Bulletin T.CXXII de l’Académie Serbe des Sciences et des Arts - 2001 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 26

A CLASS OF EXPONENTIALLY BOUNDED DISTRIBUTION SEMIGROUPS

M. MIJATOVIĆ, S. PILIPOVIĆ

(Presented at the 9th Meeting, held on December 22, 2000)

A b s t r a c t. A structural theorem for a vector valued exponentially bounded distribution is used for introducing and studyng of a class distribution semigroups. An infinitesimal generator of such a semigroup is not necessarily densely defined, but if it is the case, then it corresponds to a distribution semigroup introduced by Lions. This result is obtained by Wang and Kunstmann for a class of exponentially bounded quasi-distribution semigroups. In fact we show that our class of distribution semigroup is identical to Wang-Kunstmann’s one. Our approach is completely different and gives

new characterizations. Applications to equations \frac{∂u}{∂t} = Au + f, where A is not necessarily densely defined and f is an exponential vector valued distribution supported by [0, ∞), are given.

This paper is written much before the publishing of Wang’s and Kunstmann’s paper but because of various reasons it is published with a very long delay. Here it is given in the primary version as an original approach although some parts are consequences of published results of Wang and Kunstmann.

AMS Mathematics Subject Classification (2000): 47D03

Key Words: distribution semigroups, integrated semigroups

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 C_{0}-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 L^{p} 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. C_{0}-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)⟩,

φ ∈ K_{1}(ℝ^{2}), 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 E_{1} ⊂ D(A^{n}) continuously imbedded in E such that the part of A in E_{1} is the generator of a strongly continuous semigroup (cf. [4]). In this paper a Banach space E_{0} ⊂ E is constructed such that a 0-EDSG in E has the restriction on E_{0} forming an EDSG.

In example 1 the results are applied to equation u′ = Au + f, where E = C_{b}(ℝ) or E = L^{∞}(ℝ),

A = ∑_{j=0}^{k} a_{j}(\frac{d}{dx})^{j}, Re ∑_{j=0}^{k} a_{j}(ix)^{j} < ∞, f ∈ K′_{1}(E_{0})

and E_{0} is a suitable subspace of E. Note, A is not densely defined. In example 2 is solved

\frac{∂u}{∂t} − iH_{m}(∂)u = f, f ∈ K′_{1}(L^{p}(ℝ^{n})), suppf ⊂ [a, ∞),

where H_{m}(∂) is a pseudodifferential operator i(Δ)^{\frac{m}{2}} , m > 0. Another approach to this equation in the framework of L^{p} spaces is given in [5-6] for m = 2 and [15]. The assumption f ∈ K′_{1}(L^{p}(ℝ^{n})) justifies our approach.

1. Preliminaries from the theory of distributions

Denote by E a Banach space with a norm ∥ · ∥; L(E) = L(E, E) is the space of bounded linear operators from E into E and C(ℝ, L(E)) is the space continuous mappings from ℝ into L(E). We refer to [26-27] and [31] for the definitions of spaces D(ℝ), E(ℝ), S(ℝ), their strong duals and S′(E) = L(S(ℝ), E). Moreover, we refer to [33] for the space S_{+} = {φ; |t^{k}φ^{(v)}(t)| < C_{k,v}, t ∈ [0, ∞), k, v ∈ ℕ_{0}} (ℕ_{0} = ℕ ∪ {0}) and its dual S′_{+}, which consists of tempered distributions supported by [0, ∞). Recall ([12]), the space of exponentially decreasing test functions on the real line ℝ is defined by K_{1}(ℝ) = {φ; |e^{k|t|}φ^{(v)}(t)| < C_{k,v}, t ∈ ℝ, k, v ∈ ℕ_{0}}. The space K_{1}(ℝ^{2}) is defined in an appropriate way. This space has the same topological properties as S(ℝ). Note,

f ∈ K′_{1}(ℝ) if and only if e^{−r|x|}f ∈ S′(ℝ) for some r ∈ ℝ.  (1)

The strong dual of K_{1}(ℝ), K′_{1}(ℝ) is the space of exponential distributions. The space K′_{1^{+}} ⊂ K′_{1}(ℝ) consists of distributions which are supported by [0, ∞).

We will also use the space K_{1^{+}} = {φ; |e^{k|t|}φ^{(v)}(t)| < C_{k,v}, t ∈ [0, ∞), k, v ∈ ℕ_{0}} which has the same topological properties as S_{+}. Note, its dual space is K′_{1^{+}}.

Essential role will play spaces K_{1_{0}} and D_{0}, subspaces of K_{1} and D, respectively, which elements are supporte by [0, ∞). In the sequal, we will use the family of distributions

f_{n}(t) = \{ \begin{matrix}\frac{H(t)t^{n−1}}{(n − 1)!} , & n ∈ ℕ \\ f_{n+n_{1}}^{(n_{1})}(t), & −n ∈ ℕ_{0}, n_{1} ∈ ℕ, n + n_{1} > 0, t ∈ ℝ, \end{matrix}

where H is Heaviside’s function. Note f_{−1} = δ′.

Let K′_{1}(E) = L(K_{1}, E) denote the space of continuous linear functions K_{1} → E with respect to the topology of uniform convergence on bounded sets of K_{1}. Denote K′_{1^{+}}(E) = L(K_{1^{+}}, E). It is a subspace of K′_{1}(E) with elements supported by [0, ∞). There holds K′_{1}(E) = K′_{1}(ℝ) ˆ⊗E = L(K_{1}, E) where ˆ⊗ denotes the completion of tensor product with respect to the Etopology which is equivalent with the π-topology, since K′_{1}(ℝ) is nuclear (cf. [31]). Also we have K′_{1^{+}}(E) = K′_{1^{+}} ˆ⊗E.

The convolution of f ∈ K′_{1^{+}}(E) and g ∈ K′_{1^{+}} is defined by ⟨f ∗ g, φ⟩ = ⟨f, ǧ ∗ φ⟩, φ ∈ K_{1}(ℝ), (ǧ(t) = g(−t)). One can prove easily that f ∗ g = g ∗ f ∈ K′_{1^{+}}(E).

Let T : [0, ∞) → L(E) be strongly continuous. Then, it is exponentially bounded at infinity if there exist M ≥ 0 and ω ≥ 0 such that

∥T (t)∥ ≤ Me^{ωt}, t ≥ 0,  (2)

In this case φ → ∫_{0}^{∞}T (t)φ(t)dt, φ ∈ K_{1}(ℝ), defines an element of K′_{1^{+}}(E).

We need the following representation for elements of K′_{1^{+}}(L(E)).

