Abstract Degenerate Cauchy Problems in

Locally Convex Spaces ^{∗}

Jin Liang and Ti-Jun Xiao

Department of Mathematics, University of Science and Technology of China

Hefei 230026, People’s Republic of China e-mail:jliang@ustc.edu.cn; xiaotj@ustc.edu.cn

This paper is concerned with the abstract degenerate Cauchy problem (DCP): \frac{d}{dt}Bu(t) = Au(t) (t ≥ 0), Bu(0) = Bu_{0}, where A and B are closed linear operators in a sequentially complete locally convex space. A C-propagation family for (DCP) is introduced, leading to a general C-wellposedness result about (DCP). Moreover, conditions are given ensuring the existence of Cpropagation families for those (DCP) with differential operators, on various function spaces with Fréchet topologies, as coefficient operators. These results are new even in the case of Banach spaces.

0. INTRODUCTION

Let E be a sequentially complete locally convex space topologized by Γ of seminorms, and let A, B be closed linear operators in E. We consider the following abstract degenerate Cauchy problem

\{ \begin{matrix}\frac{d}{dt}Bu(t) = Au(t), t ≥ 0, \\ Bu(0) = Bu_{0}. \end{matrix}  (DCP)

∗This work was supported partly by the K. C. Wong Education Foundation, the Key Project of the Chinese Academy of Sciences and the National NSF of China

The (DCP) has been studied by many researchers in the setting of Banach spaces (see, e.g., [1, 3, 4, 7, 8, 10, 15] and references therein). In this paper, we introduce a C-propagation family {W (t)}_{t≥0} for (DCP) in a locally convex space, which is a generalization of the C-regularized semigroup and so the classical strongly continuous semigroup(cf., e.g., [5 - 7, 9, 11, 14]). In terms of {W (t)}_{t≥0}, we derive a general C-wellposedness result regarding (DCP) (Theorem 1.6). On the other hand, we pay attention to those (DCP) with differential operators, on various function spaces endowed with Fréchet topologies, as coefficient operators, which can not be treated by the known theory about (DCP). We obtain conditions ensuring the existence of Cpropagation families for these (DCP) (Theorems 2.2 and 2.3). The above mentioned results are new even in the case of Banach spaces. For the related discussion about the case of B = I, please see, for example, [5, 14] and references cited there.

1. DEFINITIONS AND THE GENERAL RESULTS

Definition 1.1. By a classical solution of (DCP) we mean a continuous function u(·) : [0, ∞) → D(A) ∩ D(B) such that Bu(·) is continuously differentiable and (DCP) is satisfied.

Definition 1.2. By a mild solution of (DCP) we mean a continuous function u(·) : [0, ∞) → D(B) such that Bu(·) is continuous and ∫_{0}^{t} u(s)ds ∈ D(A) for each t ≥ 0, satisfying

Bu(t) = Bu_{0} + A ∫_{0}^{t} u(s)ds, t ≥ 0.  (1.1)

Let C be a continuous and injective operator on E, with CB ⊂ BC, CA ⊂ AC, and let R(C) ⊂ R(λB − A) and (λB − A)^{−1} exists for λ large enough.

Definition 1.3. (DCP) is (exponentially) C-wellposed if for each u_{0} ∈ C(D(A)∩ D(B)), (DCP) has a unique classical solution u(·) satisfying that, to each p ∈ Γ there corresponds a continuous seminorm q such that

p(u(t)) ≤ e^{ωt}q (C^{−1}u_{0}) , t ≥ 0,

for a constant ω > 0.

Definition 1.4. A strongly continuous operator family {W (t)}_{t≥0} of D(B) to E, satisfying that {e^{−at}W (t); t ≥ 0} is equicontinuous for some a > 0, which means that to each p ∈ Γ there corresponds a continuous seminorm q such that

p(W (t)u) ≤ e^{at}q(u) for any t ≥ 0 and u ∈ D(B),

is called a (exponentially equicontinuous) C-propagation family for (DCP) if for λ > a,

(λB − A)^{−1}CBu = ∫_{0}^{∞} e^{−λt}W (t)udt, u ∈ D(B).  (1.2)

In this case, we also say that (DCP) has a C-propagation family.

