Laplace transforms for generalised functions and the abstract Cauchy problem

Working Groups: Lehrstuhl Angewandte Analysis

Collaborators (MAT): Dr. Karsten Kruse

Description

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

[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: 10.15083/00039821.
[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.
[4] K. Kruse. Vector-valued Fourier hyperfunctions. PhD thesis, Universität Oldenburg, 2014. URN: urn:nbn:de:gbv:715-oops-19095.
[5] K. Kruse. Vector-valued Fourier hyperfunctions and boundary values, 2019. arXiv:1912.03659.
[6] K. Kruse. Asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions. Funct. Approx. Comment. Math., 66(1):59-117, 2021. doi: 10.7169/facm/1955.
[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: 10.1007/s13398-022-01295-5.
[8] M. Langenbruch. Asymptotic Fourier and Laplace transformations for hyperfunctions. Stud. Math., 205(1):41-69, 2011. doi: 10.4064/sm205-1-4.
[9] G. Lumer, F. Neubrander. The asymptotic Laplace transform: New results and relation to Komatsu´s Laplace transform of hyperfunctions. In F. Mehmeti, J. von Below, S. Nicaise, editors, Partial differential equations on multistructures, volume 219 of Notes Pure Appl. Math., 147-162, Dekker, New York, 2001. doi: 10.1201/9780203902196.
[10] M. Sato. Theory of hyperfunctions, I. J. Fac. Sci. Univ. Tokyo, Sect. IA, 8:139-193, 1959. doi: 10.15083/00039918.
[11] M. Sato. Theory of hyperfunctions, II. J. Fac. Sci. Univ. Tokyo, Sect. IA, 8:387-437, 1960. doi: 10.15083/00039916.