^{-1}_{1}(u(x))\rangle =\langle [J_{1}\circ {\overline{\partial }}\circ J^{-1}_{1}](u(x)),\delta _{z}\rangle =\langle ({\overline{\partial }}_{1}\circ u)(x),\delta _{z}\rangle \\&=\langle ({\overline{\partial }}^{*}_{1}u)(x),\delta _{z}\rangle =\psi ^{-1}(g)(x)(\delta _{z}) ={^{t}({\mathcal {J}}^{-1}\circ R_{g}^{t})}(x)(\delta _{z})\\&=({\mathcal {J}}^{-1}\circ R_{g}^{t})(\delta _{z})(x) ={\mathcal {J}}^{-1}({\mathcal {J}}(g(z))(x) =g(z)(x). \end{aligned}$$

Thus ${\overline{\partial }}^{E}(\psi (u))(z)=g(z)$ for every $z\in \varOmega$, which proves the surjectivity.

(b) Let E be an ultrabornological PLS-space satisfying (PA). Since the nuclear Fréchet space $\mathcal {EV}_{{\overline{\partial }}}(\varOmega )$ is also a Schwartz space, its strong dual $\mathcal {EV}_{{\overline{\partial }}}(\varOmega )_{b}'$ is a DFS-space. By [6, Theorem 4.1, p. 577] we obtain ${\text {Ext}}^{1}_{PLS}(\mathcal {EV}_{{\overline{\partial }}}(\varOmega )_{b}',E)=0$ as the bidual $\mathcal {EV}_{{\overline{\partial }}}(\varOmega )''$ satisfies $(\varOmega )$, E is a PLS-space satisfying (PA) and condition (c) in the theorem is fulfilled because $\mathcal {EV}_{{\overline{\partial }}}(\varOmega )_{b}'$ is the strong dual of a nuclear Fréchet space. Moreover, we have ${\text {Proj}}^{1} E=0$ due to [66, Corollary 3.3.10, p. 46] because E is an ultrabornological PLS-space. Then the exactness of the sequence (5), [6, Theorem 3.4, p. 567] and [6, Lemma 3.3, p. 567] (in the lemma the same condition (c) as in [6, Theorem 4.1, p. 577] is fulfilled and we choose $H=\mathcal {EV}_{{\overline{\partial }}}(\varOmega )''$ and $F=G=\mathcal {EV}(\varOmega )''$), imply that the sequence

$$\begin{aligned} 0\rightarrow L(E_{b}',\mathcal {EV}_{{\overline{\partial }}}(\varOmega )'') \overset{i^{*}_{0}}{\rightarrow }L(E_{b}',\mathcal {EV}(\varOmega )'') \overset{{\overline{\partial }}^{*}_{1}}{\rightarrow }L(E_{b}',\mathcal {EV}(\varOmega )'')\rightarrow 0 \end{aligned}$$

is exact. The maps $i^{*}_{0}$ and ${\overline{\partial }}^{*}_{1}$ are defined like in part (a). Especially, we get that

$$\begin{aligned} {\overline{\partial }}^{*}_{1}:L(E_{b}',\mathcal {EV}(\varOmega )'')\rightarrow L(E_{b}',\mathcal {EV}(\varOmega )'') \end{aligned}$$

(7)

is surjective.

By [19, Remark 4.4, p. 1114] we have $L_{b}(\mathcal {EV}(\varOmega )_{b}',E'') \cong L_{b}(E_{b}',\mathcal {EV}(\varOmega )'')$ via taking adjoints since $\mathcal {EV}(\varOmega )$, being a Fréchet–Schwartz space, is a PLS-space and hence its strong dual an LFS-space, which is regular by [66, Corollary 6.7, $10.\Leftrightarrow 11.$, p. 114], and E is an ultrabornological PLS-space, in particular, reflexive by [17, Theorem 3.2, p. 58]. In addition, the map

$$\begin{aligned} T:L_{b}(\mathcal {EV}(\varOmega )_{b}',E'')\rightarrow L_{b}(\mathcal {EV}(\varOmega )_{b}',E), \end{aligned}$$

defined by $T(u)(y):={\mathcal {J}}^{-1}(u(y))$ for $u\in L(\mathcal {EV}(\varOmega )_{b}',E'')$ and $y\in \mathcal {EV}(\varOmega )'$, is a topological isomorphism because E is reflexive. Due to Proposition 4 (b) we obtain the topological isomorphism

$$\begin{aligned} \psi :=S\circ {\mathcal {J}}^{-1}\circ {^{t}(\cdot )}:L_{b}(E_{b}',\mathcal {EV}(\varOmega )'') \rightarrow \mathcal {EV}(\varOmega ,E),\\ \psi (u)=[S\circ {\mathcal {J}}^{-1}\circ {^{t}(\cdot )}](u)=\bigl [z\mapsto {\mathcal {J}}^{-1}({^{t}u}(\delta _{z}))\bigr ], \end{aligned}$$

with the inverse given by

$$\begin{aligned} \psi ^{-1}(f) =(S\circ {\mathcal {J}}^{-1}\circ {^{t}(\cdot )})^{-1}(f) =[{^{t}(\cdot )}\circ {\mathcal {J}}\circ S^{-1}](f) ={^{t}({\mathcal {J}}\circ {\mathcal {J}}^{-1}\circ R_{f}^{t})}={^{t}(R_{f}^{t})} \end{aligned}$$

for $f\in \mathcal {EV}(\varOmega ,E)$.

Let $g\in \mathcal {EV}(\varOmega ,E)$. Then $\psi ^{-1}(g)\in L(E_{b}',\mathcal {EV}(\varOmega )'')$ and by the surjectivity of (7) there exists $u\in L(E_{b}',\mathcal {EV}(\varOmega )'')$ such that ${\overline{\partial }}^{*}_{1}u=\psi ^{-1}(g)$. So we have $\psi (u)\in \mathcal {EV}(\varOmega ,E)$. The last step is to show that ${\overline{\partial }}^{E}\psi (u)=g$. Like in part (a) we gain for every $z\in \varOmega$

$$\begin{aligned} {\overline{\partial }}^{E}(\psi (u))(z)={\mathcal {J}}^{-1}({^{t}u} (\delta _{z}\circ {\overline{\partial }})) \end{aligned}$$

and for every $x\in E'$

$$\begin{aligned} {^{t}u}(\delta _{z}\circ {\overline{\partial }})(x)&=u(x)(\delta _{z}\circ {\overline{\partial }}) =({\overline{\partial }}^{*}_{1}u)(x)(\delta _{z}) =\psi ^{-1}(g)(x)(\delta _{z}) ={^{t}(R_{g}^{t})}(x)(\delta _{z})\\&=\delta _{z}(x\circ g) =x(g(z)) ={\mathcal {J}}(g(z))(x). \end{aligned}$$

Thus we have ${^{t}u}(\delta _{z}\circ {\overline{\partial }})={\mathcal {J}}(g(z))$ and therefore ${\overline{\partial }}^{E}(\psi (u))(z)=g(z)$ for all $z\in \varOmega$. $\square$

By Remark 2 case (a) is included in case (b) if F is a Fréchet–Schwartz space. Therefore (a) is only interesting for Fréchet spaces F which are not Schwartz spaces. In the next more technical section we will present sufficient conditions for $\mathcal {EV}_{{\overline{\partial }}}(\varOmega )$ to have $(\varOmega )$ as well as concrete examples of such spaces.

4 $(\varOmega )$ for $\mathcal {OV}$-spaces on strips and applications of the main result

In this section we give some sufficient conditions such that the assumptions of our main result Theorem 5 are fulfilled. The outline is as follows. First, we show that $\mathcal {OV}(\varOmega )$ and $\mathcal {EV}_{{\overline{\partial }}}(\varOmega )$ coincide topologically under mild assumptions on the weights ${\mathcal {V}}$ and the sequence of sets $(\varOmega _{n})$. These mild conditions also imply that $\mathcal {EV}(\varOmega )$ is nuclear, in particular Schwartz, and thus its subspace $\mathcal {EV}_{{\overline{\partial }}}(\varOmega )=\mathcal {OV}(\varOmega )$ too. Second, we reduce the problem whether the projective limit $\mathcal {OV}(\varOmega )$ has $(\varOmega )$ to the problem whether it is weakly reduced in the case that the $\varOmega _{n}$ are strips along the real axis and the weights have a certain structure. Third, we use a similar result for $\mathcal {EV}_{{\overline{\partial }}}(\varOmega )$ which was obtained in [33] to prove the weak reducibility of $\mathcal {OV}(\varOmega )$. For corresponding results in the case that $\varOmega _{n}=\varOmega$ for all $n\in \mathbb {N}$ see [20, Theorem 3, p. 56], [39, 1.3 Lemma, p. 418] and [58, Theorem 1, p. 145]. We close this section with some examples of our main result. Let us start with the sufficient conditions, guaranteeing that the projective limit $\mathcal {EV}(\varOmega )$ is nuclear (if $q=1$). They also allow to switch from $\sup$- to weighted $L^{q}$-seminorms which is important for the proof of surjectivity of the scalar-valued ${\overline{\partial }}$-operator given in [33], using Hörmander’s $L^{2}$-machinery (if $q=2$).