Remark 1.5. The uniqueness theorem for Laplace transforms tells us that (DCP) has a unique C-propagation family for a fixed C. When B = I (the identity operator on E), W (·) is exactly the (exponentially equicontinuous) C-regularized semigroup for A (see [5, 14] and the references given there).

As in [14], we write λ ↦ f(λ) ∈ LT − E, for a function f : [ω, ∞) → E, if there exists a continuous h : [0, ∞) ⟶ E with

sup_{t≥0} p (e^{−ωt}h(t)) < ∞ for any p ∈ Γ

such that

f(λ) = ∫_{0}^{∞} e^{−λt}h(t)dt for λ > ω.

Theorem 1.6. Suppose that

(i) (DCP) has a C-propagation family {W (t)}_{t≥0}.

(ii) λ ⟼ B(λB − A)^{−1}Cu ∈ LT − E for each u ∈ E.

Then (DCP) admits a unique mild solution u(t) := W (t)C^{−1}u_{0} for each u_{0} ∈ C(D(B)). Moreover for u_{0} ∈ C(D(B) ∩ D(A)), the mild solution is a classical solution of (DCP). As a consequence, (DCP) is C-wellposed.

Proof. We first let u_{0} ∈ C(D(B)) and set v_{0} = C^{−1}u_{0} (so v_{0} ∈ D(B)). By hypothesis (ii),

λ ⟼ B(λB − A)^{−1}CBv_{0} ∈ LT − E.

From this and (1.2) together with the closedness of B, we obtain, by an application of Theorem 1.1.10 of [14], that W (t)v_{0} ∈ D(B) for each t ≥ 0 and for λ large enough

B(λB − A)^{−1}CBv_{0} = ∫_{0}^{∞} e^{−λt}BW (t)v_{0}dt.  (1.3)

On the other hand, we know

λ ⟼ λ^{−1}A(λB − A)^{−1}CBv_{0} ∈ LT − E

since

λ^{−1}A(λB − A)^{−1}CBv_{0} = B(λB − A)^{−1}CBv_{0} − λ^{−1}BCv_{0}  (1.4)

for λ large enough. Notice

λ^{−1}(λB − A)^{−1}CBv_{0} = ∫_{0}^{∞} e^{−λt} (∫_{0}^{t} W (t)v_{0}ds) dt

for λ large enough, by (1.2). It follows from Theorem 1.1.10 of [14] again that

∫_{0}^{t} W (t)v_{0}ds ∈ D(A) for each t ≥ 0,

and

λ^{−1}A(λB − A)^{−1}CBv_{0} = ∫_{0}^{∞} e^{−λt}A (∫_{0}^{t} W (t)v_{0}ds) dt  (1.5)

for λ large enough. Moreover

λ^{−1}BCv_{0} = ∫_{0}^{∞} e^{−λt}BCv_{0}dt, λ > 0.  (1.6)

Combining (1.3) – (1.6), we conclude by the uniqueness theorem for Laplace transforms that

BW (t)C^{−1}u_{0} = Bu_{0} + A ∫_{0}^{t} W (s)C^{−1}u_{0}ds, t ≥ 0.  (1.7)

Therefore W (t)C^{−1}u_{0} is a mild solution of (DCP).

Next, let u_{0} ∈ C (D(A) ∩ D(B)). Then v_{0} = C^{−1}u_{0} ∈ D(A) ∩ D(B). Observe that for λ large enough,

λB(λB − A)^{−1}CBv_{0} − BCv_{0} = B(λB − A)^{−1}CAv_{0}.

