A Class of Exponentially Bounded Distribution Semigroups

54 M.Mijatović, S.Pilipović 0. Introduction One-time integrated and n-times integrated exponentially bounded semigroups (n-t.i.e.b.s., in short), n ∈ N, of operators on a Banach space introduced by Arendt, were developed in [1-2], [13-16], [19], [22], [29-30] and applied to abstract Cauchy problems with operators which do not generate C0-semigroups. More generally, α-times integrated semigroups α ∈ R+ ∪ {0} = [0,∞) are introduced and analyzed in [13-15] and in [21] in connection with certain classes of partial differential and pseudodifferential operators on Lp spaces. Distribution semigroups were introduced and analyzed by Lions (cf. [18] and after that in [5-11], [32] and many other papers. By results of Sova in [25], Arendt had proved in [2] that every exponentially bounded distribution semigroup is an n-th distributional derivative of an nt.i.e.b.s. with densely defined infinitesimal generator, where n is sufficiently large. The corresponding result for a distribution semigroup which is not exponentially bounded and which infinitesimal generator is not densely defined is proved in [35] and [17]. Local n-times integrated semigroups are introduced and analyzed in [3], [20], [23] and [28], where the relations with distribution semigroups were given. Wang and Kunstmann have introduced and analyzed in [35] and [17] a quasi-distribution semigroup, QDSG, in short; specially in the case when it is exponentially bounded, EQDSG. A G ∈ D′ +(L(E)) (the notation is given in the next section) is a QDSG if and only if G is a distributional derivative of order n of an n.t.i.e.b.s. We call it a 0-distribution semigroup; in the case when it is exponentially bounded, we denote it by 0-EDSG. C0-semigroup is among them. We will investigate 0-EDSG having not densely defined infinitesimal generators. Lions has studied in [18] distribution semigroups with densely defined generators using the structural properties and advantages of the space of tempered distributions. Wang has developed his own approach by constructing a space of generalized functions. The natural frame for such investigations is the space of exponentially bounded distributions K′ 1 ([12]). We analyze 0-EDSG using this space. Let us present the results of the paper. An infinitesimal generator A of an n-t.i.e.b.s. is the generator of a 0-EDSG and conversely, where n is sufficiently large. If A is densely defined, then a 0-EDSG is an exponentially bounded distribution semigroup, EDSG in short (cf. [18], Definition 6.1). The composition law for a 0-EDSG is given by 〈S(t + s, x), φ(t, s)〉 = 〈S(t, S(s, x)), φ(t, s)〉, A class of exponentially bounded distribution semigroups 55 φ ∈ K1(R), suppφ ⊂ [0,∞)× [0,∞). If S = G′ where G is strongly continuous and supported by [0,∞), then the above condition is sufficient for S being an 0-EDSG. Relations between a 0-EDSG and its infinitesimal generator are determined. They are not the same as in the case of an n-t.i.e.b.s. It is known that when an operator A generates an n-t.i.e.b.s. on a Banach space E, then there exists a Banach space E1 ⊂ D(An) continuously imbedded in E such that the part of A in E1 is the generator of a strongly continuous semigroup (cf. [4]). In this paper a Banach space E0 ⊂ E is constructed such that a 0-EDSG in E has the restriction on E0 forming an EDSG. In example 1 the results are applied to equation u′ = Au + f, where E = Cb(R) or E = L∞(R),