Condition (PN)

([33, 3.3 Condition, p. 7]) Let ${\mathcal {V}}:=(\nu _{n})_{n\in \mathbb {N}}$ be a directed family of continuous weights on an open set $\varOmega \subset \mathbb {R}^{2}$ and $(\varOmega _{n})_{n\in \mathbb {N}}$ a family of non-empty open sets such that $\varOmega _{n}\subset \varOmega _{n+1}$ and $\varOmega =\bigcup _{n\in \mathbb {N}} \varOmega _{n}$. For every $k\in \mathbb {N}$ let there be $\rho _{k}\in \mathbb {R}$ such that $0<\rho _{k}<\mathrm {d}^{\Vert \cdot \Vert _{\infty }}(\{x\},\partial \varOmega _{k+1})$ for all $x\in \varOmega _{k}$ and let there be $q\in \mathbb {N}$ such that for any $n\in \mathbb {N}$ there is $\psi _{n}\in L^{q}(\varOmega _{k})$, $\psi _{n}>0$, and $\mathbb {N}\ni J_{i}(n)\ge n$ and $C_{i}(n)>0$ such that for any $x\in \varOmega _{k}$:

(PN.1):

$\sup _{\zeta \in \mathbb {R}^{2},\,\Vert \zeta \Vert _{\infty }\le \rho _{k}}\nu _{n}(x+\zeta ) \le C_{1}(n)\inf _{\zeta \in \mathbb {R}^{2},\,\Vert \zeta \Vert _{\infty }\le \rho _{k}}\nu _{J_{1}(n)}(x+\zeta )$

$(PN.2)^{q}$:

$\nu _{n}(x)\le C_{2}(n)\psi _{n}(x)\nu _{J_{2}(n)}(x)$

Example 6

Let $\varOmega :=\mathbb {R}^{2}$ and $\varOmega _{n}:=\{x=(x_{i})\in \mathbb {R}^{2}\;||x_{2}|<n\}.$ Let $0<\gamma \le 1$ and $(a_{n})_{n\in \mathbb {N}}$ be strictly increasing such that $a_{n}\ge 0$ for all $n\in \mathbb {N}$ or $a_{n}\le 0$ for all $n\in \mathbb {N}$. The family ${\mathcal {V}}:=(\nu _{n})_{n\in \mathbb {N}}$ of positive continuous functions on $\varOmega$ given by

$$\begin{aligned} \nu _{n}:\varOmega \rightarrow (0,\infty ),\;\nu _{n}(x):=e^{a_{n}|x_{1}|^{\gamma }}, \end{aligned}$$

fulfils $\nu _{n}\le \nu _{n+1}$ all $n\in \mathbb {N}$ and (PN) for every $q\in \mathbb {N}$ with $\psi _{n}(x):= (1+|x|^{2})^{-2}$, $x\in \mathbb {R}^{2}$, for every $n\in \mathbb {N}$.

The space $\mathcal {OV}({\mathbb {C}})$ with this kind of weights consists of functions which are entire and exponentially growing $(a_{n}<0)$ resp. decreasing ($a_{n}>0$) with order $\gamma$ on strips along the real axis. This example of weights and many more are included in [33, 3.7 Example, p. 9]. We restrict to this particular weights because we use it in an example for our main result.

Proposition 7

Let ${\mathcal {V}}:=(\nu _{n})_{n\in \mathbb {N}}$ be a directed family of continuous weights on an open set $\varOmega \subset \mathbb {R}^{2}$ and $(\varOmega _{n})_{n\in \mathbb {N}}$ a family of non-empty open sets such that $\varOmega _{n}\subset \varOmega _{n+1}$ and $\varOmega =\bigcup _{n\in \mathbb {N}} \varOmega _{n}$. If (PN.1) is fulfilled, then

• (a) for every $n\in \mathbb {N}$ and $m\in \mathbb {N}_{0}$ there is $C>0$ such that

  $$\begin{aligned} |f|_{n,m}\le C|f|_{2J_{1}(n)},\quad f\in {\mathcal {O}}\nu _{2J_{1}(n)}(\varOmega _{2J_{1}(n)}). \end{aligned}$$

• (b) $\mathcal {EV}_{{\overline{\partial }}}(\varOmega )=\mathcal {OV}(\varOmega )$ as Fréchet spaces.

Proof

(a) Let $n\in \mathbb {N}$ and $m\in \mathbb {N}_{0}$. We note that $\varOmega _{n+1}\subset \varOmega _{2J_{1}(n)}$ and $\partial ^{\beta }f(x)=i^{\beta _{2}}f^{(|\beta |)}(x)$, $x\in \varOmega _{2J_{1}(n)}$, holds for all $\beta =(\beta _{1},\beta _{2})\in \mathbb {N}_{0}^{2}$ and $f\in {\mathcal {O}}\nu _{2J_{1}(n)}(\varOmega _{2J_{1}(n)})$ where $f^{(|\beta |)}$ is the $|\beta |$th complex derivative of f. Then we obtain via (PN.1) and Cauchy’s inequality

(b) The space $\mathcal {EV}_{{\overline{\partial }}}(\varOmega )$ is a Fréchet space since it is a closed subspace of the Fréchet space $\mathcal {EV}(\varOmega )$ by [34, Proposition 3.7, p. 240]. From part (a) and $|f|_{n}=|f|_{n,0}$ for all $n\in \mathbb {N}$ and $f\in \mathcal {EV}_{{\overline{\partial }}}(\varOmega )$ follows the statement. $\square$

Let us come to the second part. Using special weight functions, strips along the real axis as $\varOmega _{n}$ and a decomposition theorem of Langenbruch, we will see that answering the question whether $\mathcal {OV}(\varOmega )$ satisfies the property $(\varOmega )$ of Vogt boils down to answering whether the projective limit $\mathcal {OV}(\varOmega )$ is weakly reduced. The special weights we want to consider are generated by a function $\mu$ with the following properties.

Definition 8

(strong weight generator) A continuous function $\mu :{\mathbb {C}}\rightarrow [0,\infty )$ is called a weight generator if $\mu (z)=\mu (|{{\,\mathrm{Re}\,}}(z)|)$ for all $z\in {\mathbb {C}}$, the restriction $\mu _{\mid [0,\infty )}$ is strictly increasing,

$$\begin{aligned} \lim _{\begin{array}{c} x\rightarrow \infty \\ x\in \mathbb {R} \end{array}}\frac{\ln (1+|x|)}{\mu (x)}=0 \end{aligned}$$

and

$$\begin{aligned} \exists \;\varGamma>1,\,C>0\;\forall \;x\in [0,\infty ):\; \mu (x+1)\le \varGamma \mu (x)+C. \end{aligned}$$

If $\mu$ is a weight generator which fulfils the stronger condition

$$\begin{aligned} \exists \;\varGamma>1\;\forall \;n\in \mathbb {N}\;\exists \;C>0\;\forall \;x\in [0,\infty ):\; \mu (x+n)\le \varGamma \mu (x)+C, \end{aligned}$$

then $\mu$ is called a strong weight generator.

Weight generators are introduced in [46, Definition 2.1, p. 225] and strong weight generators in [60, Definition 2.2.2, p. 43] where they are simply called weight functions resp. strong weight functions. For a weight generator $\mu$ we define the space

$$\begin{aligned} H_{\tau }(S_{t}):=\{f\in {\mathcal {O}}(S_{t})\;|\;\Vert f\Vert _{\tau ,t}:=\sup _{z\in S_{t}}|f(z)|e^{\tau \mu (z)}<\infty \} \end{aligned}$$

for $t>0$ and $\tau \in \mathbb {R}$ with the strip $S_{t}:=\{z\in {\mathbb {C}}\;|\;|{{\,\mathrm{Im}\,}}(z)|<t\}$ .