Theorem 1. Let S ∈ K′_{1^{+}}(L(E)).

a) There exists n_{0} ∈ ℕ such that for every n ≥ n_{0} there exist a strongly continuous function F_{n} : ℝ → L(E), suppF_{n} ⊂ [0, ∞) and positive constants m_{n} and C_{n}, such that

∥F_{n}(t)∥ ≤ C_{n}e^{m_{n}t}, t ≥ 0, S = F_{n}^{(n)} (^{(n)} is the distributional n-th derivative).

b) Let ψ, φ ∈ K_{1}(ℝ). Then

⟨S(t, ⟨S(s, x), ψ(s)⟩), φ(t)⟩ = ∫ S_{n_{0}}(t, S_{n_{0}}(s, x))ψ^{(n_{0})}(s)φ^{(n_{0})}(t)ds dt. (3)

c) Let φ(t, s) ∈ K_{1}(ℝ^{2}) and φ_{v}(t), ψ_{v}(s) be sequences in D(ℝ) such that the product sequence φ_{v}(t) · ψ_{v}(s) converge to φ(t, s) in K_{1}(ℝ^{2}) as v → ∞. Then, the limit

lim_{v→∞}⟨S(t, ⟨S(s, x), ψ_{v}(s)⟩), φ_{v}(t)⟩

defined by the left hand side of (3) exists and defines an element of K′_{1}(ℝ^{2}) which we denote by S(t, S(s, x)) i.e.

⟨S(t, S(s, x)), φ(t, s)⟩ = lim_{v→∞} ∫ S_{n_{0}}(t, S_{n_{0}}(s, x))ψ_{v}^{(n_{0})}(s)φ_{v}^{(n_{0})}(t)ds dt, (4)

where φ ∈ K_{1}(ℝ^{2}).

d) Also, we have for φ ∈ K_{1}(ℝ^{2}) and r, p ∈ ℕ,

(i) ⟨\frac{∂^{r}}{∂t^{r}} S(t, S(s, x)), φ(t, s)⟩ = (−1)^{r}⟨S(t, S(s, x)), \frac{∂^{r}}{∂t^{r}} φ(t, s)⟩;

(ii)  ⟨\frac{∂^{r}}{∂s^{p}} S(t, S(s, x)), φ(t, s)⟩ = ⟨S(t, \frac{∂^{p}}{∂s^{p}} S(s, x)), φ(t, s)⟩

= (−1)^{p}⟨S(t, S(s, x)), \frac{∂^{p}}{∂s^{p}} φ(t, s)⟩.

P r o o f. a) Let k ∈ ℕ and K_{1,k}(ℝ) be the spaces of functions φ ∈ C^{k}(ℝ), such that lim_{|x|→∞} sup_{i≤k}{e^{k|x|}|φ^{(i)}(x)|} = 0. Then, K_{1,k}(ℝ), k ∈ ℕ, are Banach

spaces with the norms ∥φ∥ = sup{e^{k|x|}|φ^{(i)}(x)|; x ∈ ℝ, i ≤ k} and

K_{1}(ℝ) = proj lim_{k→∞} K_{1,k}(ℝ), K′_{1}(L(E)) = ind lim_{k→∞} K′_{1,k}(L(E))

in the sense of strong topologies because the inclusion mappings K_{1,k+1}(ℝ) → K_{1,k}(ℝ) are compact, k ∈ ℕ. Thus, there exists n_{1} ∈ ℕ such that S ∈ K′_{1,n_{1}}(L(E)). Also, there exists n_{0} such that for every t ∈ ℝ, f_{n_{0}}(t − ·)θ(·) ∈

K_{1,n_{1}}(ℝ), where θ ∈ C^{∞}(ℝ), θ(x) = 0, for x ≤ −1 and θ(x) = 1, for x ≥ −\frac{1}{2}.

Then

(S ∗ f_{n_{0}})(t) = ⟨S(u), \frac{(t − u)^{n_{0}−1}}{Γ(n_{0})} H(t − u)θ(u)⟩, t ∈ ℝ,

it is continuous, supported by [0, ∞), of exponential growth and (S∗f_{n_{0}})^{(n_{0})} = S. This implies the proof of assertion for every n ≥ n_{0}.

b) Assume that φ and ψ are supported by [α, β].

⟨S(s, x), ψ(x)⟩ = (−1)^{n_{0}} ∫_{α}^{β} S_{n_{0}}(s, x)ψ^{(n_{0})}(s)ds

= (−1)^{n_{0}} lim_{v→∞} ∑_{i=0}^{v} S_{n_{0}}(s_{i}, x)ψ^{(n_{0})}(s_{i})Δs_{i}.

The continuity of S_{n_{0}} implies

(−1)^{n_{0}} ∫_{α}^{β} S_{n_{0}}(t, ⟨S_{n_{0}}(s, x), ψ^{(n_{0})}(s)⟩)φ^{(n_{0})}(t)dt

= lim_{v→∞} ∑_{i=0}^{v} ∫_{α}^{β} S_{n_{0}}(t, S_{n_{0}}(s_{i}, x))ψ^{(n_{0})}(s_{i})φ^{(n_{0})}(t)Δs_{i} dt

= ∫_{α}^{β} S_{n_{0}}(t, S_{n_{0}}(s, x))ψ^{(n_{0})}(s)φ^{(n_{0})}(t)ds dt.

Note ∥S_{n_{0}}(t, S_{n_{0}}(s, x))∥ ≤ e^{k|t|}e^{k|s|}∥x∥, x ∈ E. Thus, both sides in (3) exist for φ, ψ ∈ K_{1}(ℝ). By taking sequences φ_{v} and ψ_{v} of test functions in D(ℝ) which converge in K_{1}(ℝ) to φ and ψ, respectively, we obtain (3).

c) This assertion directly follows from a) by using the integral form given in (3).

d) (i) Assertion b) and (4) imply

⟨\frac{∂^{r}}{∂t^{r}} S(t, S(s, x)), ψ(s)φ(t)⟩

= (−1)^{r} ∫ S_{n_{0}}(t, S_{n_{0}}(s, x))ψ^{(n_{0})}(s)φ^{(n_{0}+r)}(t)ds dt, φ, ψ ∈ K_{1}(ℝ) and this gives the assertion.

