Bulletin T.CXXVII de l’Académie Serbe des Sciences et des Arts - 2003 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 28

DISTRIBUTION SEMIGROUPS ON K_{1}

M.MIJATOVIĆ, S.PILIPOVIĆ

(Presented at the 8th Meeting, held on November 22, 2002)

A b s t r a c t. Distribution semigroup in the sense of Wang and Kunstmana and the properties of infinitesimal generator are considered with in exponentially bounded distributions. Results are applied on a class of equa-

tions of the form \frac{∂}{∂t}4 − An = f, f ∈ K_{1}^{+}(L(e)), where D(A) ⊂ L^{∞}(ℝ) or D(A) ⊂ E = C_{b}(ℝ).

AMS Mathematics Subject Classification (2000): 47D06, 47A10, 46F10 Key Words: Distribution semigroups, quasi-distribution semigroups, in-

tegrated semigroups, infinitesimal generator.

0. Introduction

Distribution semigroups of Lions [12] and, later introduced, n-times integrated semigroups of Arendt [2], have been studied by many authors. The aim has been applications to Cauchy problems with the luck of regularity conditions or with non-densely defined infinitesimal generators. The references contain enough informations in these sense. Wang [23] and Kunstmann [11] introduced quasi-distribution semigroups and exponentially bounded distribution semigroups which we call (DS) and (EDS).

In this paper we analyze the properties of the infinitesimal generator of such a semigroup within distribution theory. As a basic space we use the test function space K_{1}. This is a natural framework for exponentially bounded distributions. The density of D(A) in E and of a set {S(φ, x); x ∈ D(A), φ ∈ D_{0}} in D(A), where S is a (EDS) with the infinitesimal gen-

erator A, is used in solving equations of the form \frac{∂}{∂t} u − Au = f, where

f ∈ K′_{1}(D(A)), A = ∑_{j=0}^{k} a_{j} \frac{∂^{j}}{∂x^{j}} , sup Re(p(x)) < ∞, p(x) = ∑_{j=0}^{k} a_{j}(ix)^{j} and

D(A) ⊂ E = L^{∞}(ℝ) or D(A) ⊂ E = C_{b}(ℝ).

1. Preliminaries

We denote by E a Banach space with the 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 [18-19] and [21] 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 [21] for the space

S_{+} = {φ; |t^{k}φ^{(ν)}(t)| < C_{k,ν}, t ∈ [0, ∞), k, ν ∈ ℕ_{0}} (ℕ_{0} = ℕ ∪ {0})

and its dual S′_{+}, which consists of tempered distributions supported by [0, ∞). Recall ([7]), the space of exponentially decreasing test functions on the real line ℝ is defined by K_{1}(ℝ) = {φ; e^{k|t|}|φ^{(ν)}(t)| < C_{k,ν}, t ∈ ℝ, k, ν ∈ ℕ_{0}}. This space has the same topological properties as S(ℝ). The space K_{1}(ℝ^{2}) is defined in an appropriate way. 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 supported by [0, ∞). It is the dual space to K_{1+} = {φ; e^{k|t|}|φ^{(ν)}(t)| < C_{k,ν}, t ∈ [0, ∞), k, ν ∈ ℕ_{0}} which has the same topological properties as S_{+}. Spaces K′_{1}(E), K′_{1+}(E) are defined in an appropriate way. Their properties, similar to S′(E) and S′_{+}(E), are given in [15]. Note,

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

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

∥S(t)∥ ≤ Me^{ωt}, t ≥ 0.  (2)

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

Distribution semigroups on K_{1} 9

We need a representation for elements of K′_{1+}(L(E)). This is given in part a) of the next theorem.

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

\begin{matrix}⟨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. \end{matrix}  (3)

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

lim_{ν→∞}⟨S(t, ⟨S(s, x), ψ_{ν}(s)⟩), φ_{ν}(t)⟩, φ ∈ K_{1}(ℝ^{2})

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_{ν→∞}⟨S(t, ⟨S(s, x), ψ_{ν}(s)⟩), φ_{ν}(t)⟩, φ ∈ K_{1}(ℝ^{2}).

(4)

d) Let φ ∈ K_{1}(ℝ^{2}) and r, p ∈ ℕ. We have

⟨ \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)⟨;  (i)

\begin{matrix}⟨\frac{∂^{p}}{∂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)⟩. \end{matrix}  (ii)

P r o o f. Part a) can be proved in the same way as in the case of scalar valued distributions.

Parts b), c) and d) are consequences of the continuity and linearity of S ∈ K′_{1+}(L(E)), more precisely of the generalized Fubini-type theorem.

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.[21]).If f ∈ L^{1}([0, ∞), E) (which means ∥ ∫_{0}^{∞}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.

The convolution of f ∈ K′_{1+}(E) and g ∈ K′_{1+}(ℝ) 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).

