Laplace Transforms for Generalised Functions and the Abstract Cauchy Problem

Let $E$ be a (sequentially) complete complex locally convex Hausdorff space ($\mathbb{C}$-lcHs). The initial value problem

$$ \begin{align} x’(t)&=Ax(t),\quad t>0, \\ x(0)&=x_{0}\in E, \end{align} $$

is called an abstract Cauchy problem where

$$ A\colon D(A)\subset E\to E $$

is a (sequentially) closed linear operator with domain $D(A)$. We study the abstract Cauchy problem in the space of $E$-valued hyperfunctions 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 [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 Link to heading

[1] B. Bäumer. A vector-valued operational calculus and abstract Cauchy problems. PhD thesis, Louisiana State University, Baton Rouge, LA, 1997. Available at LSU Digital Commons
[2] T. Kawai. On the theory of Fourier hyperfunctions and its applications to partial differential equations with constant coefficients. J. Fac. Sci. Univ. Tokyo, Sect. IA, 17:467–517, 1970. doi
[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
[4] K. Kruse. Vector-valued Fourier hyperfunctions. PhD thesis, University of Oldenburg, 2014. urn
[5] K. Kruse. Vector-valued Fourier hyperfunctions and boundary values, 2019. arXiv (to appear in Kyoto Journal of Mathematics)
[6] K. Kruse. Asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions. Funct. Approx. Comment. Math., 66(1):59–117, 2021. doi
[7] K. Kruse. The abstract Cauchy problem in locally convex spaces. RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 116(4):1–23, 2022. doi
[8] M. Langenbruch. Asymptotic Fourier and Laplace transformations for hyperfunctions. Stud. Math., 205(1):41–69, 2011. doi
[9] G. Lumer, and F. Neubrander. The asymptotic Laplace transform: New results and relation to Komatsu’s Laplace transform of hyperfunctions. In F. Mehmeti, J. von Below, and S. Nicaise, editors, Partial differential equations on multistructures, volume 219 of Notes Pure Appl. Math., 147–162, Dekker, New York, 2001. doi
[10] M. Sato. Theory of hyperfunctions, I. J. Fac. Sci. Univ. Tokyo, Sect. IA, 8:139–193, 1959. doi
[11] M. Sato. Theory of hyperfunctions, II. J. Fac. Sci. Univ. Tokyo, Sect. IA, 8:387–437, 1960. doi