Theorem 9

[46, Theorem 2.2, p. 225]¹ Let $\mu$ be a weight generator. There are ${\widetilde{t}}$, $K_{1}$, $K_{2}>0$ such that for any $\tau _{0}<\tau <\tau _{2}$ there is $C_{0}=C_{0}({\text {sign}}(\tau ))$ such that for any $0<2t_{0}<t<t_{2}<{\widetilde{t}}$ with

$$\begin{aligned} t_{0}\le \min \Bigl [K_{1},K_{2}\sqrt{\frac{\tau -C_{0}\tau _{0}}{\tau _{2}-C_{0}\tau _{0}}}\Bigr ] \end{aligned}$$

there is $C_{1}\ge 1$ such that for any $r\ge 0$ and any $f\in H_{\tau }(S_{t})$ with $\Vert f\Vert _{\tau ,t}\le 1$ the following holds: there are $f_{2}\in {\mathcal {O}}(S_{t_{2}})$ and $f_{0}\in {\mathcal {O}}(S_{t_{0}})$ such that $f=f_{0}+f_{2}$ on $S_{t_{0}}$ and

$$\begin{aligned} \Vert f_{0}\Vert _{C_{0}\tau _{0},t_{0}}\le C_{1}e^{-Gr}\quad \text {and}\quad \Vert f_{2}\Vert _{\tau _{2},t_{2}}\le e^{r} \end{aligned}$$

where

$$\begin{aligned} G:= K_{1} \min \Bigl [1,\frac{t-t_{0}}{2{\widetilde{t}}},\frac{\tau -C_{0}\tau _{0}}{\tau _{2}-C_{0}\tau _{0}}\Bigr ]. \end{aligned}$$

To apply this theorem, we have to know the constants involved. In the following the notation of [46] is used and it is referred to the corresponding positions resp. conditions for these constants. We have

$$\begin{aligned} {\widetilde{t}}:=\frac{1}{4 \ln (\varGamma )} \end{aligned}$$

by [46, Lemma 2.4, (2.15), p. 228] with $\varGamma$ from Definition 8 such that $\varGamma \ge e^{1/4}$. The choice $\varGamma \ge e^{1/4}$ comes from wanting ${\widetilde{t}}\le 1$ in [46, Lemma 2.4, p. 228]. By [46, Corollary 2.6, p. 230-231] we have

$$\begin{aligned} C_{0}:= {\left\{ \begin{array}{ll}4 \varGamma B_{3}=\frac{64\cosh (1)}{\cos (1/2)}\varGamma ^{2}>1 &{},\tau<0,\\ \frac{1}{4 \varGamma B_{3}}=\frac{\cos (1/2)}{64\cosh (1)\varGamma ^{2}}<1 &{},\tau \ge 0, \end{array}\right. } \end{aligned}$$

where $B_{3}:=\frac{16\cosh (1)}{\cos (1/2)}\varGamma$ by [46, Lemma 2.4, p. 228–229].² To get the constants $K_{1}$ and $K_{2}$, we have to analyze the conditions for $t_{0}$ in the proof of [46, Theorem 2.2, p. 225]. By the assumptions on $\tau _{0}$, $\tau$ and $\tau _{2}$ and the choice of $C_{0}$ we obtain

$$\begin{aligned} \tau _{2}-C_{0}\tau _{0}>\tau _{2}-C_{0}\tau \ge \tau _{2}-\tau >0 \end{aligned}$$

(8)

and

$$\begin{aligned} \tau -C_{0}\tau _{0}>\tau -C_{0}\tau =\tau (1-C_{0})>0. \end{aligned}$$

(9)

By choosing $D>0$ in the proof of [46, Theorem 2.2, (2.22), p. 232–233] as $D:=\frac{\tau -C_{0}\tau _{0}}{(\tau _{2}-C_{0}\tau _{0})2\varGamma _{0}}$, the estimate

$$\begin{aligned} D=\frac{\tau -C_{0}\tau _{0}}{(\tau _{2}-C_{0}\tau _{0})2\varGamma _{0}} =\min \Bigl (\frac{1}{2{\widetilde{\varGamma }}},\frac{1}{2{\widehat{\varGamma }}}\Bigr ) \frac{\tau -C_{0}\tau _{0}}{\tau _{2}-C_{0}\tau _{0}} \underset{(8),\,(9)}{\le }\min \Bigl (\frac{1}{2{\widetilde{\varGamma }}}, \frac{1}{2{\widehat{\varGamma }}}\Bigr ) \frac{\tau -C_{0}\tau _{0}}{\tau _{2}-C_{0}\tau } \end{aligned}$$

holds where $\varGamma _{0}:=\max ({\widetilde{\varGamma }},{\widehat{\varGamma }})$ with ${\widetilde{\varGamma }}$, ${\widehat{\varGamma }}>1$ from the proof. With $\theta \ge \frac{t-t_{0}}{2{\widetilde{t}}}$ (p. 232) we get on p. 233, below (2.24), due to the condition $t_{0}\le T_{0}:=\min (\frac{t}{2},\frac{1}{4a^{2}B_{1}{\widetilde{t}}})$,

$$\begin{aligned} \min \Bigl (\frac{\theta }{2}, D, 1\Bigr )&\ge \min \Bigl (\frac{1}{2},\frac{1}{2\varGamma _{0}}\Bigr ) \min \Bigl (\theta ,\frac{\tau -C_{0}\tau _{0}}{\tau _{2}-C_{0}\tau _{0}},1\Bigr ) \ge \frac{1}{2\varGamma _{0}}\min \Bigl (\frac{t-t_{0}}{2{\widetilde{t}}},\frac{\tau -C_{0}\tau _{0}}{\tau _{2}-C_{0}\tau _{0}},1\Bigr )\\&\ge \min \Bigl (\frac{1}{2\varGamma _{0}},\frac{1}{4a^{2}B_{1}{\widetilde{t}}}\Bigr )\min \Bigl (\frac{t-t_{0}}{2{\widetilde{t}}}, \frac{\tau -C_{0}\tau _{0}}{\tau _{2}-C_{0}\tau _{0}},1\Bigr )\\&=\underbrace{\min \Bigl (\frac{1}{2\varGamma _{0}},\frac{1}{2\cosh (1)\ln (\varGamma )}\Bigr )}_{=:K_{1}} \min \Bigl (\frac{t-t_{0}}{2{\widetilde{t}}},\frac{\tau -C_{0}\tau _{0}}{\tau _{2}-C_{0}\tau _{0}},1\Bigr )=:G \end{aligned}$$

where $a:=\ln (\varGamma )$ (in the middle of p. 231) and $B_{1}:=2\cosh (1)$ by the proof of [46, Lemma 2.3, p. 226–227]. The assumptions $2t_{0}<t$ and $t_{0}\le K_{1}$ in Theorem 9 guarantee that the condition $t_{0}\le T_{0}$ is satisfied. Looking at the condition $t_{0}\le T_{1}:=\sqrt{\frac{D}{a^{2}B_{1}}}$ (p. 232), we derive

$$\begin{aligned} T_{1}=\frac{1}{\sqrt{2\varGamma _{0}a^{2}B_{1}}} \sqrt{\frac{\tau -C_{0}\tau _{0}}{\tau _{2}-C_{0}\tau _{0}}} =\underbrace{\frac{1}{2\sqrt{\cosh (1)\varGamma _{0}} \ln (\varGamma )}}_{=:K_{2}}\sqrt{\frac{\tau -C_{0}\tau _{0}}{\tau _{2}-C_{0}\tau _{0}}}. \end{aligned}$$

For the subsequent theorem we merge and modify the proofs of [60, Satz 2.2.3, p. 44]³ ($a_{n}=n$, $n\in \mathbb {N}$, and $\mu$ a strong weight generator) and [32, 5.20 Theorem, p. 84] ($a_{n}=-1/n$, $n\in \mathbb {N}$, and $\mu =|{{\,\mathrm{Re}\,}}(\cdot )|$).

Theorem 10

Let $\mu$ be a strong weight generator, $a_{n}\nearrow 0$ or $a_{n}\nearrow \infty$, ${\mathcal {V}}:=(\exp (a_{n}\mu ))_{n\in \mathbb {N}}$ and $\varOmega _{n}:=S_{n}$ for all $n\in \mathbb {N}$. If $\mathcal {OV}({\mathbb {C}})$ is weakly reduced, then $\mathcal {OV}({\mathbb {C}})$ satisfies $(\varOmega )$.