Since

λ ⟼ B(λB − A)^{−1}CAv_{0} ∈ LT − E,

it follows from (1.3) that BW (t)v_{0} is continuously differentiable in t ≥ 0, in view of Theorem 1.1.9 of [14]. Therefore we have by (1.7) that

\{ \begin{matrix}\frac{d}{dt}BW (t)v_{0} = AW (t)v_{0}, t ≥ 0, \\ BW (0)v_{0} = Bu_{0}. \end{matrix}

This says that W (t)C^{−1}u_{0} is a solution of (DCP).

Finally, we show the uniqueness. To this end, let u(·) be a mild solution of (DCP) with Bu(0) = 0. Put v(t) = Cu(t), t ≥ 0. Then one sees easily

CBu(t) = Bv(t) = A ∫_{0}^{t} v(s)ds, t ≥ 0.

Using this, we obtain by integrating by parts that for λ large enough

(B − λ^{−1}A) ∫_{0}^{t} e^{λ(t−s)}v(s)ds
= ∫_{0}^{t} e^{λ(t−s)} (Bv(s) − A ∫_{0}^{s} v(σ)dσ) ds − λ^{−1}A ∫_{0}^{t} v(σ)dσ
= −λ^{−1}Bv(t), t ≥ 0.

Hence for any t ≥ 0, p ∈ Γ,

lim_{λ→+∞} e^{−λ}p (∫_{0}^{t} e^{λ(t−s)}v(s)ds)

= − lim_{λ→+∞} e^{−λ}p ((λB − A)^{−1}CBu(t))
= 0,

in view of (1.2). It follows that

lim_{λ→+∞} ∫_{0}^{t−1} e^{λ(t−1−s)}v(s)ds = 0, t ≥ 1,

which gives v(t) ≡ 0 with the aid of Lemma 4.1.1 of [11]. So u(t) ≡ 0 by the injectivity of C and this completes the proof.

2. (DCP) CONNECTING WITH DIFFERENTIAL OPERATORS

Let R^{n} be the n-dimensional Euclidean space. An n-tuple of nonnegative integers α = (α_{1}, α_{2}, ⋯, α_{n}) is called a multiindex which we denote by α ∈ N_{0}^{n} and we define

|α| = ∑_{i=1}^{n} α_{i}, D^{α} = (\frac{∂}{∂ξ_{1}} )^{α_{1}} ⋯ (\frac{∂}{∂ξ_{n}})^{α_{n}}.

By S(R^{n}), we denote the space of all rapidly decreasing functions on R^{n}. The Fourier transform and its inverse transform are denoted by

(Ff)(ξ) = f̂(ξ) := ∫_{R^{n}} e^{−i⟨η, ξ⟩}f(η)dη

and

(F^{−1}f)(η) := (2π)^{−n} ∫_{R^{n}} e^{i⟨η, ξ⟩}f(ξ)dξ.

Let X be one of the spaces L^{p}(R^{n}) (1 ≤ p ≤ ∞), C_{0}(R^{n}), UC_{b}(R^{n}) or C_{b}(R^{n}). Let l ∈ N_{0} with 0 ≤ l ≤ n. A multiindex α = (α_{1}, α_{2}, ⋯, α_{n}) will be also denoted by α ∈ N_{0}^{l} if α_{l+1} = ⋯ = α_{n} = 0. Define

E_{l} = {f ∈ X; D^{β}f ∈ X, for any β ∈ N_{0}^{l}},

and q_{β}(f) = ∥D^{β}f∥_{X}. It is not difficult to verify (cf. the proof of Lemma 1.5.6 of [14]) that the totality Γ of these seminorms corresponding to all multiindices β ∈ N_{0}^{l} induces a Fréchet topology on E_{l}. Letting F be a subspace of E_{l}, we will denote by B_{Γ}(F, E_{l}) the normed space of those continuous operators B of F to E_{l}, with the norm

∥B∥_{Γ} := sup{q_{β}(Bx); q_{β} ∈ Γ, x ∈ F with q_{β}(x) ≤ 1} < ∞,

and B_{Γ}(E_{l}) := B_{Γ}(E_{l}, E_{l}).

