Asymptotic Fourier and Laplace transformations for hyperfunctions

Abstract

We develop an elementary theory of Fourier and Laplace transformations for exponentially decreasing hyperfunctions. Since any hyperfunction can be extended to an exponentially decreasing hyperfunction, this provides simple notions of asymptotic Fourier and Laplace transformations for hyperfunctions, improving the existing models. This is used to prove criteria for the uniqueness and solvability of the abstract Cauchy problem in Fréchet spaces.

The abstract Cauchy problem in locally convex spaces

“…In Theorem 3.4 we give a sufficient condition for the solvability of the abstract Cauchy problem in terms of asymptotic right resolvents. Our results on the solvability extend the ones from [44].…”

“…Hence [u] = 0 by Theorem 1.5. Now, we generalise Langenbruch's sufficient criterion [44,Theorem 7.2,p. 62] for the uniqueness property, which itself is a generalisation of Lyubich's uniqueness theorem [49,Theorem 9.2,p.…”

“…A common problem in the mentioned approaches to the abstract Cauchy problem is the development of a suitable notion of a Laplace transform for vector-valued generalised functions. In [4,5,47] an appropriate (asymptotic) Laplace transform was developed for Banach-valued locally integrable functions and applied to abstract Cauchy problems, and in [27][28][29][30][31] under minimal regularity assumptions for Banach-valued hyperfunctions as well as in [44] for Fréchet valued hyperfunctions. This was extended in [38] beyond the class Fréchet spaces to a large variety of locally convex Hausdorff spaces containing common spaces of distributions.…”

“…After recalling the necessary notions and results from [38], we characterise the uniqueness of the solutions of the abstract Cauchy problem in Section 2, in particular, we derive necessary and sufficient conditions in Theorem 2.2. We use these conditions to phrase sufficient conditions for the uniqueness of solutions in terms of asymptotic left resolvents in Theorem 2.3 and Theorem 2.4, generalising corresponding results from [44] (for general notions of resolvents in locally convex Hausdorff spaces see [2] and [12]). In Section 3 we turn to the solvability of the abstract Cauchy problem for vector-valued hyperfunctions and present necessary and sufficient conditions for the solvability in Theorem 3.3.…”

On the nuclearity of weighted spaces of smooth functions

“…where the supremum in (1) is taken over all α ∈ N d 0 or all n ∈ N, conditions for nuclearity are known due to Komatsu 1, p. 485], and are also the basic spaces for the theory of Fourier hyperfunctions, see e.g. [9], [10], [12], [17], [21] and [22]. In addition, an affirmative answer to the question of nuclearity of EV(Ω) transfers the surjectivity of a linear partial differential operator P (∂)∶ EV(Ω) → EV(Ω) with smooth coefficients to its corresponding vector-valued counterpart on smooth weighted functions with values in certain locally convex spaces E, for example, in the case that EV(Ω) and E are Fréchet spaces by [13,Satz 10.24,p.…”