In the sequal, we will use the family of distributions

f_{α}(t) = \{ \begin{matrix}\frac{H(t)t^{α−1}}{Γ(α)} , & t ∈ ℝ, α > 0, \\ f_{α+n}^{(n)}(t), & t ∈ ℝ, α ≤ 0, α + n > 0, n > 0, \end{matrix}

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

Let S ∈ K′_{1+}(L(E)) and R(λ) = L(S)(λ), Reλ > ω (cf . [2]). 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, suppS_{n_{0}} ⊂ [0, ∞) and, for φ, ψ ∈ K_{1},

\begin{matrix}⟨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)⟩ \\ = ⟨\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}  (6)

The next definition is equivalent to the one given by Kunstmann and Wang, with D instead of K_{1}.

Definition 1.Let S ∈ K′_{1+}(L(E)). Then S is called exponentially bounded distribution semigroup (EDS), 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, satisfies (6) and it is non-degenerate: ⟨S(t, x), φ(t)⟩ = 0 for all φ ∈ K_{1}, implies x = 0.

We will also use the notation (S(t))_{t≥0} for an (EDS). If (6) holds for ψ ∈ D(−∞, a) for some a > 0, (S(t))_{t≥0} is called a local distribution semigroup. This definition coincides with the definition of (DS) of Kunstmann and Wang but with D instead of K_{1}.

Also the next definition is equivalent to the known one of cited authors.

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

As in case with D instead of K_{1}, one can simply prove the next theorem.

Theorem 2. Let A be a generator of a (EDS) (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) ⟨S(t, x), φ(t)⟩ ∈ D(A), x ∈ E.

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.

We refer to Definition 6.1 in [12] for a distribution semigroup, (DS-L) in short and exponentially distribution semigroups (EDS-L) in short in the sense of Lions. If D(A) is dense in E, then these notions coincide with (DS) and (EDS).

2. Comments on generators

Let (S(t))_{t≥0} be a (DS) or (EDS). Recall S(T, ·), T ∈ E′(ℝ) is defined as follows:

y = S(T, x), if S(T ∗ ψ, x) = S(ϕ, y), ϕ ∈ D_{0}. The set of x ∈ E for which this holds is denoted by D(T). It follows that the domain of S(−δ′, ·) is D(A) and S(−δ′, x) = Ax, x ∈ D(A).

Let S_{n}(·, x) = S(·, x) ∗ f_{n}, x ∈ E be an n-times integrated semigroup determined by the (EDS), (S(t))_{t≥0} with the infinitesimal generator A.

One can simply prove

S_{n}(t, x) = lim_{ν→∞}⟨S_{n}(s, x), ρ_{ν}(t − s)⟩, t ≥ 0,

S_{n}(φ^{(n)}, x) = (−1)^{n}S(φ, x), φ ∈ D_{0}, x ∈ E,

where {ρ_{ν}} is δ sequences in D_{0}, (ρ_{ν} → δ, ν → ∞).

Theorem 3. Let (S(t))_{t≥0} be an (EDS) and (S_{n}(t) = (S ∗ f_{n}(t))_{t≥0} be an n-times integrated exponentially bounded semigroup, n ∈ ℕ_{0} with the infinitesimal generator A. Then

a) D(S(f)) = D(S_{n}(f^{(n)})), f ∈ E′(ℝ), suppf ⊂ [0, ∞) and

S_{n}(f^{(n)}, x) = (−1)^{n}S(f, x), x ∈ D(S(f)),
S_{n}(h, x) = S_{n}(δ(t − h), x), x ∈ E, h > 0,

In particular

(−1)^{n}S_{n}(δ^{(n)}, x) = x, x ∈ E,
(−1)^{n}S_{n}(−δ^{(n+1)}, x) = Ax, x ∈ D(A).

b)

Ax = (n + 1)! lim_{h↓0} \frac{S_{n}(h)x − \frac{h^{n}}{n!} x}{h^{n+1}}, x ∈ D(A).  (7)

P r o o f. We will prove only b). Let φ ∈ D. Since,

\frac{(n + 1)!}{h^{n+1}} (φ(h) − \frac{h^{n}}{n!} φ^{(n)}(0)) → φ^{(n+1)}(0), as h → 0^{+},

it follows

\frac{(n + 1)!}{h^{n+1}} ⟨δ(t − h) − \frac{h^{n}}{n!} (−1)^{n}δ^{(n)}(t), φ(t)⟩ →

→ (−1)^{n+1}⟨δ^{(n+1)}(t), φ(t)⟩, φ ∈ D as h → 0^{+}.

Then, for x ∈ D(A) we have,

(n+1)! lim_{h↓0} \frac{S_{n}(h, x) − \frac{h^{n}}{n!} x}{h^{n+1}} = (n+1)! lim_{h↓0} \frac{S_{n}(δ(t − h), x) − S_{n}(\frac{h^{n}}{n!} (−1)^{n}δ^{(n)}(t), x)}{h^{n+1}}

= (n + 1)! lim_{h↓0} \frac{S_{n}(δ(t − h) − \frac{h^{n}}{n!} (−1)^{n}δ^{(n)}(t), x)}{h^{n+1}}

= S_{n}((−1)^{n+1}δ^{(n+1)}(t), x) = S(−δ′, x) = Ax.