Proof

Since $\mathcal {OV}({\mathbb {C}})$ is weakly reduced, for every $n\in \mathbb {N}$ there exists $m_{n}\in \mathbb {N}$ such that $\pi _{n}(\mathcal {OV}({\mathbb {C}}))$ is dense in ${\mathcal {O}}\nu _{m_{n}}(\varOmega _{m_{n}})$ w.r.t. the topology of ${\mathcal {O}}\nu _{n}(\varOmega _{n})$ where

$$\begin{aligned} \pi _{n}:\mathcal {OV}({\mathbb {C}})\rightarrow {\mathcal {O}}\nu _{n} (\varOmega _{n}),\;\pi _{n}(f):=f_{\mid \varOmega _{n}}, \end{aligned}$$

is the canonical projection. Let $p,k\in \mathbb {N}$. As $(a_{n})_{n\in \mathbb {N}}$ is strictly increasing and $\lim _{n\rightarrow \infty }a_{n}=0$ or $\lim _{n\rightarrow \infty }a_{n}=\infty$, we may choose $q\in \mathbb {N}$ such that $a_{m_{p}}/C_{0}<a_{q}$ and $2m_{p}<q$. To use the decomposition from Theorem 9, we need a linear transformation between strips to get the decomposition on the desired strip $S_{m_{p}}$. We choose $\varGamma \ge e^{1/4}$ and $T\in \mathbb {R}$ such that

$$\begin{aligned} 0<T<\frac{1}{4\max (q+1,m_{k})\ln (\varGamma )} \end{aligned}$$

(10)

which also fulfils

$$\begin{aligned} T\le \frac{1}{m_{p}}\min \Biggl (\frac{1}{2\varGamma _{0}},\frac{1}{2\cosh (1)\ln (\varGamma )}, \frac{1}{2\sqrt{\cosh (1)\varGamma _{0}}\ln (\varGamma )} \sqrt{\frac{a_{q}-a_{m_{p}}}{\max (a_{q+1},a_{m_{k}})-a_{m_{p}}}}\Biggr ).\nonumber \\ \end{aligned}$$

(11)

Let

$$\begin{aligned} \tau _{0}&:=\frac{a_{m_{p}}}{C_{0}},&\tau :=a_{q},&\tau _{2}:=\max (a_{q+1},a_{m_{k}}),\\ t_{0}&:= m_{p}T,&t:=qT,&t_{2}:=\max (q+1,m_{k})T. \end{aligned}$$

By the choice of q we have

$$\begin{aligned} \tau _{0}=\frac{a_{m_{p}}}{C_{0}}<a_{q}=\tau <\max (a_{q+1},a_{m_{k}})=\tau _{2}. \end{aligned}$$

By the choice of q and (10) we get

$$\begin{aligned} 0<2t_{0}=2m_{p}T<qT=t<\max (q+1,m_{k})T=t_{2}<\frac{1}{4\ln (\varGamma )}={\widetilde{t}}. \end{aligned}$$

Further, we deduce from (11) that

$$\begin{aligned} t_{0}=m_{p}T\le \min \Bigl [K_{1},K_{2} \sqrt{\frac{\tau -C_{0}\tau _{0}}{\tau _{2}-C_{0}\tau _{0}}}\Bigr ]. \end{aligned}$$

Let $r\ge 0$ and $f\in \mathcal {OV}({\mathbb {C}})$ such that $|f|_{q}=\Vert f\Vert _{a_{q},q}\le 1$. We set ${\widetilde{f}}:S_{qT}\rightarrow {\mathbb {C}}$, ${\widetilde{f}}(z):=f(z/T)$, and define

$$\begin{aligned} H_{\tau }^{\sim }(S_{t}):=\{g\in {\mathcal {O}}(S_{t})\;|\;\Vert g\Vert _{\tau ,t}^{\sim }:=\sup _{z\in S_{t}}|g(z)|e^{\tau {\widetilde{\mu }}(z)}<\infty \} \end{aligned}$$

where ${\widetilde{\mu }}:=\mu (\cdot /T)$. We note that for ${\widetilde{n}}:=\lceil 1/T\rceil$, where $\lceil \cdot \rceil$ is the ceiling function, there is $C>0$ such that for all $x\ge 0$

$$\begin{aligned} {\widetilde{\mu }}(x+1)=\mu \Bigl (\frac{x+1}{T}\Bigr ) \le \mu \Bigl (\frac{x}{T}+\Bigl \lceil \frac{1}{T}\Bigr \rceil \Bigr ) =\mu \Bigl (\frac{x}{T}+{\widetilde{n}}\Bigr ) \le \varGamma \mu \Bigl (\frac{x}{T}\Bigr )+C =\varGamma {\widetilde{\mu }}(x)+C \end{aligned}$$

because $\mu$ is a strong weight generator. We conclude that ${\widetilde{\mu }}$ is also a weight generator with the same $\varGamma$ as $\mu$ which is independent of T. Moreover, from

$$\begin{aligned} \Vert {\widetilde{f}}\Vert _{\tau ,t}^{\sim }=\sup _{z\in S_{qT}}|{\widetilde{f}}(z)|e^{a_{q}{\widetilde{\mu }}(z)} =\sup _{z\in S_{q}}|f(z)|e^{a_{q}\mu (z)}=|f|_{q}\le 1 \end{aligned}$$

it follows by Theorem 9 that there are ${\widetilde{f}}_{j}\in {\mathcal {O}}(S_{t_{j}})$, $j\in \{0,2\}$, such that

$$\begin{aligned} {\widetilde{f}}(z)={\widetilde{f}}_{0}(z)+{\widetilde{f}}_{2}(z),\quad z\in S_{t_{0}}, \end{aligned}$$

(12)

and

$$\begin{aligned} C_{1}e^{-Gr} \ge \Vert {\widetilde{f}}_{0}\Vert _{C_{0}\tau _{0},t_{0}}^{\sim } =\sup _{z\in S_{t_{0}/T}}|\underbrace{{\widetilde{f}}_{0}(Tz)}_{=:f_{0}(z)} |e^{C_{0}\tau _{0}{\widetilde{\mu }}(Tz)} =\sup _{z\in S_{m_{p}}}|f_{0}(z)|e^{a_{m_{p}}\mu (z)}=|f_{0}|_{m_{p}},\nonumber \\ \end{aligned}$$

(13)

where $f_{0}\in {\mathcal {O}}(S_{m_{p}})$, as well as

$$\begin{aligned} e^{r} \ge \Vert {\widetilde{f}}_{2}\Vert _{\tau _{2},t_{2}}^{\sim } =\sup _{z\in S_{t_{2}/T}}|\underbrace{{\widetilde{f}}_{2}(Tz)}_{=:f_{2}(z)}|e^{\tau _{2}{\widetilde{\mu }}(Tz)} \ge \sup _{z\in S_{m_{k}}}|f_{2}(z)|e^{a_{m_{k}}\mu (z)}=|f_{2}|_{m_{k}} \end{aligned}$$

(14)

where $f_{2}\in {\mathcal {O}}(S_{t_{2}/T})\subset {\mathcal {O}}(S_{m_{k}})$ and the inclusion is justified by the identity theorem. Furthermore, for $z\in S_{t_{0}/T}=S_{m_{p}}$ the equation

$$\begin{aligned} f(z)={\widetilde{f}}(Tz)\underset{(12)}{=}{\widetilde{f}}_{0} (Tz)+{\widetilde{f}}_{2}(Tz)=f_{0}(z)+f_{2}(z) \end{aligned}$$

holds, thus $f=f_{0}+f_{2}$ on $S_{m_{p}}$. By virtue of the weak reducibility of $\mathcal {OV}({\mathbb {C}})$ and the choice of $m_{p},m_{k}$ the following is valid:

$$\begin{aligned} \forall \;\varepsilon >0\;\exists \;{\widehat{f}}_{0},\, {\widehat{f}}_{2}\in \mathcal {OV}({\mathbb {C}}):\; (i)\;\;|{\widehat{f}}_{0}-f_{0}|_{p}<\varepsilon \quad \text {and}\quad (ii)\;\;|{\widehat{f}}_{2}-f_{2}|_{k}<\varepsilon . \end{aligned}$$

(15)

Now, we have to consider two cases. Let $\varepsilon :=C_{1}e^{-Gr}$. For $k\le p$ we get via (15) (i) that $f={\widehat{f}}_{0}+(f_{2}+f_{0}-{\widehat{f}}_{0})$ on $S_{m_{p}}$ so

$$\begin{aligned} f_{2}+f_{0}-{\widehat{f}}_{0}=f-{\widehat{f}}_{0}=:{\overline{f}}_{2} \quad \text {on}\;S_{m_{p}} \end{aligned}$$

