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

Let Ω be an open subset of 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 EV (Ω) of smooth functions and on the space E to ensure that the vector-valued Cauchy–Riemann operator ∂ : EV (Ω, E ) → EV (Ω, 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.


Introduction
Let E be a linear space of functions on a set U and P(∂) : F (Ω) → F (Ω) be a linear partial differential operator with constant coefficients which acts continuously on a locally convex Hausdorff space of (generalized) differentiable scalar-valued functions F (Ω) on an open set Ω ⊂ R n . We call the elements of U parameters and say that a family ( f λ ) λ∈U in F (Ω) depends on a parameter w.r.t. E if the map λ → f λ (x) is an element of E for every x ∈ Ω. The question of parameter dependence is whether for every family ( f λ ) λ∈U in F (Ω) depending on a parameter w.r.t. E there is a family (u λ ) λ∈U in F (Ω) with the same kind of parameter dependence which solves the partial differential equation In particular, it is the question of C k -smooth (holomorphic, distributional, etc.) parameter dependence if E is the space The question of parameter dependence has been subject of extensive research varying in the choice of the spaces E, F (Ω) and the properties of the partial differential operator P(∂), e.g. being (hypo)elliptic, parabolic or hyperbolic. Even partial differential differential operators P λ (∂) where the coefficients also depend C k ([0, 1])-smoothly [49], C ∞ -smoothly [61], holomorphically [50,61] or differentiable resp. real analytic [13] on the parameter λ were considered. The case that the coefficients of the partial differential differential operator P(x, ∂) are non-constant functions in x ∈ Ω was treated for F (Ω) = A (R n ), the space of real analytic functions on R n , as well [3].
The answer to the question of C k -smooth (holomorphic, distributional, etc.) parameter dependence is obviously affirmative if P(∂) has a linear continuous right inverse. The problem to determine those P(∂) which have such a right inverse was posed by Schwartz in the early 1950s (see [21, p. 680]). In the case that F (Ω) is the space of C ∞ -smooth functions or distributions on an open set Ω ⊂ 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 F (Ω) is a weighted space of C ∞ -smooth functions on Ω = R n or its dual was handled in [40], even for some P(x, ∂) with smooth coefficients, the case of tempered distributions in [38] and of Fourier (ultra-)hyperfunctions in [44,45]. For Hörmander's spaces B loc p,κ (Ω) as F (Ω) the problem was studied in [25]. The necessary condition of surjectivity of the partial differential operator P(∂) was studied in many papers, e.g. in [1,23,28,48,67] on C ∞ -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 However, if P(∂) : C ∞ (Ω) → C ∞ (Ω), Ω ⊂ R n open, is elliptic, then P(∂) has a linear right inverse (by means of a Hamel basis of C ∞ (Ω)) 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(∂) has no linear continuous right inverse if n ≥ 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 is surjective where C ∞ (Ω, E) is the space of C ∞ -smooth E-valued functions on Ω and P(∂) E the version of P(∂) for E-valued functions. From Grothendieck's classical theory of tensor products [24] and the surjectivity of P(∂) it follows that P(∂) E is also surjective if E is a Fréchet space. In general, Grothendieck's theory of tensor products can be applied if P(∂) is surjective and F (Ω) a nuclear Fréchet space. However, the surjectivity of P(∂) E , n ≥ 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, ker P(∂) has the property (Ω) and E is the dual of a Fréchet space with the property (DN ) or an ultrabornological PLS-space with the property (P A). The splitting theory of Bonet and Domański can also be applied if F (Ω) is a non-Fréchet PLS-space and for PLH-spaces F (Ω), e.g. D L 2 and B loc 2,κ (Ω) 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 F (Ω) 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 F (Ω) is the space of C ∞ -smooth functions or distributions.
Notably, the preceding results imply that the inhomogeneous Cauchy-Riemann equation with a right-hand side f ∈ E(Ω, E) := C ∞ (Ω, E), where Ω ⊂ R 2 is open and E a locally convex Hausdorff space over C whose topology is induced by a system of seminorms ( p α ) α∈A , given by or if E is an ultrabornological PLS-space having the property (P A). Among these spaces E are several spaces of distributions like 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 for every n ∈ N, m ∈ N 0 and α ∈ A. Let us call the space of such functions EV(Ω, E). Our interest is in conditions on V and (Ω n ) n∈N such that there is a solution u ∈ EV(Ω, E) of (1), i.e. we search for conditions that guarantee the surjectivity of 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 (Ω n ) n∈N , we concentrate on the case that it is a sequence of strips along the real axis, i.e. Ω n := {z ∈ C | | 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 (Ω), (DN ) and (P A) and provide some examples of spaces E having these invariants. Then we prove our main result on the surjectivity of Cauchy-Riemann operator on EV(Ω, E) which depends on these invariants (see Theorem 5). To apply our main result, the kernel ker ∂ needs to have (Ω) and in Sect. 4 we provide sufficient conditions on the weights and the sequence (Ω n ) n∈N which guarantee (Ω) (see Theorem 10 and Corollary 13). We close this section with a special case of our main theorem where (Ω n ) n∈N is a sequence of strips along the real axis (see Corollary 17) and for example ν n (z) := exp(a n | Re(z)| γ ) for some 0 < γ ≤ 1 and a n 0 (see Corollary 18).

