Hyperfunction solutions of the abstract Cauchy problem

Hyperfunction semigroups

“…For the sake of simplicity, we shall also say, in that case, that A generates a hyperfunction semigroup G. The next assertion is proved in [44]:…”

“…The definition of a hyperfunction fundamental solution G for a closed linear operator A can be found in the paper [44] of S.Ōuchi. For the sake of simplicity, we shall also say, in that case, that A generates a hyperfunction semigroup G. The next assertion is proved in [44]:…”

“…S.Ōuchi [44] was the first who introduced the class of hyperfunction semigroups, more general than that of distribution and ultradistribution semigroups and in [45] he considered the abstract Cauchy problem in the space of hyperfunctions. Furthermore, generators of hyperfunction semigroups in the sense of [44] are not necessarily densely defined. A.N.…”

“…We analyze Fourier hyperfunction semigroups with non-densely defined generators continuing over the investigations of Roumieu type ultradistribution semigroups and constructed examples of tempered ultradistribution semigroups [32] as well as of Fourier hyperfunction semigroups with non-densely defined generators. An analysis of R. Beals [4,Theorem 2'] gives an example of a densely defined operator A in the Hardy space H 2 (C + ) which generates a hyperfunction semigroup of [44] but this operator is not a generator of any ultradistribution semigroup, and any (local) integrated C-semigroups, C ∈ L(H 2 (C + )). Our main interest is the existence of fundamental solutions for the Cauchy problems with initial data being hyperfunctions.…”

Differential and Analytical Properties of Semigroups of Operators

“…Differential and analytical properties of hyperfunction fundamental solutions can be found in the papers of Kochubei [52] andŌuchi [91,92]. Finally, we would like to propose the following questions.…”