(16)

where the function ${\overline{f}}_{2}\in \mathcal {OV}({\mathbb {C}})$ and thus is a holomorphic extension of the left-hand side on ${\mathbb {C}}$. Hence we clearly have $f={\widehat{f}}_{0}+{\overline{f}}_{2}$ and

$$\begin{aligned} |{\widehat{f}}_{0}|_{p} \le |{\widehat{f}}_{0}-f_{0}|_{p}+|f_{0}|_{p} \underset{(15) (i)}{\le }\varepsilon +|f_{0}|_{p} \le \varepsilon +|f_{0}|_{m_{p}} \underset{(13)}{\le }2C_{1}e^{-Gr}=:C_{2}e^{-Gr} \end{aligned}$$

(17)

as well as

(18)

Analogously, for $k>p$ we obtain via (15) (ii) that $f={\widehat{f}}_{2}+(f_{0}+f_{2}-{\widehat{f}}_{2})$ on $S_{m_{p}}$ so

$$\begin{aligned} f_{0}+f_{2}-{\widehat{f}}_{2}=f-{\widehat{f}}_{2} =:{\overline{f}}_{0}\quad \text {on}\;S_{m_{p}} \end{aligned}$$

(19)

where the function ${\overline{f}}_{0}\in \mathcal {OV}({\mathbb {C}})$ and thus is a holomorphic extension of the left-hand side on ${\mathbb {C}}$. Hence we clearly have $f={\overline{f}}_{0}+{\widehat{f}}_{2}$ and

(20)

as well as

$$\begin{aligned} |{\widehat{f}}_{2}|_{k} \le |{\widehat{f}}_{2}-f_{2}|_{k}+|f_{2}|_{k} \underset{(15) (ii)}{\le }\varepsilon +|f_{2}|_{m_{k}} \underset{(14)}{\le } C_{1}e^{-Gr}+e^{r}\le C_{3}e^{r}. \end{aligned}$$

(21)

Next, we set $n:=\lceil 1/G \rceil$ and $C:=C_{3}e^{\ln (C_{2})/G}$. Let ${\widetilde{r}}>0$. For ${\widetilde{r}}\ge 1$ there is $r\ge 0$ such that

$$\begin{aligned} {\widetilde{r}}=e^{Gr-\ln (C_{2})}=\frac{e^{Gr}}{C_{2}} \end{aligned}$$

and we have by (17) and (18) for $k\le p$

$$\begin{aligned} |{\widehat{f}}_{0}|_{p}\le C_{2}e^{-Gr}=\frac{1}{{\widetilde{r}}}, \quad |{\overline{f}}_{2}|_{k} \le C_{3}e^{r}=C_{3}e^{\frac{1}{G}\ln (C_{2})}e^{\frac{1}{G}(Gr-\ln (C_{2}))} =C\,{\widetilde{r}}^{\,\frac{1}{G}}\underset{{\widetilde{r}}\ge 1}{\le }C\,{\widetilde{r}}^{\;n}, \end{aligned}$$

as well as by (20) and (21) for $k>p$

$$\begin{aligned} |{\overline{f}}_{0}|_{p}\le \frac{1}{{\widetilde{r}}},\quad |{\widehat{f}}_{2}|_{k}\le C\,{\widetilde{r}}^{\;n}. \end{aligned}$$

For $0<{\widetilde{r}}<1$ we have, since $q\ge p$,

$$\begin{aligned} |f|_{p}\le |f|_{q}\le 1<\frac{1}{{\widetilde{r}}}. \end{aligned}$$

Thus our statement is proved. $\square$

Let us remark that the choice of the sequence $(a_{n})_{n\in \mathbb {N}}$ in the preceding theorem does not really matter.

Remark 11

Let $\mu :{\mathbb {C}}\rightarrow [0,\infty )$ be continuous, $a_{n}\nearrow 0$ or $a_{n}\nearrow \infty$, ${\mathcal {V}}:=(\exp (a_{n}\mu ))_{n\in \mathbb {N}}$ and $\varOmega _{n}:=S_{n}$ for all $n\in \mathbb {N}$. Set ${\mathcal {V}}_{-}:=(\exp ((-1/n)\mu ))_{n\in \mathbb {N}}$ and ${\mathcal {V}}_{+}:=(\exp (n\mu ))_{n\in \mathbb {N}}$. Then

$$\begin{aligned} \mathcal {OV}({\mathbb {C}})\cong \mathcal {OV}_{-}({\mathbb {C}}),\quad \text {if}\;a_{n}\nearrow 0,\quad \text {and}\quad \mathcal {OV}({\mathbb {C}})\cong \mathcal {OV}_{+}({\mathbb {C}}),\quad \text {if}\;a_{n}\nearrow \infty , \end{aligned}$$

which is easily seen. Thus one may choose the most suitable sequence $(a_{n})_{n\in \mathbb {N}}$ for one’s purpose without changing the space.

Let us turn to the third part. The following quite technical conditions guarantee a kind of weak reducibility of the projective limit $\mathcal {EV}(\varOmega )$ and in combination with (PN.1) the weak reducibility of $\mathcal {OV}(\varOmega )$ too.

Condition (WR)

Let ${\mathcal {V}}:=(\nu _{n})_{n\in \mathbb {N}}$ be a directed family of continuous weights on an open set $\varOmega \subset \mathbb {R}^{2}$ and $(\varOmega _{n})_{n\in \mathbb {N}}$ a family of non-empty open sets such that $\varOmega _{n}\ne \mathbb {R}^{2}$, $\varOmega _{n}\subset \varOmega _{n+1}$ for all $n\in \mathbb {N}$, $\mathrm {d}_{n,k}:=\mathrm {d}^{|\cdot |}(\varOmega _{n},\partial \varOmega _{k})>0$ for all $n,k\in \mathbb {N}$, $k>n$, and $\varOmega =\bigcup _{n\in \mathbb {N}} \varOmega _{n}$.

(WR.1) For every $n\in \mathbb {N}$ let there be $g_{n}\in {\mathcal {O}}({\mathbb {C}})$ with $g_{n}(0)=1$ and $\mathbb {N}\ni I_{j}(n)> n$ such that

• (a) for every $\varepsilon >0$ there is a compact set $K\subset {\overline{\varOmega }}_{n}$ with $\nu _{n}(x)\le \varepsilon \nu _{I_{1}(n)}(x)$ for all $x\in \varOmega _{n}{\setminus } K$.

• (b) there is an open set $X_{I_{2}(n)}\subset \mathbb {R}^{2}{\setminus } {\overline{\varOmega }}_{I_{2}(n)}$ such that there are $R_{n},r_{n}\in \mathbb {R}$ with $0<2R_{n}<\mathrm {d}^{|\cdot |}(X_{I_{2}(n)},\varOmega _{I_{2}(n)}):=\mathrm {d}_{X,I_{2}(n)}$ and $R_{n}<r_{n}<\mathrm {d}_{X,I_{2}(n)}-R_{n}$ as well as $A_{2}(\cdot ,n):X_{I_{2}(n)}+\mathbb {B}_{R_{n}}(0)\rightarrow (0,\infty )$, $A_{2}(\cdot ,n)_{\mid X_{I_{2}(n)}}$ locally bounded, satisfying

  $$\begin{aligned} \max \{|g_{n}(\zeta )|\nu _{I_{2}(n)}(z)\;|\;\zeta \in \mathbb {R}^{2},\,|\zeta -(z-x)|=r_{n}\}\le A_{2}(x,n) \end{aligned}$$

  for all $z\in \varOmega _{I_{2}(n)}$ and $x\in X_{I_{2}(n)}+\mathbb {B}_{R_{n}}(0)$.

• (c) for every compact set $K\subset \mathbb {R}^{2}$ there is $A_{3}(n,K)>0$ with

  $$\begin{aligned} \int _{K}{\frac{|g_{n}(x-y)|\nu _{n}(x)}{|x-y|}\mathrm {d}y}\le A_{3}(n,K),\quad x\in \varOmega _{n}. \end{aligned}$$

(WR.2) Let (WR.1a) be fulfilled. For every $n\in \mathbb {N}$ let there be $\mathbb {N}\ni I_{4}(n)>n$ and $A_{4}(n)>0$ such that

$$\begin{aligned} \int _{\varOmega _{I_{4}(n)}}{\frac{|g_{I_{14}(n)}(x-y)|\nu _{p}(x)}{|x-y|\nu _{k}(y)}\mathrm {d}y}\le A_{4}(n), \quad x\in \varOmega _{p}, \end{aligned}$$

for $(k,p)=(I_{4}(n),n)$ and $(k,p)=(I_{14}(n),I_{14}(n))$ where $I_{14}(n):=I_{1}(I_{4}(n))$.