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 ⊂ R 2 w.r.t. a norm · on R 2 via Moreover, we denote by · ∞ the sup-norm, by | · | the Euclidean norm on R 2 , 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 L 1 (Ω) we denote the space of (equivalence classes of) C-valued Lebesgue integrable functions on a measurable set Ω ⊂ R 2 and by L q (Ω), q ∈ N, the space of functions f such that f q ∈ L 1 (Ω). If (a n ) n∈N is a strictly increasing real sequence, we write a n 0 resp. a n ∞ if a n < 0 for all n ∈ N and lim n→∞ a n = 0 resp. a n ≥ 0 for all n ∈ N and lim n→∞ a n = ∞.
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 := {| · |}. 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 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 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 = σ , on the precompact subsets of F if t = γ , on the absolutely convex, compact subsets of F if t = κ and on the bounded subsets of The so-called ε-product of Schwartz is defined by where L(F κ , E) is equipped with the topology of uniform convergence on equicontinuous subsets of F . This definition of the ε-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 : Ω → E on an open set Ω ⊂ R 2 to E is called continuously partially differentiable ( f is C 1 ) if for the n-th unit vector e n ∈ R 2 the limit β n -times f if β n = 0 as well as 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 ∂ β f : exists in E for every z 0 ∈ Ω and the space of such functions is denoted by O(Ω, E). The exact definition of the spaces from the introduction is as follows.
The subscript α in the notation of the seminorms is omitted in the C-valued case. The letter E is omitted in the case E = C as well, e.g. we write Eν n (Ω n ) := Eν n (Ω n , C) and EV(Ω) := EV(Ω, C) .
A projective limit F of a sequence of locally convex Hausdorff spaces (F n ) n∈N is called weakly reduced if for every n ∈ N there is m ∈ N such that π n (F) is dense in F m w.r.t. the topology of F n where π n : F → F n is the canonical projection. The spaces FV(Ω, E), F = E, O, are projective limits, namely, we have where the spectral maps are given by the restrictions