(ii) The same arguments imply

⟨\frac{∂^{p}}{∂s^{p}} S(t, S(s, x)), ψ(s)φ(t)⟩

= (−1)^{p} ∫ S_{n_{0}}(t, S_{n_{0}}(s, x))ψ^{(n_{0}+p)}(s)φ^{(n_{0})}(t)ds dt

= ⟨S_{n_{0}}(t, ⟨S_{n_{0}}(s, x), ψ^{(n_{0}+p)}(s)⟩), φ^{(n_{0})}(t)⟩, φ, ψ ∈ K_{1}(ℝ). □

By using (1) one can prove easily

f ∈ K′_{1^{+}}(L(E)) if and only if e^{−r|x|}f ∈ S′_{+}(L(E)) for some r ≥ 0. (5)

Let f satisfy (5). Then the Laplace transformation of f is defined by

L(f)(λ) = f̂(λ) = ⟨f(t), e^{−λt}η(t)⟩, Reλ > r,

where η ∈ ℂ^{∞}(ℝ), suppη = [−ϵ, ∞), ϵ > 0 and η ≡ 1 on [0, ∞). As in the case of tempered distributions, one can easily show that this definition does not depend on η (cf. [27]). If f ∈ L^{1}([0, ∞), E) (which means ∥ ∫ f(t)dt∥_{E} < ∞), then

f̂(λ) = ∫_{0}^{∞}e^{−λt}f(t)dt = ⟨f(t), e^{−λt}⟩, Reλ > 0,

where integral is taken in Bochner’s sense.

2. 0-exponentially bounded distribution semigroup

Let T : (0, ∞) → L(E) be strongly continuous, integrable in a neighborhood of 0, i.e., integrable on (0, ϵ) for some ϵ > 0 and exponentially bounded at infinity i.e. satisfies (2) for some M ≥ 0 and ω ∈ ℝ. The Laplace trans-

formation of T is defined by L(T)(λ) = R(λ) = ∫_{0}^{∞}e^{−λt}T (t)dt, Reλ > ω,

where the integral is understood in Bochner’s sense.

Let (T (t))_{t≥0} be a C_{0}-semigroup. It is well-known that A is the infinitesimal generator of this semigroup if and only if there exists ω ∈ ℝ such that (ω, ∞) ⊂ ρ(A) and R : {λ ∈ ℂ; Reλ > ω} → L(E), defined by R(λ) = (λI − A)^{−1}, Reλ > ω, is the Laplace transformation of (T (t))_{t≥0}.

Arendt ([2]) defined an n-t.i.e.b.s. as follows.

Let (S(t))_{t≥0} be a strongly continuous (on [0, ∞)) exponentially bounded family in L(E) and n ∈ ℕ. Then, it is called n-t.i.e.b.s. if S(0, x) = 0 and

\begin{matrix}S(t, S(s, x)) = \frac{1}{(n−1)!}[ ∫_{t}^{t+s}(t + s − r)^{n−1}S(r, x)dr \\ − ∫_{0}^{s}(t + s − r)^{n−1}S(r, x)dr], t, s ≥ 0, x ∈ E. \end{matrix}  (6)

Let S : (0, ∞) → L(E) be strongly continuous, exponentially bounded, integrable in a neighbourhood of 0, satisfy (2) for some M > 0, ω ∈ ℝ and

R(λ) = λ^{n} ∫_{0}^{∞}e^{−λt}S(t)dt, Reλ > ω, n ∈ ℕ.  (7)

Then, Arendt proved that (R(λ))_{Reλ>ω} is a pseudoresolvent if and only if (6) holds.

Theorem 2. Let S ∈ K′_{1^{+}}(L(E)) and R(λ) = L(S)(λ), Reλ > ω.

a) Then, (R(λ))_{Reλ≥ω} is a pseudoresolvent if and only if there exists n_{0} ∈ ℕ such that S_{n_{0}}(t) = (S ∗ f_{n_{0}})(t), t ∈ ℝ, is continuous, S_{n_{0}}(0) = 0 and satisfies

⟨S(t, S(s, x)), φ(t)ψ(s)⟩ = ⟨(S_{n_{0}}(t, S_{n_{0}}(s, x)))^{(n_{0},n_{0})}, φ(t)ψ(s)⟩

\begin{matrix}⟨\frac{1}{(n_{0}−1)!}( ∫_{t}^{t+s}(t + s − r)^{n_{0}−1}S_{n_{0}}(r, x)dr \\ − ∫_{0}^{s}(t + s − r)^{n_{0}−1}S_{n_{0}}(r, x)dr)^{(n_{0},n_{0})}, φ(t)ψ(s)⟩, \end{matrix}  (8)

for every φ, ψ ∈ K_{1_{0}}(ℝ) and x ∈ E.

b) If (R(λ))_{Reλ>ω} is a pseudoresolvent then

⟨S(t, S(s, x)), φ(t, s)⟩ = ⟨S(t + s, x), φ(t, s)⟩, φ ∈ K_{1_{0}}(ℝ^{2}), x ∈ E. (9)

If (9) holds and S = G′, where G is strongly continuous, G : ℝ → L(E) and suppG ⊂ [0, ∞), then (R(λ))_{Reλ>ω} is a pseudoresolvent.

Remark. If (8) holds, then it holds for every n ≥ n_{0} with S_{n} = S ∗ f_{n} because S_{n_{0}} = S_{n_{0}+(n−n_{0})}^{(n−n_{0})}.

P r o o f.

a) Since the necessity simply follows, we will prove the sufficiency of (8). We have S = S_{n_{0}}^{(n_{0})}. Let x ∈ E. Then, (8) implies

(S_{n_{0}}(t, S_{n_{0}}(s, x)))^{(n_{0},n_{0})} = (F (t, s, x))^{(n_{0},n_{0})} (in the sense of distributions),

in open sets not intersecting lines x = 0 and y = 0, where

F (t, s, x) = \frac{1}{(n_{0} − 1)!}[ ∫_{t}^{t+s}(t + s − r)^{n_{0}−1}S_{n_{0}}(r, x)dr

− ∫_{0}^{s}(t + s − r)^{n_{0}−1}S_{n_{0}}(r, x)dr], t, s ≥ 0,

F (t, s, x) = 0, t ≤ 0 or s ≤ 0.

Since both sides are supported by [0, ∞) and S_{n_{0}}(t, S_{n_{0}}(s, x)) and F (t, s, x), t, s ∈ ℝ are continuous, it follows that

S_{n_{0}}(t, S_{n_{0}}(s, x)) = F (t, s, x), t, s ≥ 0.