The following lemma is well known for l = 0.

Lemma 2.1. Let X = L^{p} (R^{n}) (1 ≤ p < ∞). Then S(R^{n}) is dense in E_{l} for every l = 0, 1, 2, …, n.

Proof. For any r ∈ (0, ∞), set

B_{r} = \{ξ ∈ R^{n}; |ξ| < r\}, \overline{B}_{r} = \{ξ ∈ R^{n}; |ξ| ≤ r\}.

Let ϕ be a C_{c}^{∞}(R^{n})-function such that

\{ \begin{matrix}ϕ(ξ) = 1, & if ξ ∈ \overline{B}_{1} \\ ϕ(ξ) = 0, & if ξ ∉ \overline{B}_{2}. \end{matrix}  (2.1)

Clearly, for any β ∈ N_{0}^{l}, there exists a M > 0 such that

|D^{β}ϕ(ξ)| ≤ M, ξ ∈ R^{n}.  (2.2)

For any k ∈ N, define

ϕ_{k}(ξ) = ϕ (\frac{1}{k} ξ) .  (2.3)

Then for each k ∈ N,

|D^{β}ϕ_{k}(ξ)| ≤ M (\frac{1}{k} )^{|β|} ≤ M, ξ ∈ R^{n}.  (2.4)

For every f ∈ E_{l}, it is clear that

f_{k} := ϕ_{k}f ∈ E_{l}, k ∈ N,  (2.5)
suppf_{k} ⊂ \overline{B}_{2k}, k ∈ N.  (2.6)

Thus, it follows from (2.1), (2.4) and (2.5) that for any f ∈ E_{l}, k ∈ N, β ∈ N_{0}^{l},

|D^{β}f_{k}| = | ∑_{α≤β} ( \begin{matrix}β \\ α \end{matrix} ) D^{α}fD^{β−α}ϕ_{k}| ≤ M ∑_{α≤β} ( \begin{matrix}β \\ α \end{matrix} ) |D^{α}f| .  (2.7)

Therefore for any k ∈ N, β ∈ N_{0}^{l},

∥D^{β}f_{k}∥_{L^{p}(R^{n})} ≤ const ∥D^{β}f∥_{L^{p}(R^{n})} ,  (2.8)
∥D^{β}f_{k} − D^{β}f∥_{L^{p}(R^{n})} = \{∫_{R^{n}−\overline{B}_{k}} |D^{β}f_{k}(ξ) − D^{β}f(ξ)|^{p} dξ\}^{\frac{1}{p}}
≤ const \{∫_{R^{n}−\overline{B}_{k}} |D^{β}f(ξ)|^{p} dξ\}^{\frac{1}{p}} .

Consequently,

D^{β}f_{k} ⟶ D^{β}f in L^{p}(R^{n}) as k → ∞.  (2.9)

For every j ∈ N, put

J_{j}(ξ) =\{ \begin{matrix}j^{n}e^{− \frac{1}{1−|jξ|^{2}}} (∫_{B_{1}} e^{− \frac{1}{1−|ξ|^{2}}} dξ)^{−1} , & if ξ ∈ B_{\frac{1}{j}} , \\ 0, & if ξ ∉ B_{\frac{1}{j}} . \end{matrix}  (2.10)

Then J_{j}(ξ) is a mollifier. For each k ∈ N, let

ψ_{jk}(ξ) = ∫_{R^{n}} J_{j}(ξ − η)f_{k}(η)dη = J_{j} ∗ f_{k}.  (2.11)

Then, according to the fundamental properties of mollifiers (cf. [2]), we know that (i) J_{j}(ξ − η) is an infinitely differentiable function of ξ,