(WR.3) Let (WR.1a), (WR.1b) and (WR.2) be fulfilled. For every $n\in \mathbb {N}$, every closed subset $M\subset {\overline{\varOmega }}_{n}$ and every component N of $M^{C}$ we have

$$\begin{aligned} N\cap {\overline{\varOmega }}_{n}^{C}\ne \varnothing \;\Rightarrow \; N\cap X_{I_{214}(n)}\ne \varnothing \end{aligned}$$

where $I_{214}(n):=I_{2}(I_{14}(n))$, fulfilling $I_{214}(n)\ge I_{14}(n+1)$.

(WR) is [33, 4.2 Condition, p. 10] combined with the assumption $I_{214}(n)\ge I_{14}(n+1)$, $n\in \mathbb {N}$. We will see that $\varOmega _{n}:=\{z\in {\mathbb {C}}\;|\;|{{\,\mathrm{Im}\,}}(z)|<n\}$ and $\nu _{n}(z):=\exp (a_{n}|{{\,\mathrm{Re}\,}}(z)|^{\gamma })$ for some $0<\gamma \le 1$ and $a_{n}\nearrow 0$ or $a_{n}\nearrow \infty$ fulfil the conditions above with $g_{n}(z):=\exp (-z^2)$.

Theorem 12

[33, 4.3 Theorem, p. 10] Let $n\in \mathbb {N}$. Then $\pi _{I_{214}(n),n}({\mathcal {E}}\nu _{I_{214}(n),{\overline{\partial }}} (\varOmega _{I_{214}(n)}))$ is dense in $\pi _{I_{14}(n),n}({\mathcal {E}}\nu _{I_{14}(n), {\overline{\partial }}}(\varOmega _{I_{14}(n)}))$ w.r.t. $(|\cdot |_{n,m})_{m\in \mathbb {N}_{0}}$ if (WR) is fulfilled.

As a consequence of this theorem, whose proof does not need the assumption $I_{214}(n)\ge I_{14}(n+1)$, we obtain that the projective limit $\mathcal {OV}(\varOmega )$ is weakly reduced, which is a generalisation of [32, 5.6 Corollary, p. 69] and [32, 5.11 Corollary, p. 75].

Corollary 13

$\mathcal {OV}(\varOmega )$ is weakly reduced if (WR) and (PN.1) are satisfied.

Proof

Let $n\in \mathbb {N}$. We show that $\pi _{n}(\mathcal {OV}(\varOmega ))$ is dense in $\pi _{2J_{1}I_{14}(n),n}({\mathcal {O}} \nu _{2J_{1}I_{14}(n)}(\varOmega _{2J_{1}I_{14}(n)}))$ w.r.t. $|\cdot |_{n}$ where $J_{1}I_{14}(n):=J_{1}(I_{14}(n))$ and

$$\begin{aligned} \pi _{n}:\mathcal {OV}(\varOmega )\rightarrow {\mathcal {O}}\nu _{n} (\varOmega _{n}),\;\pi _{n}(f):=f_{\mid \varOmega _{n}}. \end{aligned}$$

We omit the restriction maps in our proof. Due to Proposition 7 (a) the restrictions to $\varOmega _{I_{14}(n)}$ of functions from ${\mathcal {O}}\nu _{2J_{1}I_{14}(n)}(\varOmega _{2J_{1}I_{14}(n)})$ are elements of ${\mathcal {E}}\nu _{I_{14}(n),{\overline{\partial }}}(\varOmega _{I_{14}(n)})$. Let $\varepsilon >0$ and $f_{0}\in {\mathcal {O}}\nu _{2J_{1}I_{14}(n)}(\varOmega _{2J_{1}I_{14}(n)})$. For every $j\in \mathbb {N}$ there exists

• (i) $f_{j}\in {\mathcal {E}}\nu _{I_{214}(n+j-1),{\overline{\partial }}}(\varOmega _{I_{214}(n+j-1)})$ with

• (ii) ${f_{j}}_{\mid \varOmega _{I_{14}(n+j)}}\in {\mathcal {E}}\nu _{I_{14}(n+j),{\overline{\partial }}}(\varOmega _{I_{14}(n+j)}) \subset {\mathcal {O}}\nu _{I_{14}(n+j)}(\varOmega _{I_{14}(n+j)})$

such that

$$\begin{aligned} |f_{j}-f_{j-1}|_{n+j-1}=|f_{j}-f_{j-1}|_{n+j-1,0}<\frac{\varepsilon }{2^{j+1}} \end{aligned}$$

(22)

by Theorem 12 and the condition $I_{214}(k)\ge I_{14}(k+1)$ for all $k\in \mathbb {N}$ from (WR). Therefore we obtain for every $k\in \mathbb {N}$

(23)

Now, let $\varepsilon _{0}>0$ and $l\in \mathbb {N}$. We choose $l_{0}\in \mathbb {N}$, $l_{0}\ge l$, such that $\frac{\varepsilon }{2^{l_{0}+1}}<\varepsilon _{0}$. Similarly, we get for all $p\ge k\ge l_{0}$

Hence $(f_{k})_{k\ge n_{0}}$ is a Cauchy sequence in the Banach space ${\mathcal {O}}\nu _{I_{14}(n+n_{0})}(\varOmega _{I_{14}(n+n_{0})})$ for every $n_{0}\in \mathbb {N}_{0}$ and thus has a limit $F_{n_{0}}\in {\mathcal {O}}\nu _{I_{14}(n+n_{0})}(\varOmega _{I_{14}(n+n_{0})})$. These limits coincide on their common domain because for every $n_{1},n_{2}\in \mathbb {N}_{0}$ with $I_{14}(n+n_{1})<I_{14}(n+n_{2})$ and $\varepsilon _{1}>0$ there exists $N\in \mathbb {N}$ such that for all $k\ge N$

$$\begin{aligned} |F_{n_{1}}-F_{n_{2}}|_{I_{14}(n+n_{1})}&\le |F_{n_{1}}-f_{k}|_{I_{14}(n+n_{1})}+|f_{k}-F_{n_{2}}|_{I_{14}(n+n_{1})}\\&\le |F_{n_{1}}-f_{k}|_{I_{14}(n+n_{1})}+|f_{k}-F_{n_{2}}|_{I_{14}(n+n_{2})} <\frac{\varepsilon _{1}}{2}+\frac{\varepsilon _{1}}{2}=\varepsilon _{1}. \end{aligned}$$

We deduce that the glued limit function f given by $f:=F_{n_{0}}$ on $\varOmega _{I_{14}(n+n_{0})}$ for all $n_{0}\in \mathbb {N}_{0}$ is well-defined and we have $f\in \bigcap _{n_{0}\in \mathbb {N}_{0}}{\mathcal {O}}\nu _{I_{14}(n+n_{0})}(\varOmega _{I_{14}(n+n_{0})}) =\mathcal {OV}(\varOmega )$ since $I_{14}(n+n_{0})\ge n+n_{0}$. By the definition of f there exists $N\in \mathbb {N}$ such that for every $k\ge N$

$$\begin{aligned} |f-f_{0}|_{n}\le |f-f_{k}|_{n}+|f_{k}-f_{0}|_{n} \underset{n\le I_{14}(n+0)}{<}\frac{\varepsilon }{2}+|f_{k}-f_{0}|_{n} \underset{(23)}{\le }\frac{\varepsilon }{2}+\frac{\varepsilon }{2}=\varepsilon , \end{aligned}$$

which proves our statement. $\square$

Combining Theorem 10 and Corollary 13, we obtain the following corollary.

Corollary 14

Let $a_{n}\nearrow 0$ or $a_{n}\nearrow \infty$, ${\mathcal {V}}:=(\exp (a_{n}\mu ))_{n\in \mathbb {N}}$ and $\varOmega _{n}:=S_{n}$ for all $n\in \mathbb {N}$ where

$$\begin{aligned} \mu :{\mathbb {C}}\rightarrow [0,\infty ),\;\mu (z):=|{{\,\mathrm{Re}\,}}(z)|^{\gamma }, \end{aligned}$$

for some $0<\gamma \le 1$. Then $\mathcal {OV}({\mathbb {C}})$ satisfies $(\varOmega )$.