Thus, R(λ) = λ^{n_{0}}L(S_{n_{0}})(λ), Reλ > ω, is a pseudoresolvent.

b) Let (R(λ))_{Reλ>ω} be a pseudoresolvent. By assertion a) and (4)

⟨S(t, S(s, x)), φ(t, s)⟩ = ⟨S_{n_{0}}(t, S_{n_{0}}(s, x)), φ^{(n_{0},n_{0})}(t, s)⟩

= ⟨D_{t}^{n_{0}}D_{s}^{n_{0}}(\frac{1}{(n_{0} − 1)!}[ ∫_{0}^{t+s}(t+s−r)^{n_{0}−1}S_{n_{0}}(r, x)dr−∫_{0}^{t}(t+s−r)^{n_{0}−1}S_{n_{0}}(r, x)dr

− ∫_{0}^{s}(t + s − r)^{n_{0}−1}S_{n_{0}}(r, x)dr]), φ(t, s)⟩, φ ∈ K_{1_{0}}(ℝ^{2}),

where D_{t}^{n_{0}} = \frac{∂^{n_{0}}}{∂t^{n_{0}}} . Then,

⟨S(t, S(s, x)), φ(t, s)⟩

= ⟨D_{t}^{n_{0}}D_{s}^{n_{0}}(\frac{1}{(n_{0} − 1)!}[ ∫_{0}^{t+s}(t+s−r)^{n_{0}−1}S_{n_{0}}(r, x)dr−∫_{0}^{t}(t+s−r)^{n_{0}−1}S_{n_{0}}(r, x)dr

62 M.Mijatović, S.Pilipović

− ∫_{0}^{s}(t + s − r)^{n_{0}−1}S_{n_{0}}(r, x)dr]), φ(t, s)⟩

= ⟨D_{t}^{n_{0}}(S_{n_{0}}(t + s − r) − S_{n_{0}}(s, x) − ∑_{j=1}^{n_{0}−2} \frac{t^{j}}{j!}S^{(j+1)}(s, x)), φ(t, s)⟩

= ⟨S(t + s, x), φ(t, s)⟩.

This implies

⟨S(t, S(s, x)), φ(t, s)⟩ = ⟨S(t + s, x), φ(t, s)⟩, φ ∈ K_{1_{0}}(ℝ^{2}). Now we prove the second part of b). By (9) we have

S(φ ∗ ψ, x) = S(φ, S(ψ, x)), φ, ψ ∈ K_{1_{0}}, x ∈ E.  (10)

Let φ, ψ ∈ K_{1_{0}} and x ∈ E. Then by using G(φ, G(ψ, x)) = ⟨S(t, S(s, x)), φ′(t)ψ′(s)⟩,

G(φ ∗ ψ, x) = −⟨S(t + s, x)), φ′(t)ψ(s)⟩,

we obtain

∫ ∫ S(t, S(s, x))φ′(t)ψ′(s)dt ds = − ∫ ∫ S(t + s, x))φ′(t)ψ(s)dt ds

= ∫ ∫ ( ∫_{t}^{t+s}S(t + s − r, x)dr − ∫_{0}^{s} S(r, x)dr)φ′(t)ψ′(s)dt ds.

Put P (t, s, x) = 0 if t ≤ 0 or s ≤ 0 and

P (t, s, x) = ∫_{t}^{t+s}S(t + s − r, x)dr − ∫_{0}^{s} S(r, x)dr − S(t, S(s, x)), t > 0, s > 0.

This is a continuous function on ℝ^{2} and P (t, 0, x) = 0, t ∈ ℝ, P (0, s, x) = 0, s ∈ ℝ. We have

⟨P (t, s, x), φ′(t)ψ′(s)⟩ = 0, φ, ψ ∈ K_{1_{0}}(ℝ).  (11)

Put, for fixed φ,

p(s, x) = ∫ P (t, s, x)φ′(t)dt, s ∈ ℝ.

It is continuous on ℝ and p(0, x) = 0.

Now by (11) we have

⟨p(s, x), ψ′(s)⟩ = 0, ψ ∈ K_{1_{0}}(ℝ).

The continuity argument implies p(s, x) = c.

Thus, we have ⟨P (t, s, x), φ′(t)⟩ = 0, ∀φ ∈ K_{1_{0}}. This implies P (t, s, x) = C(s), t > 0, s > 0, where C is a continuous function. With the same procedure, but first applying P (t, s, x) on ψ′(s) we obtain

P (t, s, x) = C_{1}(t), t > 0, s > 0,

where C_{1} is a continuous function.

This implies C_{1}(t) = C(s), t > 0, s > 0. Thus C_{1}(t) = C(s) and P (t, s, x) = 0, t > 0, s > 0. □

Definition 1. Let S ∈ K′_{1^{+}}(L(E)). Then, S is called a 0-exponentially bounded distribution semigroup, a 0-EDSG, in short, if there exists n_{0} ∈ ℕ, such that S_{n_{0}} = S∗f_{n_{0}} is continuous on ℝ, supported by [0, ∞), exponentially bounded and satisfies (8). It is called non-degenerate if ⟨S(t, x), φ(t)⟩ = 0 for all φ ∈ K_{1_{0}}, implies x = 0. We will also use the notation (S(t))_{t≥0} for an 0-EDSG.

Clearly, C_{0}-semigroup is a 0-EDSG.

Let (S(t))_{t≥0} be a 0-EDSG, and R(λ) = L(S)(λ), where Reλ > ω. Then, by the resolvent equation, kerR(λ) is independent of Reλ > ω. Hence, by the uniqueness theorem R(λ) is injective if and only if (S(t))_{t≥0} is non-degenerate. In that case there exists a unique operator A : D(A) → E, (D(A) ⊂ E) satisfying (ω, ∞) ⊂ ρ(A) such that R(λ) = (λI−A)^{−1}, Reλ > ω. This operator is called the generator of (S(t))_{t≥0}. We put this in the following definition.

Definition 2. A closed operator A is the generator of a 0-EDSG (S(t))_{t≥0} if (a, ∞) ⊂ ρ(A) for some a ∈ ℝ and the function λ ↦ (λI − A)^{−1} = L(S)(λ), Reλ > a, and it is injective, where the Laplace transformation is understood in the sense of distribution theory.

Theorem 2 and the above definition directly imply the next Proposition.

Proposition 1. Let (S_{n}(t))_{t≥0}, n ∈ ℕ be an n-t.i.e.b.s. Then S_{n} ∗ f_{−n} is a 0-EDSG. If (S(t))_{t≥0} is a 0-EDSG, then there is n_{0} ∈ ℕ such that S ∗f_{n} is an n-t.i.e.b.s. for every n ≥ n_{0}.

Also, A is the generator of an n-t.i.e.b.s. (S(_{n}(t))_{t≥0} if and only if A is the generator of a 0-EDSG, S_{n} ∗ f_{−n}.

The relations of a 0-EDSG (S(t))_{t≥0} and its infinitesimal generator A are slightly different then in the case of an n-t.i.e.b.s. However, in the proofs we have to use results for n-t.i.e.b.s. and after that to apply the n-th distributional derivative.

