is a (sequentially) closed linear operator with domain $D(A)$. We study the abstract Cauchy problem in the space of $E$-valued [hyperfunctions](https://en.wikipedia.org/wiki/Hyperfunction) with support in $[0,\infty)$. Hyperfunctions were introduced by Sato [10,11] and extended to Fourier hyperfunctions by Kawai [2]. Hyperfunctions form a quite large class of generalised functions, containing locally integrable functions, distributions and ultradistributions.
We study Fourier and Laplace transforms for Fourier hyperfunctions with values in a $\mathbb{C}$-lcHs. Since any hyperfunction with values in a wide class of locally convex Hausdorff spaces can be extended to a Fourier hyperfunction [6,7], this gives simple notions of asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions [4], which improves the existing models of Komatsu [3], Bäumer [1], Lumer and Neubrander [9] and Langenbruch [8]. We apply our theory of asymptotic Laplace transforms to prove existence and uniqueness results for solutions of the abstract Cauchy problem in a wide class of locally convex Hausdorff spaces, containing Fréchet spaces and several spaces of distributions [5].
We study Fourier and Laplace transforms for Fourier hyperfunctions with values in a $\mathbb{C}$-lcHs. Since any hyperfunction with values in a wide class of locally convex Hausdorff spaces can be extended to a Fourier hyperfunction [4,5], this gives simple notions of asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions [6], which improves the existing models of Komatsu [3], Bäumer [1], Lumer and Neubrander [9] and Langenbruch [8]. We apply our theory of asymptotic Laplace transforms to prove existence and uniqueness results for solutions of the abstract Cauchy problem in a wide class of locally convex Hausdorff spaces, containing Fréchet spaces and several spaces of distributions [7].
## References
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@@ -33,13 +33,13 @@ We study Fourier and Laplace transforms for Fourier hyperfunctions with values i
[3] H. Komatsu. Laplace transforms of hyperfunctions -- A new foundation of the Heaviside calculus. *J. Fac. Sci. Univ. Tokyo, Sect. IA*, 34:805--820, 1987. doi: [10.15083/00039471](https://doi.org/10.15083/00039471).
[4] K. Kruse. Asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions, 2021. [arXiv:2104.02682](https://arxiv.org/abs/2104.02682).
[4] K. Kruse. Vector-valued Fourier hyperfunctions. PhD thesis, Universität Oldenburg, 2014. URN: [urn:nbn:de:gbv:715-oops-19095](http://nbn-resolving.org/urn:nbn:de:gbv:715-oops-19095)
[5] K. Kruse. The abstract Cauchy problem in locally convex spaces. *In preparation*.
[5] K. Kruse. Vector-valued Fourier hyperfunctions and boundary values, 2019. [arXiv:1912.03659](https://arxiv.org/abs/1912.03659).
[6] K. Kruse. Vector-valued Fourier hyperfunctions. PhD thesis, Universität Oldenburg, 2014. URN: [urn:nbn:de:gbv:715-oops-19095](http://nbn-resolving.org/urn:nbn:de:gbv:715-oops-19095)
[6] K. Kruse. Asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions, 2021. [arXiv:2104.02682](https://arxiv.org/abs/2104.02682).
[7] K. Kruse. Vector-valued Fourier hyperfunctions and boundary values, 2019. [arXiv:1912.03659](https://arxiv.org/abs/1912.03659).
[7] K. Kruse. The abstract Cauchy problem in locally convex spaces. *In preparation*.
[8] M. Langenbruch. Asymptotic Fourier and Laplace transformations for hyperfunctions. *Stud. Math.*, 205(1):41--69, 2011. doi: [10.4064/sm205-1-4](https://doi.org/10.4064/sm205-1-4).
It is a classical idea to represent vector-valued functions by continuous linear operators [3]. Let $\mathcal{F}(\Omega)$ be a locally convex Hausdorff space (lcHs) of functions from a set $\Omega$ to the field $\mathbb{K}$ of real or complex numbers and $E$ an lcHs over $\mathbb{K}$. Then Schwartz' $\varepsilon$-product of $\mathcal{F}(\Omega)$ and $E$ is defined as the space of continuous linear operators
It is a classical idea to represent vector-valued functions by continuous linear operators [13]. Let $\mathcal{F}(\Omega)$ be a locally convex Hausdorff space (lcHs) of functions from a set $\Omega$ to the field $\mathbb{K}$ of real or complex numbers and $E$ an lcHs over $\mathbb{K}$. Then Schwartz' $\varepsilon$-product of $\mathcal{F}(\Omega)$ and $E$ is defined as the space of continuous linear operators
$$
\mathcal{F}(\Omega)\varepsilon E :=L_{e}(\mathcal{F}(\Omega)_{\kappa}',E).
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@@ -20,11 +20,11 @@ $$
is a well-defined topological isomorphism.
In [11] we derive sufficient conditions on $E$ and on the properties and structures of the functions and function spaces $\mathcal{F}(\Omega)$ and $\mathcal{F}(\Omega,E)$ such that the map $S$ is a topological isomorphism. Once the isomorphism $S$ is established, the famous [approximation property](https://en.wikipedia.org/wiki/Approximation_property) of a space $\mathcal{F}(\Omega)$ is equivalent to the property that every function in $\mathcal{F}(\Omega,E)$ can be approximated by functions with values in finite dimensional subspaces of $E$ for any lcHs $E$, which we investigate in [4] for weighted spaces of $\mathcal{C}^{k}$-smooth functions. In [7] we study the stronger property that $\mathcal{F}(\Omega)$ is [nuclear](https://en.wikipedia.org/wiki/Nuclear_space) in the case of weighted $\mathcal{C}^{\infty}$-smooth functions.