Main result
In this section we prove our main result that the surjectivity of the vector-valued Cauchy-Riemann operator on EV(Ω, E) is inherited from the surjectivity on EV(Ω) if the kernel EV ∂ (Ω) in the scalar-valued case has (Ω), and E := F b where F is a Fréchet space satisfying the condition (DN ) or E is an ultrabornological PLS-space having the property (P A). Therefore we recall the definitions of the topological invariants (Ω), (DN ) and (P A) and give some examples.
A Fréchet space F with an increasing fundamental system of seminorms where the X N given by inductive limits are DFS-spaces (which are also called LS-spaces), and it satisfies  Let us summarise some examples of ultrabornological PLS-spaces satisfying (P A) 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].
The following spaces are ultrabornological PLS-spaces with property (P A): , Y has (Ω) and both are nuclear Fréchet spaces.
(c) The following spaces are strong duals of a Fréchet space satisfying (DN ): Since we will use the ε-product EV(Ω)ε E to pass the surjectivity from ∂ to ∂ 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
where the topological isomorphism is the identity map.

Theorem 5 Let EV(Ω) be a Schwartz space and EV ∂ (Ω) a nuclear subspace satisfying property (Ω). Assume that the scalar-valued operator
is surjective.
Proof Throughout this proof we use the notation X : is a topological isomorphism by [36, 3.21 Example b), p. 14] where δ z is the point-evaluation at z ∈ Ω. We denote by J : E → E * the canonical injection in the algebraic dual E * of the topological dual E and for f ∈ EV(Ω, E) we set Then where i means the inclusion, is a topologically exact sequence of Fréchet spaces because ∂ is surjective by assumption. Let us denote by J 0 : EV ∂ (Ω) → EV ∂ (Ω) and J 1 : EV(Ω) → EV(Ω) the canonical embeddings which are topological isomorphisms since EV ∂ (Ω) and EV(Ω) are reflexive. Then the exactness of (4) implies that , is an exact topological sequence. Topological as the (strong) bidual of a Fréchet space is again a Fréchet space by [54,Corollary 25.10 Combined with the exactness of (5) this implies that the sequence EV(Ω) ). In particular, we obtain that is surjective. Via E = F b and Proposition 4 (X = EV(Ω) and Y = F) we have the topological isomorphism and the inverse Let g ∈ EV(Ω, E). Then ψ −1 (g) ∈ L(F, EV(Ω) ) and by the surjectivity of (6) there is u ∈ L(F, EV(Ω) ) such that ∂ * 1 u = ψ −1 (g). So we get ψ(u) ∈ EV(Ω, E). Next, we show that ∂ E ψ(u) = g is valid. Let x ∈ F, z ∈ Ω and h ∈ R, h = 0, and e k denote the kth unit vector in R 2 . From for every f ∈ EV(Ω) it follows that Thus ∂ E (ψ(u))(z) = g(z) for every z ∈ Ω, which proves the surjectivity.
(b) Let E be an ultrabornological PLS-space satisfying (P A). Since the nuclear Fréchet space EV ∂ (Ω) is also a Schwartz space, its strong dual EV ∂ (Ω) b is a DFS-space. By is exact. The maps i * 0 and ∂ * 1 are defined like in part (a). Especially, we get that is surjective.
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 EV ∂ (Ω) to have (Ω) as well as concrete examples of such spaces.

(Ä) for 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 OV(Ω) and EV ∂ (Ω) coincide topologically under mild assumptions on the weights V and the sequence of sets (Ω n ). These mild conditions also imply that EV(Ω) is nuclear, in particular Schwartz, and thus its subspace EV ∂ (Ω) = OV(Ω) too. Second, we reduce the problem whether the projective limit OV(Ω) has (Ω) to the problem whether it is weakly reduced in the case that the Ω n are strips along the real axis and the weights have a certain structure. Third, we use a similar result for EV ∂ (Ω) which was obtained in [33] to prove the weak reducibility of OV(Ω). For corresponding results in the case that Ω n = Ω for all n ∈ N see [ that the projective limit EV(Ω) 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 ∂-operator given in [33], using Hörmander's L 2 -machinery (if q = 2).