Theorem 3. Let A be a generator of a 0-EDSG (S(t))_{t≥0}. Then, for all φ ∈ K_{1}, we have

a) A⟨S(t, x), φ(t)⟩ = ⟨S(t, Ax), φ(t)⟩, x ∈ D(A)

b) For every x ∈ E, ⟨S(t, x), φ(t)⟩ ∈ D(A).

c) ⟨S(t, x), φ(t)⟩ = ⟨f_{1}(t, x), φ(t)⟩+⟨(f_{1} ∗S)(t, Ax), φ(t)⟩, x ∈ D(A) and

A⟨(f_{1} ∗ S)(t, x), φ(t)⟩ = ⟨S(t, x), φ(t)⟩ − ⟨f_{1}(t, x), φ(t)⟩, x ∈ E.

In particular

A⟨S(t, x), φ(t)⟩ = −⟨S(t, x), φ′(t)⟩ − φ(0)x, x ∈ E.

Remark. We will use also the notation A⟨S(t, x), φ(t)⟩ = ⟨AS(t, x), φ(t)⟩.

P r o o f.

a) Let n_{0} ∈ ℕ such that S_{n_{0}} = S ∗ f_{n_{0}} is an n_{0}-t.i.e.b.s. Let φ ∈ D(ℝ) and x ∈ D(A). Then,

⟨S(t, x), φ(t)⟩ = (−1)^{n_{0}}⟨S_{n_{0}}(t, x), φ^{(n_{0})}(t)⟩, n_{0} ∈ ℕ.

Proposition 3.3 in [2] implies S_{n_{0}}(t, x) ∈ D(A) and AS_{n_{0}}(t, x) = S_{n_{0}}(t, Ax). This and the continuity of A imply

A⟨S(t, x), φ(t)⟩ = (−1)^{n_{0}}A⟨S_{n_{0}}(t, x), φ^{(n_{0})}(t)⟩

= (−1)^{n_{0}}A ∫ S_{n_{0}}(t, x)φ^{(n_{0})}(t)dt = (−1)^{n_{0}}A lim_{v→∞} ∑_{i=1}^{v} S_{n_{0}}(t_{j}, x)φ^{(n_{0})}(t_{j})Δt_{j}

= (−1)^{n_{0}} lim_{v→∞} ∑_{j=1}^{v} AS_{n_{0}}(t_{j}, x)φ^{(n_{0})}(t_{j})Δt_{j}

= (−1)^{n_{0}}⟨AS_{n_{0}}(t, x), φ^{(n_{0})}(t)⟩ = ⟨S(t, Ax), φ(t)⟩, x ∈ E, φ ∈ D where ( ∑_{j=1}^{v} S_{n_{0}}(t_{j}, x)φ^{(n_{0})}(t_{j})Δt_{j}), v ∈ ℕ is a sequence of integral sums for

∫ S_{n_{0}}(t, x)φ^{(n_{0})}(t)dt.

Let φ ∈ K_{1} and φ_{v} be a sequence in D(ℝ) which converges to φ in K_{1}. Then,

A⟨S(t, x), φ(t)⟩ = lim_{v→∞}⟨S(t, Ax), φ_{v}(t)⟩ = ⟨S(t, Ax), φ(t)⟩.

This implies the assertion.

b) By Proposition 3.3 in [2], we have ∫_{0}^{t} S_{n_{0}}(s, x)ds ∈ D(A) for every

x ∈ E. Thus, ⟨ ∫_{0}^{t} S_{n_{0}}(s, x)ds, φ(t)⟩ ∈ D(A) for every φ ∈ K_{1} and x ∈ E. By

putting φ^{(n_{0}+1)} instead of φ, we obtain ⟨S(·, x), φ⟩ ∈ D(A) for every φ ∈ K_{1}.

c) Note, for x ∈ D(A), t ≥ 0, S_{n_{0}}(t, x) ∈ D(A), AS_{n_{0}}(t, x) = S_{n_{0}}(t, Ax),

S_{n_{0}}(t, x) = \frac{t^{n_{0}}}{n_{0}!}x + ∫_{0}^{t} S_{n_{0}}(s, Ax)ds (Proposition 3.3. in [2]). This implies

⟨S(t, x), φ(t)⟩ = (−1)^{n_{0}}⟨S_{n_{0}}(t, x), φ^{(n_{0})}(t)⟩
(−1)^{n_{0}}⟨f_{n_{0}+1}(t, x), φ^{(n_{0})}(t)⟩ + (−1)^{n_{0}}⟨f_{1} ∗ S_{n_{0}})(t, Ax), φ^{(n_{0})}(t)⟩
= ⟨f_{1}(t, x), φ(t)⟩ + ⟨(f_{1} ∗ S_{n_{0}}^{(n_{0})})(t, Ax), φ(t)⟩
= ⟨f_{1}(t, x), φ(t)⟩ + ⟨(f_{1} ∗ S)(t, Ax), φ(t)⟩, x ∈ D(A), φ ∈ K_{1},

which gives the first assertion.

Again by using the quoted Proposition 3.3 in [2], it follows

A⟨(f_{1} ∗ S)(t, x), φ(t)⟩ = (−1)^{n_{0}}⟨A(f_{1} ∗ S_{n_{0}})(t, x), φ^{(n_{0})}(t)⟩
= (−1)^{n_{0}}⟨S_{n_{0}}(t, x), φ^{(n_{0})}(t)⟩ − (−1)^{n_{0}}⟨f_{n_{0}+1}(t, x), φ^{(n_{0})}(t)⟩

= ⟨S_{n_{0}}^{(n_{0})}(t, x), φ(t)⟩ − ⟨f_{1}(t, x), φ(t)⟩ = ⟨S(t, x), φ(t)⟩ − ⟨f_{1}(t, x), φ(t)⟩,

which gives the second assertion. The particular case follows by putting φ′ in the second relation in assertion c). □

3. Relations with distribution semigroup of Lions

We follow Definition 6.1 in [18] of a distribution semigroup. Note, instead of S(ℝ) which is used as the basic space in [18], we use in this paper the space K_{1}(ℝ).

Let us recall conditions for a G ∈ D′_{+}(L(E)) to be a distribution semigroup.

(D.1) G(φ ∗ ψ, ·) = G(φ, G(ψ, ·)), φ, ψ ∈ D_{0};

(D.2) ⋂_{φ∈D_{0}} N(G(φ, ·)) = {0};

(D.3) The linear hull R of ⋃_{φ∈D_{0}} R(G(φ, ·)) is dense in E;

(D.4) for every x ∈ R there exists a function u : ℝ → E such that suppu ⊂ [0, ∞), u(0) = x and u is continuous for t ≥ 0 and G(φ, x) =

∫_{0}^{∞}φ(t)u(t)dt for any φ ∈ D_{0}.