In [10] we derive sufficient conditions on $E$ and on the properties and structures of the functions and function spaces $\mathcal{F}(\Omega)$ and $\mathcal{F}(\Omega,E)$ such that the map $S$ is a topological isomorphism. Once the isomorphism $S$ is established, the famous [approximation property](https://en.wikipedia.org/wiki/Approximation_property) of a space $\mathcal{F}(\Omega)$ is equivalent to the property that every function in $\mathcal{F}(\Omega,E)$ can be approximated by functions with values in finite dimensional subspaces of $E$ for any lcHs $E$, which we investigate in [4] for weighted spaces of $\mathcal{C}^{k}$-smooth functions. In [7] we study the stronger property that $\mathcal{F}(\Omega)$ is [nuclear](https://en.wikipedia.org/wiki/Nuclear_space) in the case of weighted $\mathcal{C}^{\infty}$-smooth functions.
Nuclearity can be used to transfer the surjectivity of a continuous linear map $T\colon \mathcal{F}(\Omega)\to\mathcal{F}(\Omega)$ to the $\varepsilon$-product $T\varepsilon \operatorname{id}_{E}\colon \mathcal{F}(\Omega)\varepsilon E\to\mathcal{F}(\Omega)\varepsilon E$ for Fréchet spaces $\mathcal{F}(\Omega)$ and $E$ by Grothendieck's classical [tensor product](https://en.wikipedia.org/wiki/Topological_tensor_product) theory [1]. In combination with the topological isomorphism $S$ this implies that the surjectivity of a continuous linear partial differential operator can be transfered from the scalar-valued to the vector-valued case, which we study for the Cauchy-Riemann operator $T=\overline{\partial}$ on weighted spaces of $\mathcal{C}^{\infty}$-smooth functions in [2,5,6,8] even for $E$ beyond the class of Fréchet spaces.
Another application of the topological isomorphism $S$ lies in lifting series representations from scalar-valued to $E$-valued functions [9], for instance power series representations of holomorphic functions [10], and the extension of $E$-valued functions via weak extensions [3,12], i.e. to answer the question:
Another application of the topological isomorphism $S$ lies in lifting series representations from scalar-valued to $E$-valued functions [12], for instance power series representations of holomorphic functions [9], and the extension of $E$-valued functions via weak extensions [3,11], i.e. to answer the question:
Let $\Lambda$ be a subset of $\Omega$ and $G$ a linear subspace of $E'$. Let $f\colon \Lambda\to E$ be such that for every $e'\in G$, the function $e'\circ f\colon\Lambda\to \mathbb{K}$ has an extension in $\mathcal{F}(\Omega)$. When is there an extension $F\in\mathcal{F}(\Omega,E)$ of $f$, i.e. $F_{\mid \Lambda} = f$ ?
[8] K. Kruse. Parameter dependence of solutions of the Cauchy-Riemann equation on spaces of weighted smooth functions. *RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat.*, 114(141):1--24, 2020. doi: [10.1007/s13398-020-00863-x](https://doi.org/10.1007/s13398-020-00863-x).
[9] K. Kruse. Series representations in spaces of vector-valued functions via Schauder decompositions. *Math. Nachr.*, 294(2):354--376, 2021. doi: [10.1002/mana.201900172](https://doi.org/10.1002/mana.201900172).
[9] K. Kruse. Vector-valued holomorphic functions in several variables. *Funct. Approx. Comment. Math.*, 63(2):247--275, 2020. doi: [10.7169/facm/1861](https://doi.org/10.7169/facm/1861).
[10] K. Kruse. Vector-valued holomorphic functions in several variables. *Funct. Approx. Comment. Math.*, 63(2):247--275, 2020. doi: [10.7169/facm/1861](https://doi.org/10.7169/facm/1861).
[10] K. Kruse. Weighted spaces of vector-valued functions and the $\varepsilon$-product, *Banach J. Math. Anal.*, 14(4):1509--1531, 2020. doi: [10.1007/s43037-020-00072-z](https://doi.org/10.1007/s43037-020-00072-z).
[11] K. Kruse. Weighted spaces of vector-valued functions and the $\varepsilon$-product, *Banach J. Math. Anal.*, 14(4):1509--1531, 2020. doi: [10.1007/s43037-020-00072-z](https://doi.org/10.1007/s43037-020-00072-z).
[11] K. Kruse. Extension of vector-valued functions and weak-strong principles for differentiable functions of finite order, 2021. [arXiv:1910.01952](https://arxiv.org/abs/1910.01952).
[12] K. Kruse. Extension of vector-valued functions and weak-strong principles for differentiable functions of finite order, 2021. [arXiv:1910.01952](https://arxiv.org/abs/1910.01952).
[12] K. Kruse. Series representations in spaces of vector-valued functions via Schauder decompositions. *Math. Nachr.*, 294(2):354--376, 2021. doi: [10.1002/mana.201900172](https://doi.org/10.1002/mana.201900172).
[13] L. Schwartz. Espaces de fonctions différentiables à valeurs vectorielles. *J. Analyse Math.*, 4:88--148, 1955. doi: [10.1007/BF02787718](https://doi.org/10.1007/BF02787718).