Theorem 4. Let (S(t))_{t≥0} be an (EDS) with the infinitesimal generator A and F = {S(φ, x), x ∈ D(A), φ ∈ D_{0}}. Then F is dense in D(A).

P r o o f. Let x ∈ D(A). Since

S(−δ′_{ν}, x) = lim_{h↓0} ⟨\frac{S(t + h, x) − S(t, x)}{h}, δ_{ν}(t)⟩,

there exists a sequence (h_{ν})_{ν∈ℕ}, h_{ν} → 0^{+}, such that

⟨ \frac{S(t + h_{ν}, x) − S(t, x)}{h_{ν}}, δ_{ν}(t)⟩ → 0, ν → ∞

and therefore

⟨S(t + h_{ν}, x) − S(t, x), δ_{ν}(t)⟩ → 0, ν → ∞.

But we have ⟨S(t, x), δ_{ν}(t)⟩ → x, ν → ∞, which implies that

(⟨S(t, x), δ_{ν}(t − h)⟩)_{ν∈ℕ}

is a sequence in F which converges to x. Thus F is dense in D(A).

Theorem 4 implies that there exists a closed subspace E_{0} of E such that (S(t))_{t≥0} is an (EDS -L) on E_{0}, where E_{0} is the closure in E of the set F = {S(φ, x); x ∈ D(A), φ ∈ D_{0}}.

The following theorem is proved by Wang and Kunstmann. We reformulate it with K_{1} instead of D.

Theorem 5. Let (S(t))_{t≥0} be an (EDS) with the infinitesimal generator A. Then the restriction of (S(t))_{t≥0} on E_{0}×K_{1}, (S_{|E_{0}×K_{1}}), is an (EDS−L).

3. Applications

Example. Let E = C_{b}(ℝ), or L^{∞}(ℝ) and A be defined by Af =

∑_{j=0}^{k} α_{j}D^{j}f, where D^{j} = \frac{∂^{j}}{∂x^{j}} , α_{0}, ..., α_{k} ∈ ℂ, p(x) = ∑_{j=0}^{k} α_{j}(ix)^{j}, k ≥ 1, α_{k} ≠

0, sup_{x∈ℝ} Re(p(x)) < ∞, where D(A) = {f ∈ E, ∑_{j=0}^{k} α_{j}D^{j}f ∈ E, distributionally }. It is known that D(A) is not dense in E (cf.[10]).

Let S_{t}(f) = \frac{1}{\sqrt{2π}} F^{−1}(e^{p(x)t}) ∗ f. Here F denotes the Fourier transfor-

mation and F^{−1} denotes is inverse; F(f)(λ) = ∫_{ℝ} e^{−iλt}f(t)dt, λ ∈ ℝ. Then

it is an (EDS) because S_{t}(f) = \frac{1}{\sqrt{2π}} F^{−1}(∫_{0}^{1} e^{p(x)s}ds) ∗ f, ∈ D(A), is 1-

time integrated semigroup. Moreover, since the set {⟨S(t, f), φ(t)⟩, f ∈ D(A), φ ∈ D_{0}} is dense in D(A) it follows that S_{t} is (EDS-L) on the subspace E_{0} = \{\overline{⟨S(t, f), φ(t)⟩, f ∈ D(A), φ ∈ K_{0}\}}. Note ([16]), A generates

a norm continuous α-times integrated semigroup for α ∈ (\frac{1}{2}, 1] equal to S_{t} ∗ f_{α}, t ≥ 0.

Recall, for U ∈ K′_{1+}(L(E, D(A))), V ∈ K′_{1+}(L(D(A), E)) and supp U ⊂ [a, ∞), supp V ⊂ [b, ∞), a, b ∈ ℝ. Then U ∗V and V ∗U are defined as in [19]. Moreover, they are elements of K′_{1+}(L(D(A))) and K′_{1+}(L(E)) respectively and their supports are bounded on the left.

Now we apply our results to equation

u′ = Au + T, T ∈ K_{1+}(L(E_{0})).

We refer to this equations in the case T ∈ S′_{1+}(L(E_{0})) to ([12],[15]).

Theorem 6. Let (S(t))_{t≥0} be an (EDS-L) with the infinitesimal generator A. Then

( − A + \frac{∂}{∂t}) ∗ S = I_{E_{0}}, S ∗ ( − A + \frac{∂}{∂t}) = I_{D(A)},  a)

where

−A + \frac{∂}{∂t} = δ ⊗ A + δ′ ⊗ I.

b) u = S ∗ T in K′_{1+}(L(E_{0})) is a unique solution of (11)

Remark. In particular, with the notation given above this theorem gives the unique solution to \frac{∂}{∂t}u(t, x) − ∑_{j=0}^{k} a_{j} \frac{∂^{j}}{∂x^{j}} u(t, x) = f, f ∈ K′_{1+}(L(E_{0})) in K′_{1+}(L(E_{0})).

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Institute of Mathematics University of Novi Sad Trg Dositeja Obradovića 4 21000 Novi Sad Serbia and Montenegro