Condition (PN) ([33, 3.3 Condition, p. 7])
Let V := (ν n ) n∈N be a directed family of continuous weights on an open set Ω ⊂ R 2 and (Ω n ) n∈N a family of non-empty open sets such that Ω n ⊂ Ω n+1 and Ω = n∈N Ω n . For every k ∈ N let there be ρ k ∈ R such that 0 < ρ k < d · ∞ ({x}, ∂Ω k+1 ) for all x ∈ Ω k and let there be q ∈ N such that for any n ∈ N there is ψ n ∈ L q (Ω k ), ψ n > 0, and N J i (n) ≥ n and C i (n) > 0 such that for any x ∈ Ω k :
The space OV(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 γ 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.

(b) EV ∂ (Ω) = OV(Ω) as Fréchet spaces.
Proof (a) Let n ∈ N and m ∈ N 0 . We note that Ω n+1 ⊂ Ω 2J 1 (n) and is the |β|th complex derivative of f . Then we obtain via (P N.1) and Cauchy's inequality Let us come to the second part. Using special weight functions, strips along the real axis as Ω n and a decomposition theorem of Langenbruch, we will see that answering the question whether OV(Ω) satisfies the property (Ω) of Vogt boils down to answering whether the projective limit OV(Ω) is weakly reduced. The special weights we want to consider are generated by a function μ with the following properties.

Definition 8 (strong weight generator) A continuous function
If μ is a weight generator which fulfils the stronger condition then μ is called a strong weight generator.
for t > 0 and τ ∈ R with the strip S t := {z ∈ C | | Im(z)| < t} . Theorem 9 [46, Theorem 2.2, p. 225] 1 Let μ be a weight generator. There are t, K 1 , K 2 > 0 such that for any τ 0 < τ < τ 2 there is C 0 = C 0 (sign(τ )) such that for any 0 < 2t 0 < t < t 2 < t with 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  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 τ 0 , τ and τ 2 and the choice of C 0 we obtain and where a := ln(Γ ) (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 ≤ K 1 in Theorem 9 guarantee that the condition t 0 ≤ T 0 is satisfied. Looking at the condition t 0 ≤ T 1 := D a 2 B 1 (p. 232), we derive For the subsequent theorem we merge and modify the proofs of [60, Satz 2.2.3, p. 44] 3 (a n = n, n ∈ N, and μ a strong weight generator) and [32, 5.20 Theorem, p. 84] (a n = −1/n, n ∈ N, and μ = | Re(·)|).

Theorem 10
Let μ be a strong weight generator, a n 0 or a n ∞, V := (exp(a n μ)) n∈N and Ω n := S n for all n ∈ N. If OV(C) is weakly reduced, then OV(C) satisfies (Ω).

Proof
Since OV(C) is weakly reduced, for every n ∈ N there exists m n ∈ N such that π n (OV(C)) is dense in Oν m n (Ω m n ) w.r.t. the topology of Oν n (Ω n ) where is the canonical projection. Let p, k ∈ N. As (a n ) n∈N is strictly increasing and lim n→∞ a n = 0 or lim n→∞ a n = ∞, we may choose q ∈ 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 Γ ≥ e 1/4 and T ∈ R such that which also fulfils Let t 0 := m p T , t := qT , t 2 := max(q + 1, m k )T .
By the choice of q we have By the choice of q and (10) we get Further, we deduce from (11) that where μ := μ(·/T ). We note that for n := 1/T , where · is the ceiling function, there is C > 0 such that for all x ≥ 0 because μ is a strong weight generator. We conclude that μ is also a weight generator with the same Γ as μ which is independent of T . Moreover, from it follows by Theorem 9 that there are f j ∈ O(S t j ), j ∈ {0, 2}, such that and where f 0 ∈ O(S m p ), as well as where and the inclusion is justified by the identity theorem. Furthermore, for z ∈ S t 0 /T = S m p the equation holds, thus f = f 0 + f 2 on S m p . By virtue of the weak reducibility of OV(C) and the choice of m p , m k the following is valid: Now, we have to consider two cases. Let ε := C 1 e −Gr . For k ≤ p we get via (15) where the function f 2 ∈ OV(C) and thus is a holomorphic extension of the left-hand side on C. Hence we clearly have f = f 0 + f 2 and 2C 1 e −Gr =: C 2 e −Gr (17) as well as C 1 e −Gr + e r ≤ (C 1 + 1)e r =: C 3 e r .
Analogously, for k > p we obtain via (15) where the function f 0 ∈ OV(C) and thus is a holomorphic extension of the left-hand side on C. Hence we clearly have f = f 0 + f 2 and 2C 1 e −Gr = C 2 e −Gr (20) as well as C 1 e −Gr + e r ≤ C 3 e r .
Next, we set n := 1/G and C := C 3 e ln(C 2 )/G . Let r > 0. For r ≥ 1 there is r ≥ 0 such that r = e Gr−ln(C 2 ) = e Gr C 2 and we have by (17) and (18) for k ≤ p as well as by (20) and (21) for k > p For 0 < r < 1 we have, since q ≥ p, Thus our statement is proved.
Let us remark that the choice of the sequence (a n ) n∈N in the preceding theorem does not really matter.