If, in addition, there exists ξ_{0} ∈ ℝ such that (D.5) e^{−ξt}G ∈ S′_{+}(L(E)), for ξ > ξ_{0}, then it is called an exponentially

bounded distribution semigroup, EDSG in short.

With another convolution φ ⊗ ψ = ∫_{0}^{t} φ(x − t)ψ(t)dt, φ, ψ ∈ D, Wang

[35] and Kunstmann [17] have introduced a quasi-distribution semigroup, QDSG in short, as an element of D′(L(E)) satisfying

(Q.D.1) G(φ ⊗ ψ, ·) = G(φ, G(ψ, ·)), φ, ψ ∈ D, (Q.D.2) = (D.2). One can simply show that (Q.D.1) implies G ∈ D′_{+}(L(E)). If, more-

over, (D.5) holds, than it is called exponentially bounded quasi-distribution semigroup, EQDSG in short. In fact in [35] is used another definition of exponential boundedness.

Let (S_{0}(t))_{t≥0} be a 0-EDSG with an infinitesimal generator which is not densely defined. We recall:

S_{0}(φ, x) = 0 for every φ ∈ D_{0} ⇒ x = 0.  (12)

As in [18], we extend (S_{0}(t))_{t≥0} on T ∈ E′(ℝ), suppT ⊂ [0, ∞) using δ sequences {ρ_{v}} in D_{0}, (ρ_{v} → δ). We denote by D(S_{0}(T)) the set of x ∈ E such that:

(i) S_{0}(ρ_{v}, x) → x, v → ∞, (ii) lim_{v→∞} S_{0}(T ∗ ρ_{v}, x) exists, and does not depend on ρ_{v} for which (i)

holds.

This means that if ρ̃_{v} is another δ sequence in D_{0} for which (i) holds, then S_{0}(T ∗ ρ̃_{v}, x) − S_{0}(T ∗ ρ_{v}, x) → 0 as v → ∞. This limit defines S_{0}(T, x).

Because of (12), we can define the closure of S_{0}(T, ·) which will be denoted by \overline{S_{0}(T, ·)}.

In the next theorem we summerize basic relations between C_{0}-semigroups and various distribution exponentially bounded semigroups.

Theorem 4. a) Let (T (t))_{t≥0} be a C_{0}-semigroup and A its infinitesimal generator. Then by

J (φ, x) = (T ∗ φ̌)(0)(x) = ⟨T (t, x), φ(t)⟩, φ ∈ K_{1^{+}}, x ∈ E

is defined an EDSG, Ax = \overline{J (−δ′, x)}, x ∈ D(A) and

AJ (φ, x) = − ∫_{0}^{∞}T (t, x)φ′(t)dt − ⟨δ(t, x), φ(t)⟩ =

= − ∫_{0}^{∞}T (t, x)φ′(t)dt − φ(0)x, x ∈ D(A), φ ∈ K_{1^{+}}.

b) Let (S_{0}(t))_{t≥0} be a 0-EDSG with a densely defined infinitesimal generator A. Then, by

J_{0} : φ → ⟨S_{0}(t, ·), φ(t)⟩, φ ∈ K_{1^{+}},

is defined an EDSG. Moreover, A = J_{0}(−δ′, ·).

If A is not densely defined, then by J_{0} is defined an element of K′_{1^{+}}(L(E)) which satisfies all the properties for an EDSG except that {J_{0}(φ, x); φ ∈ D_{0}, x ∈ E} is dense in E.

c) Let (S_{n}(t))_{t≥0}, n ∈ ℕ_{0} be an n-t.i.e.b.s. Assume that its infinitesimal generator A is densely defined. Then,

J_{n}(φ, x) = (S_{n} ∗ φ̌)(0)(x), φ ∈ K_{1^{+}},  (13)

defines an element of K′_{1^{+}}(L(E)) which is an EDSG if and only if n = 0. d) Let S ∈ D′_{+}(L(E)). It is a 0-EDSG if and only if it is an EQDSG.

P r o o f.

a) Clearly J ∈ K′_{1^{+}}(L(E)). Then,

(T ∗ (φ̌ ∗ ψ))(0)(x) = ((T ∗ ψ̌) ∗ φ̌)(0)(x), φ, ψ ∈ D_{0}(ℝ) and this implies J (φ ∗ ψ, x) = J (φ, J (ψ, x)), x ∈ E, φ, ψ ∈ D_{0}(ℝ).

Put y = J (ψ, x), where ψ ∈ D_{0}(ℝ) and x ∈ E are fixed. Then, a distribution φ ↦ J (φ, y), φ ∈ D_{0}(ℝ) is determined by a continuous function u : [0, ∞) → E, u(0) = y. In fact, we have u(t) = (T ∗ ψ̌)(t).

Let x ∈ E be fixed. If J (φ, x) = 0 for every φ ∈ D_{0}, then it follows

x = 0 or x ≠ 0 and T = ∑_{α=0}^{m} a_{α}δ^{(α)} ⊗ I, where I is the identity operator

and a_{α} ∈ ℂ, α = 0, 1, ..., m. But since T could not be of this form, we have x = 0. Let us prove that {⟨T (t, x), φ(t)⟩; φ ∈ D_{0}, x ∈ E} is dense in E. For every t_{0} ∈ [0, ∞) and every x ∈ E,

⟨T (t, x), δ_{n}(t − t_{0})⟩ converges in E to T (t_{0}, x) as n → ∞,

where δ_{n}(t − t_{0}) is a δ - sequence. For example, we can take δ_{n}(t − t_{0}) = nφ(n(t − t_{0})), where φ ∈ D_{0}(ℝ), ∫ φ = 1. Since the set {T (t_{0}, x); t_{0} ∈ [0, ∞), x ∈ E} is dense in E, the assertions follows. Thus, we verify all the conditions which J has to satisfy to be an EDSG.

We have to prove that Ax = \overline{J (−δ′, x)}, x ∈ D(A); this implies that A is the infinitesimal generator of semigroup J .

By noting that for x ∈ D(A), T (·, x) is differentiable, we have

\overline{J (−δ′, x)} = (T ∗ δ′)(0)(x) = T′(0)(x) = Ax.

Let x ∈ D(A), then we have

AJ (φ, x) = J (−δ′ ∗ φ, x) = J (−φ′, x) = ∫_{0}^{∞}T′(t, x)φ(t)dt, φ ∈ K_{1^{+}}.