Proof

We only need to check that the conditions of Theorem 10 are fulfilled. Obviously, $\mu (z)=\mu (|{{\,\mathrm{Re}\,}}(z)|)$ for all $z\in {\mathbb {C}}$, $\mu$ is strictly increasing on $[0,\infty )$ and $\lim _{x\rightarrow \infty ,\, x\in \mathbb {R}}\frac{\ln (1+|x|)}{\mu (x)}=0$. The observation

$$\begin{aligned} \mu (x+n)-\mu (x)=|x+n|^{\gamma }-|x|^{\gamma }\le |x+n-n|^{\gamma }=n^{\gamma },\quad n\in \mathbb {N},\;x\in [0,\infty ), \end{aligned}$$

implies that $\mu$ is a strong weight generator with any $\varGamma >1$ and $C:=n^{\gamma }$ by Definition 8. Let us turn to the conditions (WR) and (PN.1) which we need for the weak reducibility of $\mathcal {OV}({\mathbb {C}})$ by Corollary 13. Condition (PN.1) is fulfilled by Example 6. If $a_{n}<0$ for all $n\in \mathbb {N}$, then (WR) is fulfilled by [33, 4.10 Example a), p. 22] where we used ${\widetilde{\mu }}(z):=|z|^{\gamma }$ instead of $\mu$, which does not make a difference since

$$\begin{aligned} |{{\,\mathrm{Re}\,}}(z)|^{\gamma }\le |z|^{\gamma }\le |{{\,\mathrm{Re}\,}}(z)|^{\gamma }+n^{\gamma },\quad z\in \varOmega _{n}=S_{n}. \end{aligned}$$

If $a_{n}\ge 0$ for all $n\in \mathbb {N}$, we only have to modify [33, 4.10 Example a), p. 22] a bit. We choose $I_{j}(n):=2n$ for $j\in \{1,2,4\}$ and define the open set $X_{I_{2}(n)}:={\overline{S}}_{4n}^{C}$. Then we have

$$\begin{aligned} I_{214}(n)=8n\ge 4n+4=I_{14}(n+1),\quad n\in \mathbb {N}. \end{aligned}$$

Furthermore, we have $\mathrm {d}_{n,k}=|n-k|$ for all $n,k\in \mathbb {N}$.

(WR.1a) and (WR.3): Verbatim as in [33, 4.10 Example a), p. 22].

(WR.1b): We have $\mathrm {d}_{X,I_{2}}=2n$. We choose $g_{n}:{\mathbb {C}}\rightarrow {\mathbb {C}}$, $g_{n}(z):=\exp (-z^2)$, as well as $r_{n}:=1/(4n)$ and $R_{n}:=1/(6n)$ for $n\in \mathbb {N}$. Let $z=z_{1}+iz_{2}\in \varOmega _{I_{2}(n)}=S_{2n}$ and $x\in X_{I_{2}(n)}+\mathbb {B}_{R_{n}}(0)$. For $\zeta =\zeta _{1}+i\zeta _{2}\in {\mathbb {C}}$ with $|\zeta -(z-x)|=r_{n}$ we have

$$\begin{aligned} |g_{n}(\zeta )|e^{a_{2n}\mu (z)}&=e^{-{{\,\mathrm{Re}\,}}(\zeta ^{2})}e^{a_{2n}|{{\,\mathrm{Re}\,}}(z)|^{\gamma }} \le e^{-\zeta _{1}^{2}+\zeta _{2}^{2}}e^{a_{2n}(1+|z_{1}|)}\\&\le e^{(r_{n}+|z_{2}|+|x_{2}|)^{2}+a_{2n}(1+r_{n}+|x_{1}|)}e^{-|\zeta _{1}|^{2}+a_{2n}|\zeta _{1}|}\\&\le e^{(r_{n}+2n+|x_{2}|)^{2}+a_{2n}(1+r_{n}+|x_{1}|)}\sup _{t\in \mathbb {R}}e^{-t^{2}+a_{2n}t}\\&= e^{(r_{n}+2n+|x_{2}|)^{2}+a_{2n}(1+r_{n}+|x_{1}|)+a_{2n}^{2}/4}=:A_{2}(x,n) \end{aligned}$$

and observe that $A_{2}(\cdot ,n)$ is continuous and thus locally bounded on $X_{I_{2}(n)}$.

(WR.1c): Let $K\subset {\mathbb {C}}$ be compact and $x=x_{1}+ix_{2}\in \varOmega _{n}$. Then there is $b>0$ such that $|y|\le b$ for all $y=y_{1}+iy_{2}\in K$ and from polar coordinates and Fubini’s theorem it follows that

$$\begin{aligned}&\int _{K}\frac{|g_{n}(x-y)|}{|x-y|}\mathrm {d}y\\&\quad \le \underbrace{\sup _{w\in K}e^{a_{2n}|{{\,\mathrm{Re}\,}}(w)|}}_{=:C_{1}} \int _{K}\frac{e^{-{{\,\mathrm{Re}\,}}((x-y)^{2})}}{|x-y|}e^{-a_{2n}|y_{1}|}\mathrm {d}y\\&\quad \le C_{1}\bigl ( \int _{\mathbb {B}_{1}(x)}\frac{e^{-{{\,\mathrm{Re}\,}}((x-y)^{2})}}{|x-y|}e^{-a_{2n}|{{\,\mathrm{Re}\,}}(y)|}\mathrm {d}y +\int _{K{\setminus } \mathbb {B}_{1}(x)}\frac{e^{-{{\,\mathrm{Re}\,}}((x-y)^{2})}}{|x-y|}e^{-a_{2n}|{{\,\mathrm{Re}\,}}(y)|}\mathrm {d}y\bigr )\\&\quad \le C_{1}\bigl (\int _{0}^{2\pi }\int _{0}^{1}\frac{e^{-r^{2}\cos (2\varphi )}}{r}e^{-a_{2n}|x_{1}+r\cos (\varphi )|}r\mathrm {d}r\mathrm {d}\varphi + \int _{K{\setminus } \mathbb {B}_{1}(x)}e^{-{{\,\mathrm{Re}\,}}((x-y)^{2})}e^{-a_{2n}|{{\,\mathrm{Re}\,}}(y)|}\mathrm {d}y\bigr )\\&\quad \le C_{1}\bigl (2\pi e^{1+a_{2n}}e^{-a_{2n}|x_{1}|} +\int _{-b}^{b}e^{(x_{2}-y_{2})^{2}}\mathrm {d}y_{2} \int _{\mathbb {R}}e^{-(x_{1}-y_{1})^{2}+a_{2n}|x_{1}-y_{1}|}\mathrm {d}y_{1}e^{-a_{2n}|x_{1}|}\bigr )\\&\quad \le C_{1}\bigl (2\pi e^{1+a_{2n}}+2be^{(|x_{2}|+b)^{2}}\int _{\mathbb {R}}e^{-y_{1}^{2}+a_{2n}|y_{1}|}\mathrm {d}y_{1}\bigr )e^{-a_{2n}|x_{1}|}\\&\quad = C_{1}\bigl (2\pi e^{1+a_{2n}}+2be^{(|x_{2}|+b)^{2}}e^{a_{2n}^{2}/4} \int _{\mathbb {R}}e^{-(|y_{1}|-a_{2n}/2)^{2}}\mathrm {d}y_{1}\bigr )e^{-a_{2n}|x_{1}|}\\&\quad = C_{1}\bigl (2\pi e^{1+a_{2n}}+4be^{(|x_{2}|+b)^{2}}e^{a_{2n}^{2}/4} \int _{-a_{2n}/2}^{\infty }e^{-y_{1}^{2}}\mathrm {d}y_{1}\bigr )e^{-a_{2n}|x_{1}|}\\&\quad \le C_{1}\bigl (2\pi e^{1+a_{2n}}+4\sqrt{\pi }be^{(n+b)^{2}+a_{2n}^{2}/4}\bigr )e^{-a_{2n}|x_{1}|}. \end{aligned}$$

We conclude that (WR.1c) holds since

$$\begin{aligned} e^{-a_{2n}|x_{1}|}e^{a_{n}|{{\,\mathrm{Re}\,}}(x)|^{\gamma }}\le e^{(a_{n}-a_{2n})|x_{1}|+a_{n}}\le e^{a_{n}}. \end{aligned}$$