Remark 11
Let μ : C → [0, ∞) be continuous, a n 0 or a n ∞, V := (exp(a n μ)) n∈N and Ω n := S n for all n ∈ N. Set V − := (exp((−1/n)μ)) n∈N and V + := (exp(nμ)) n∈N . Then which is easily seen. Thus one may choose the most suitable sequence (a n ) n∈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 EV(Ω) and in combination with (P N.1) the weak reducibility of OV(Ω) too. (W R.1) For every n ∈ N let there be g n ∈ O(C) with g n (0) = 1 and N I j (n) > n such that (a) for every ε > 0 there is a compact set K ⊂ Ω n with ν n (x) ≤ εν I 1 (n) (x) for all x ∈ Ω n \K .
As a consequence of this theorem, whose proof does not need the assumption I 214 (n) ≥ I 14 (n + 1), we obtain that the projective limit OV(Ω) is weakly reduced, which is a generalisation of [32, 5.6 Corollary, p. 69] and [32,5.11 Corollary,p. 75].
and observe that A 2 (·, n) is continuous and thus locally bounded on X I 2 (n) .
(W R.1c): Let K ⊂ C be compact and x = x 1 + i x 2 ∈ Ω n . Then there is b > 0 such that |y| ≤ b for all y = y 1 + iy 2 ∈ K and from polar coordinates and Fubini's theorem it follows that We conclude that (W R.1c) holds since e −a 2n |x 1 | e a n | Re(x)| γ ≤ e (a n −a 2n )|x 1 |+a n ≤ e a n .
(W R.2): Let p, k ∈ N with p ≤ k. For all x = x 1 + i x 2 ∈ Ω p and y = y 1 + iy 2 ∈ Ω I 4 (n) we note that because (a n ) n∈N is non-negative and increasing and 0 < γ ≤ 1. Like before we deduce that for (k, p) = (I 4 (n), n) and (k, p) = (I 14 (n), I 14 (n)) as (a n ) n∈N is non-negative and increasing.
We close this section with a special case of our main result on the surjectivity of the Cauchy-Riemann operator on EV(Ω, E). We recall the corresponding result for E = 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 ∂-problem in weighted A consequence of this theorem is the following corollary. [33, 4.10 Example a), p. 22] Let (a n ) n∈N be strictly increasing, a n < 0 for all n ∈ N, V := (exp(a n μ)) n∈N and Ω n := {z ∈ C | | Im(z)| < n} for all n ∈ N where The restriction to negative a n comes from the condition that − ln ν n should be subharmonic. We note that the E-valued versions of Theorem 15 and Corollary 16 where E is a Fréchet space over 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.

K. Kruse
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