Distribution semigroups and abstract Cauchy problems

Abstract

Abstract. We present a new definition of distribution semigroups, covering in particular non-densely defined generators. We show that for a closed operator A in a Banach space E the following assertions are equivalent: (a) A generates a distribution semigroup; (b) the convolution operator δ ⊗ I − δ ⊗ A has a fundamental solution in D (L(E, D)) where D denotes the domain of A supplied with the graph norm and I denotes the inclusion D → E; (c) A generates a local integrated semigroup. We also show that every generator of a distribution semigroup generates a regularized semigroup.

Regularization of some classes of ultradistribution semigroups and sines

“…The assertions (i), (iv) and (v) of the next theorem can be attributed to Straub [28]. Herein we notice that the denseness of is not used in the proofs of Propositions 2.2, 2.5, 2.6 and 2.8 as well as Lemmas 2.7 and 2.10 of [28] and that the assertion (v) extends [3, Lemma 1] and some estimates used in the proof of [18,Lemma 5.4] (cf. also [1, Lemma II-1, Theorem II-3]).…”

“…The first comprehensive analysis of ultradistribution semigroups and sines was obtained by Komatsu [13]. The definition of ultradistribution semigroup and its generator employed therein has been recently reconsidered in [16] following the approaches of Kunstmann [18] and Wang [31] for distribution semigroups.…”

“…In such a way, we obtain an extension of the well known result of Kunstmann (cf. [18,Section 5]) which asserts that every generator of a distribution semigroup of [18] is also the generator of a global -semigroup, and refine several estimates given in [3,Lemma 1], [6,Remark 2.6,Corollary 4.3], [9,Example 22.31] as well as in the proof of [18,Theorem 5.5]. Although far from being optimal, we will see in Remark 2.1 how these improvements can be used in a more detailed analysis of the incomplete higher-order Cauchy problems (cf.…”

“…Remark 2.1. Suppose generates a distribution semigroup of [18]. Then one can employ [18,Corollary 3.12] in order to conclude that, for every > 0, there exist > 0, > 0, ∈ N and ∈ R such that (2.7) and (2.8) hold.…”

“…Suppose generates a distribution semigroup of [18]. Then one can employ [18,Corollary 3.12] in order to conclude that, for every > 0, there exist > 0, > 0, ∈ N and ∈ R such that (2.7) and (2.8) hold. Hence, for every ∈ (0, 1) and ∈ (︀ 0, arctan(cos( 2 )) )︀ , generates a global ( )-semigroup, where we define ( ) as before; let us remind that Kunstmann [18] proved that this statement holds for every ∈ (0, 1) and ∈ (︀ 0, (1− )…”