Now, by partial integration, we obtain

AJ (φ, x) = T (t, x)φ(t)|_{0}^{∞} − ∫_{0}^{∞}T (t, x)φ′(t)dt = − ∫_{0}^{∞}T (t, x)φ(t)dt − φ(0)x.

b) Let (S_{0}(t))_{t≥0} be a 0-EDSG with a densely defined infinitesimal generator A. Let S_{0}(φ, x), φ ∈ K_{1^{+}}, x ∈ E, be defined by

S_{0}(φ, x) = (S_{0}(·, x) ∗ φ̌)(0) = ⟨S_{0}(t, x), φ(t)⟩.  (14)

Let φ, ψ ∈ D_{0}. Then,

(S_{0}(·, x) ∗ (φ̌ ∗ ψ))(0) = ⟨S_{0}(t, x), (φ ∗ ψ)(t)⟩

= ⟨S_{0}(t + s, x), φ(t)ψ(s)⟩ = ⟨S_{0}(t, ⟨S_{0}(s, x), ψ(s)⟩), φ(t)⟩.

This implies

S_{0}(φ ∗ ψ, x) = S_{0}(φ, S_{0}(ψ, x)), x ∈ E, φ, ψ ∈ D_{0}.

Clearly, for fixed ψ ∈ D_{0} and x ∈ E, D_{0} ∋ φ ↦ S_{0}(φ, S_{0}(ψ, x)) is determined by u(t) = (S_{0}(·, x) ∗ ψ̌)(t) which is continuous and supported by [0, ∞).

Since S_{0} is a non-degenerate one and not of the form ∑_{α=0}^{m} a_{α}δ^{(α)} ⊗ I, it

follows

(S_{0}(φ, x) = 0 for every φ ∈ K_{1_{0}}) ⇒ (x = 0).

Similary as in [18], pp. 152-153, we will prove that {S_{0}(φ, x); φ ∈ D_{0}, x ∈ E} is dense in E. This will imply that (13) defines an EDSG.

The duals of D(A) and E, D(A)′ and E′ are equal, since D(A) is dense in E. Thus,

H(φ, ·) =^{t} ⟨S_{0}(t, ·), φ(t)⟩ ∈ D(A)′, φ ∈ K_{1^{+}}, where superscript t denotes a transposed operator. Moreover, H ∈ K′_{1}(L(D(A)′, E)), suppH ⊂ [0, ∞). Theorem 3, a) and c) implies

−H(φ)^{t}A − H(φ′) = φ(0)I_{E′}
−^{t}AH(φ) − H(φ′) = φ(0)I_{D(A)′}.

Let x′ ∈ E′ and

⟨⟨S_{0}(t, x), φ(t)⟩, x′⟩ = 0, for every φ ∈ D_{0}, x ∈ E.  (15)

We have to prove that x′ = 0 which means, by Hahn-Banach theorem, that {⟨S_{0}(t, x), φ(t)⟩; φ ∈ D_{0}, x ∈ E} is dense in E. Now (15) implies ⟨H(φ, ·), x′⟩ = 0 which implies x′ = 0.

We define A_{0}x = \overline{S_{0}(−δ′, x)}. Then, A_{0} = A because the resolvent for both operators are determined by L(S_{0})(λ), Reλ > ω, (cf. [18]).

Let \overline{D(A)} ≠ E. Then, except of density of \{S_{0}(φ, x); φ ∈ D_{0}, x ∈ E\}, one can simply prove that S_{0} verifies all the other properties which should be satisfied by an EDSG.

c) Let J be an EDSG. As it is remarked by Arendt, Theorem 4.3 in [2] and Theorem 3.2 in [25] imply that there exists an n-t.i.e.b.s., (S_{n})(t))_{t≥0}, n ∈ ℕ, such that

J (φ, x) = ⟨S_{n}^{(n)}(t, x), φ(t)⟩ = (S_{n}^{(n)}(·, x) ∗ φ̌)(0), φ ∈ D, x ∈ E.

This implies S_{n}^{(n)} = S_{n} ∗ f_{−n} = S_{0} = J , where S_{0} satisfies Theorem 2.

If for some n ∈ ℕ, (13) defines an EDSG then (S_{n}(t))_{t≥0} determine exponentially bounded distribution semigroups which is impossible by the uniqueness of an EDSG with the given infinitesimal generator A.

d) The properties of a 0 − EDSG given in Theorems 1,2, Proposition 1 and in [35], Theorem 3.8, as well as arguments in [35], Section 4, imply the assertion. □

The last part of Theorem 4 shows that our investigations give another approach and insight into the theory of EQDSG.

With the asumptions as in the second part of Theorem 4 b), we have the following theorem.

Theorem 5.Let (S_{0}(t)_{t≥0} be a 0−EDSG with an infinitesimal generator A such that \overline{D(A)} ≠ E. Denote by E_{0} the closure in E of the set Ẽ = {S_{0}(φ, x); x ∈ E, φ ∈ D_{0}}. Then the set {S_{0}(φ, x); x ∈ E_{0}, φ ∈ D_{0}} is dense in E_{0} and S_{0|E_{0}×K_{1+}} defines an EDSG with the infinitesimal generator A = A_{|D(A)∩E_{0}}.

P r o o f. Let y ∈ E_{0} and y_{n} = S_{0}(φ_{n}, x_{n}) ∈ Ẽ ⊂ E_{0} be a sequence which converges to y in E_{0}. We have S_{0}(φ_{n}, x_{n}) = lim_{m→∞} S_{0}(φ_{n} ∗ ρ_{m}, x_{n}), where ρ_{m} is a δ sequence in D_{0}. Since

S_{0}(φ_{n} ∗ ρ_{m}, x_{n}) = S_{0}(ρ_{m}, (S_{0}(φ_{n}, x_{n})) = S_{0}(ρ_{m}, y_{n}), m, n ∈ ℕ,

we obtain that y is an limit point for {S_{0}(φ, x); x ∈ E_{0}, φ ∈ D_{0}} and y_{n} ∈ Ẽ.

Since {S_{0}(φ, x), x ∈ E_{0}, φ ∈ D_{0}} ⊂ D(A), we have that D(A) ∩ E_{0} is dense in E_{0}. This completes the proof. □

4. Applications

Let A be an operator on E, and T ∈ K′_{1} (E), suppT ⊂ [a, ∞), a ≥ 0. Then, u ∈ K′_{1} (E) is a solution to equation

u′ = Au + T, in K′_{1}(E)  (16)

if ⟨u(t), φ(t)⟩ ∈ D(A) for every φ ∈ K′_{1+}(ℝ) and (16) holds.