(WR.2): Let $p,k\in \mathbb {N}$ with $p\le k$. For all $x=x_{1}+ix_{2}\in \varOmega _{p}$ and $y=y_{1}+iy_{2}\in \varOmega _{I_{4}(n)}$ we note that

$$\begin{aligned} a_{p}|{{\,\mathrm{Re}\,}}(x)|^{\gamma }-a_{k}|{{\,\mathrm{Re}\,}}(y)|^{\gamma } \le a_{k}|x_{1}-y_{1}|^{\gamma }\le a_{k}(1+|x_{1}-y_{1}|) \end{aligned}$$

because $(a_{n})_{n\in \mathbb {N}}$ is non-negative and increasing and $0<\gamma \le 1$. Like before we deduce that

$$\begin{aligned}&\int _{\varOmega _{I_{4}(n)}}\frac{|g_{n}(x-y)|\nu _{p}(x)}{|x-y|\nu _{k}(y)}\mathrm {d}y\\&=\int _{\varOmega _{2n}}\frac{e^{-{{\,\mathrm{Re}\,}}((x-y)^{2})}}{|x-y|}e^{a_{p}|{{\,\mathrm{Re}\,}}(x)|^{\gamma }-a_{k}|{{\,\mathrm{Re}\,}}(y)|^{\gamma }}\mathrm {d}y \le \int _{\varOmega _{2n}}\frac{e^{-{{\,\mathrm{Re}\,}}((x-y)^{2})}}{|x-y|}e^{a_{k}|{{\,\mathrm{Re}\,}}(x)-{{\,\mathrm{Re}\,}}(y)|^{\gamma }}\mathrm {d}y\\&\le \int _{0}^{2\pi }\int _{0}^{1}\frac{e^{-r^{2}\cos (2\varphi )}}{r}e^{a_{k}r^{\gamma }}r\mathrm {d}r\mathrm {d}\varphi + \int _{\varOmega _{2n}{\setminus } \mathbb {B}_{1}(x)}e^{-{{\,\mathrm{Re}\,}}((x-y)^{2})}e^{a_{k}|{{\,\mathrm{Re}\,}}(x)-{{\,\mathrm{Re}\,}}(y)|^{\gamma }}\mathrm {d}y\\&\le 2\pi e^{1+a_{k}}+e^{a_{k}}\int _{-2n}^{2n}e^{(x_{2}-y_{2})^{2}}\mathrm {d}y_{2} \int _{\mathbb {R}}e^{-(x_{1}-y_{1})^{2}+a_{k}|x_{1}-y_{1}|}\mathrm {d}y_{1}\\&\le 2\pi e^{1+a_{k}}+8\sqrt{\pi }ne^{a_{k}+(|x_{2}|+2n)^{2}+a_{k}^{2}/4}\\&\le 2\pi e^{1+a_{I_{14}(n)}}+8\sqrt{\pi }ne^{a_{I_{14}(n)}+(I_{14}(n)+2n)^{2}+a_{I_{14}(n)}^{2}/4} \end{aligned}$$

for $(k,p)=(I_{4}(n),n)$ and $(k,p)=(I_{14}(n),I_{14}(n))$ as $(a_{n})_{n\in \mathbb {N}}$ is non-negative and increasing. $\square$

We close this section with a special case of our main result on the surjectivity of the Cauchy–Riemann operator on $\mathcal {EV}(\varOmega ,E)$. We recall the corresponding result for $E={\mathbb {C}}$ which we will need for the application of our main result. It is a consequence of the approximation Theorem 12 in combination with Hörmander’s solution of the ${\overline{\partial }}$-problem in weighted $L^{2}$-spaces [27, Theorem 4.4.2, p. 94] and the Mittag–Leffler procedure.

Theorem 15

[33, 4.8 Theorem, p. 20] Let (PN) with $\psi _{n}(z):=(1+|z|^{2})^{-2}$, $z\in \varOmega$, and (WR) be fulfilled and $-\ln \nu _{n}$ be subharmonic on $\varOmega$ for every $n\in \mathbb {N}$. Then

$$\begin{aligned} {\overline{\partial }}:\mathcal {EV}(\varOmega )\rightarrow \mathcal {EV}(\varOmega ) \end{aligned}$$

is surjective.

A consequence of this theorem is the following corollary.

Corollary 16

[33, 4.10 Example a), p. 22] Let $(a_{n})_{n\in \mathbb {N}}$ be strictly increasing, $a_{n}<0$ for all $n\in \mathbb {N}$, ${\mathcal {V}}:=(\exp (a_{n}\mu ))_{n\in \mathbb {N}}$ and $\varOmega _{n}:=\{z\in {\mathbb {C}}\;|\;|{{\,\mathrm{Im}\,}}(z)|<n\}$ for all $n\in \mathbb {N}$ where

$$\begin{aligned} \mu :{\mathbb {C}}\rightarrow [0,\infty ),\;\mu (z):=|{{\,\mathrm{Re}\,}}(z)|^{\gamma }, \end{aligned}$$

for some $0<\gamma \le 1$. Then

$$\begin{aligned} {\overline{\partial }}:\mathcal {EV}({\mathbb {C}})\rightarrow \mathcal {EV}({\mathbb {C}}) \end{aligned}$$

is surjective.

The restriction to negative $a_{n}$ comes from the condition that $-\ln \nu _{n}$ should be subharmonic. We note that the E-valued versions of Theorem 15 and Corollary 16 where E is a Fréchet space over ${\mathbb {C}}$ hold as well by the classical theory of tensor products for nuclear Fréchet spaces (see [33, 4.9 Corollary, p. 21]). Now, we use the results obtained so far to obtain a special case of our main result.

Corollary 17

Let $\mu$ be a subharmonic strong weight generator and ${\mathcal {V}}:=(\exp (a_{n}\mu ))_{n\in \mathbb {N}}$ with $a_{n}\nearrow 0$. Let (PN) with $\psi _{n}(z):=(1+|z|^{2})^{-2}$, $z\in {\mathbb {C}}$, and (WR) with $\varOmega _{n}:=\{z\in {\mathbb {C}}\;|\;|{{\,\mathrm{Im}\,}}(z)|<n\}$ for all $n\in \mathbb {N}$ be fulfilled. If

• (a) $E:=F_{b}'$ where F is a Fréchet space over ${\mathbb {C}}$ satisfying (DN), or

• (b) E is an ultrabornological PLS-space over ${\mathbb {C}}$ satisfying (PA),

then

$$\begin{aligned} {\overline{\partial }}^{E}:\mathcal {EV}({\mathbb {C}},E)\rightarrow \mathcal {EV}({\mathbb {C}},E) \end{aligned}$$

is surjective.

Proof

The space $\mathcal {EV}({\mathbb {C}})$ is nuclear, in particular Schwartz, by [37, Theorem 3.1, p. 188], [37, Remark 2.7, p. 178-179] and [37, Remark 2.3 (b), p. 177] because (PN.1) and $(PN.2)^{1}$ from (PN) are fulfilled. Hence the subspace $\mathcal {EV}_{{\overline{\partial }}}({\mathbb {C}})=\mathcal {OV}({\mathbb {C}})$ is nuclear by Proposition 7 (b) as well. Further, $\mathcal {OV}({\mathbb {C}})$ is weakly reduced by Corollary 13 due to (WR) and thus satisfies $(\varOmega )$ by Theorem 10. Therefore, the assertion is a consequence of the surjectivity of ${\overline{\partial }}$ in the ${\mathbb {C}}$-valued case by Theorem 15 and our main result Theorem 5. $\square$

Corollary 17 generalises a part of [32, 5.24 Theorem, p. 95] ($K=\varnothing$) which is the case $\gamma =1$ of the next corollary.

Corollary 18

Let $a_{n}\nearrow 0$, ${\mathcal {V}}:=(\exp (a_{n}\mu ))_{n\in \mathbb {N}}$ and $\varOmega _{n}:=\{z\in {\mathbb {C}}\;|\;|{{\,\mathrm{Im}\,}}(z)|<n\}$ for all $n\in \mathbb {N}$ where

$$\begin{aligned} \mu :{\mathbb {C}}\rightarrow [0,\infty ),\;\mu (z):=|{{\,\mathrm{Re}\,}}(z)|^{\gamma }, \end{aligned}$$

for some $0<\gamma \le 1$. If

• (a) $E:=F_{b}'$ where F is a Fréchet space over ${\mathbb {C}}$ satisfying (DN), or

• (b) E is an ultrabornological PLS-space over ${\mathbb {C}}$ satisfying (PA),

then

$$\begin{aligned} {\overline{\partial }}^{E}:\mathcal {EV}({\mathbb {C}},E)\rightarrow \mathcal {EV}({\mathbb {C}},E) \end{aligned}$$

is surjective.

Proof

Follows from Corollary 17, (the proof of) Corollary 14 and Example 6. $\square$