Splitting and parameter dependence in the category of $$\hbox {PLH}$$ PLH spaces

Dierolf, Bernhard; Sieg, Dennis

doi:10.1007/s13398-017-0424-5

“…Proposition 25.19, p. 303], there exist n ∈ ℕ and 𝜆 > 0 with M ⊂ 𝜆B n . We derive from (7) that proving the continuity of H.…”

Section: Propositionmentioning

confidence: 90%

“…The parameter dependence problem for a variety of partial differential operators on several spaces of (generalised) differentiable functions has been extensively studied, see e.g. [4,6,7,16,31,32] and the references and background in [3,22]. The answer to this problem for the Cauchy-Riemann operator is affirmative since the Cauchy-Riemann operator on the space C ∞ ( , E) of E-valued smooth functions is surjective if E = C ∞ (U) ( O(U) , D(V) � ) by [8, Corollary 3.9, p. 1112] which is a consequence of the splitting theory of Bonet and Domański for PLS-spaces [3,4], the topological isomorphy of C ∞ ( , E) to Schwartz' -product C ∞ ( ) E and the fact that ∶ C ∞ ( ) → C ∞ ( ) is surjective on the nuclear Fréchet space C ∞ ( ) (with its usual topology).…”

Section: Introductionmentioning

confidence: 99%

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

Kruse

¹

2021

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This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator $$\overline{\partial }$$ ∂ ¯ on spaces $${\mathcal {E}}{\mathcal {V}}(\varOmega ,E)$$ E V ( Ω , E ) of $${\mathcal {C}}^{\infty }$$ C ∞ -smooth vector-valued functions whose growth on strips along the real axis with holes K is induced by a family of continuous weights $${\mathcal {V}}$$ V . Vector-valued means that these functions have values in a locally convex Hausdorff space E over $${\mathbb {C}}$$ C . We derive a counterpart of the Grothendieck-Köthe-Silva duality $${\mathcal {O}}({\mathbb {C}}\setminus K)/{\mathcal {O}}({\mathbb {C}})\cong {\mathscr {A}}(K)$$ O ( C \ K ) / O ( C ) ≅ A ( K ) with non-empty compact $$K\subset {\mathbb {R}}$$ K ⊂ R for weighted holomorphic functions. We use this duality and splitting theory to prove the surjectivity of $$\overline{\partial }:{\mathcal {E}} {\mathcal {V}}(\varOmega ,E)\rightarrow {\mathcal {E}}{\mathcal {V}} (\varOmega ,E)$$ ∂ ¯ : E V ( Ω , E ) → E V ( Ω , E ) for certain E. This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on $${\mathcal {E}}{\mathcal {V}}(\varOmega ,{\mathbb {C}})$$ E V ( Ω , C ) .

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“…Then the quotient space, equipped with these seminorms, is a locally convex space (but maybe not Hausdorff). Since (7) holds for every representative F of f , we obtain for every…”

Section: Dualitymentioning

confidence: 99%

“…The parameter dependence problem for a variety of partial differential operators on several spaces of (generalised) differentiable functions has been extensively studied, see e.g. [4,6,7,37,38,18] and the references and background in [3,26]. The answer to this problem for the Cauchy-Riemann operator is affirmative since the Cauchy-Riemann operator…”

Section: Introductionmentioning

confidence: 99%

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

Kruse

2019

Preprint

0

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This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator on spaces EV(Ω, E) of C ∞ -smooth vector-valued functions whose growth on strips along the real axis with holes K is induced by a family of continuous weights V. Vector-valued means that these functions have values in a locally convex Hausdorff space E over C. We characterise the weights V which give a counterpart of the Grothendieck-Köthe-Silva duality O(C ∖ K) O(C) ≅ A (K) with non-empty compact K ⊂ R for weighted holomorphic functions. We use this duality to prove that the kernel ker ∂ of the Cauchy-Riemann operator ∂ in EV(Ω) ∶= EV(Ω, C) has the property (Ω) of Vogt. Then an application of the splitting theory of Vogt for Fréchet spaces and of Bonet and Domański for (PLS)-spaces in combination with some previous results on the surjectivity of the Cauchy-Riemann operator ∂∶ EV(Ω) → EV(Ω) yields the surjectivity of the Cauchy-Riemann operator on EV(Ω, E) if E ∶= F ′ b with some Fréchet space F satisfying the condition (DN ) or if E is an ultrabornological (PLS)-space having the property (P A). This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on EV(Ω).

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Kruse

¹

2020

RACSAM

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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.

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Splitting and parameter dependence in the category of $$\hbox {PLH}$$ PLH spaces

Cited by 4 publications

References 47 publications

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

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

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

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

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