Let A be an infinitesimal generator of a 0−EDSG(S_{0}(t))_{t≥0.} We assume that D(A) is supplied with the graph norm ∥x∥ + ∥Ax∥, x ∈ D(A).

Let U ∈ K′_{1}(L(E, D(A))), V ∈ K′_{1}(L(D(A), E)) and suppU ⊂ [a, ∞), supp V ⊂ [b, ∞), a, b ≥ 0. Then, U ∗ V and V ∗ U are elements of K′_{1}(L(D(A))) and K′_{1}(L(E)), respectively and their supports are contained in [a + b, ∞) (cf. [28]).

Theorem 6.Let A be an infinitesimal generator of a 0−EDSG (S_{0}(t))_{t≥0.} Then (S_{0}(t))_{t≥0} is an EDSG with an infinitesimal generator A on E_{0}×K′_{1+}, where E = \overline{\{S_{0}(φ, x); φ ∈ D_{0}, x ∈ E\}} and

( − A + \frac{∂}{∂t}) ∗ S_{0} = δ ⊗ I_{E_{0}}, S_{0} ∗ ( − A + \frac{∂}{∂t}) = δ ⊗ I_{D(A)∩E_{0}}.

Let T ∈ K′_{1}(E_{0}) and suppT ⊂ [a, ∞) for some a ≥ 0. Then, u = S_{0} ∗ T is a unique solution to

u′ = Au + T in K′_{1}(L(E_{0})), suppu ⊂ [a, ∞).

The proof of Theorem 6 is the same as the proof of Theorem 4.1 in [18].

Example 1. Let A = ∑_{j=0}^{k} a_{j}(\frac{d}{dx})^{j}, p(x) = ∑_{j=0}^{k} a_{j}(ix)^{j}(i^{2} = −1), Rep(x) < ∞, a_{j} ∈ ℂ, j = 0, 1, ..., k, E = C_{b}(ℝ) or E = L^{∞}(ℝ),

Af = ∑_{j=0}^{k} a_{j}(\frac{d}{dx})^{j}f, where

D(A) = \{f ∈ E; ∑_{j=0}^{k} a_{j}(\frac{d}{dx})^{j}f ∈ E distributionally \}.

It is proved in [16] that

S_{t}f = \frac{1}{\sqrt{2π}} F^{−1}(ϕ_{t}) ∗ f, t ≥ 0, f ∈ E,

where (ϕ_{t}(ξ) = f_{1}(t) ∗ e^{p(ξ)t})_{t≥0}, is an 1-t.i.e.b.s. with the generator A such that \overline{D(A)} ≠ E. Here F denotes the Fourier transformation and F^{−1} denotes its inverse; F(f)(λ) = f̃(λ) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{+∞} e^{−iλt}f(t)dt, λ ∈ ℝ. We have

⟨f_{−1}(t) ∗ ϕ_{t}(ξ), α(t))⟩ = ⟨e^{p(ξ)t}, α(t)⟩, α ∈ K′_{1+}(ℝ).

Thus F^{−1}(e^{p(ξ)t}) = \frac{d}{dt}(\frac{1}{\sqrt{2π}} F^{−1}(ϕ_{t})), in the sense of distributions, defines a 0 − EDSG which is not an EDSG. If T ∈ K′_{1}(E_{0}), suppT ⊂ [a, ∞), a ≥ 0, then the solution to equation u′ = Au + T is given by F^{−1}(e^{p(ξ)t}) ∗ T (t) (cf. Theorem 6).

Example 2. Let m > 0. Consider

\frac{∂}{∂t}u(x, t) − iΔ^{\frac{m}{2}} u(x, t) = f(x, t), f ∈ K′_{1}(L^{p}(ℝ^{n})), suppf ⊂ [a, ∞), a ≥ 0.

With m = 2 and f ∈ T′(L^{p}(ℝ^{n})) this equation is analyzed in [5] and [6]. We recall the definition of the space M_{p} of Fourier multipliers on L^{p}(ℝ^{n}),

1 ≤ p ≤ ∞ (cf. [25]): a ∈ M_{p} if and only if a is a tempered distribution and ∥a∥_{M_{p}} = sup_{∥f∥_{L^{p}}=1} ∥F^{−1}(af̃)∥_{L^{p}} < ∞.

Let m > 0, k > 0 and H_{m}(ξ) = |ξ|^{m}, ξ ∈ ℝ^{n}. Then, the following asertion is proved in [25]:

a_{H_{m},k}(ξ, t) = \frac{k}{t^{k}} ∫_{0}^{t}(t − s)^{k−1}e^{−isH_{m}(ξ)}ds ∈ M_{p}, t > 0, ξ ∈ ℝ^{n},

if:

for m ≠ 1, k ≥ n|\frac{1}{p} − \frac{1}{2}|, for m = 1, k ≥ (n − 1)|\frac{1}{p} − \frac{1}{2}|.  (17)

Moreover, ∥a_{H_{m},k}(ξ, t)∥_{M_{p}} does not depend on t.

Define, for φ ∈ K_{1}(ℝ), u ∈ S(ℝ^{n}),

G_{0}(φ, u)(x) = F^{−1}( ∫_{ℝ} e^{−itH_{m}(ξ)}φ(t)dtũ(ξ))(x), x ∈ ℝ^{n}.

Let k be an integer which satisfies (17). Then,

G_{0}(φ, u)(x) = \frac{(−1)^{k}}{k!} ∫_{ℝ} t^{k}φ^{(k)}(t)F^{−1}(a_{H_{m,}k}(ξ, t)ũ(ξ))dt(x), x ∈ ℝ^{n}.

It implies that for given φ ∈ K_{1}(ℝ) and S(ℝ^{n}) there exists C_{φ} > 0 such that

∥G_{0}(φ, u)∥_{L^{p}},

which shows that it can be extended on L^{p}(ℝ^{n}) as a continuous linear operator.

Moreover, G_{0} is an EDSG with the infinitesimal generator

A = iH_{m}(∂) = i(\frac{∂^{2}}{∂x_{1}^{2}} + ... + \frac{∂^{2}}{∂_{n}^{2}} )^{\frac{m}{2}}

which is defined on the closure in L^{p} of the set {u ∈ S; F^{−1}(|ξ|^{m}ũ) ∈ L^{p}}. Thus we have that u = G_{0} ∗ f is the solution to equation

\frac{∂}{∂t}u − iH_{m}(∂)u = f, f ∈ K′_{1}(L^{p}), suppf ⊂ [a, ∞).

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Institute of Mathematics University of Novi Sad Trg Dositeja Obradovića 4 21000 Novi Sad Yugoslavia