(ii) J_{j}(ξ − η) = 0, if |ξ − η| ≥ \frac{1}{j},

(iii) D^{β}ψ_{jk} = ∫_{R^{n}} D_{ξ}^{β}J_{j}(ξ − η)f_{k}(η)dη (noting (2.6)). So,

ψ_{jk} ∈ C_{c}^{∞} (B_{2k+1}) ⊂ S(R^{n}), j, k ∈ N.  (2.12)

Further, the properties of mollifiers and (2.6) together imply that if f ∈ E_{l}, then for any k ∈ N, β ∈ N_{0}^{l},

J_{j} ∗ D^{β}f_{k} ∈ C_{c}^{∞}(R^{n}), j ∈ N,  (2.13)

and

lim_{j→0} ∥J_{j} ∗ D^{β}f_{k} − D^{β}f_{k}∥_{L^{p}(R^{n})} = 0.  (2.14)

On the other hand, for any j ∈ N, β ∈ N_{0}^{l} and test function g, we have by Fubini’s theorem that

∫_{R^{n}} J_{j} ∗ f_{k}(ξ)D^{β}g(ξ)dξ = ∫_{R^{n}} ∫_{R^{n}} f_{k}(ξ − η)J_{j}(η)D^{β}g(ξ)dξdη
= (−1)^{|β|} ∫_{R^{n}} ∫_{R^{n}} D_{ξ}^{β}f_{k}(ξ − η)J_{j}(η)g(ξ)dξdη
= (−1)^{|β|} ∫_{R^{n}} J_{j} ∗ D^{β}f_{k}(ξ)g(ξ)dξ, k ∈ N;

that is, for any j ∈ N, β ∈ N_{0}^{l},

D^{β}J_{j} ∗ f_{k}(ξ) = J_{j} ∗ D^{β}f_{k}(ξ), k ∈ N.  (2.15)

Combining (2.15) with (2.14), we get that for any j ∈ N, β ∈ N_{0}^{l},

lim_{j→0} ∥D^{β}J_{j} ∗ f_{k} − D^{β}f_{k}∥_{L^{p}(R^{n})} = lim_{j→0} ∥D^{β}ψ_{jk} − D^{β}f_{k}∥_{L^{p}(R^{n})} = 0.  (2.16)

(2.9) and (2.16) imply that for any f ∈ E_{l}, there exists a sequence {h_{k}} ⊂ S(R^{n}) such that

lim_{k→0} ∥D^{β}h_{k} − D^{β}f∥_{L^{p}(R^{n})} = 0 (for any β ∈ N_{0}^{l}).  (2.17)

The proof is then complete.

We write M_{p} (1 < p < ∞) for the Banach space of all Fourier multipliers u with the norm

∥u∥_{M_{p}} := sup {∥F^{−1} (uϕ̂)∥_{L^{p}} ; ϕ ∈ S(R^{n}), ∥ϕ∥_{L^{p}} ≤ 1} .

With a given

u ∈ \{ \begin{matrix}M_{p}, & if X = L^{p}(R^{n}) (1 < p < ∞) \\ FL^{1}, & otherwise, \end{matrix}

we associate a bounded linear operator T(u) on X (cf., e.g., [12 – 14]) as follows:

(i) If X = L^{p}(R^{n}) (1 < p < ∞), then T(u) is defined as the bounded linear extension to X of the operator on S(R^{n}): f ⟼ F^{−1} (uf̂);

(ii) Otherwise, then T(u) is defined as

T(u)f := F^{−1}u ∗ f, for any f ∈ X;

clearly,

F^{−1}u ∗ f = F^{−1} (uf̂) , for any f ∈ X.

For each β ∈ N_{0}^{l} and

f ∈ \{ \begin{matrix}S(R^{n}), & if X = L^{p}(R^{n}) (1 < p < ∞) \\ E_{l}, & otherwise, \end{matrix}

we have

D^{β}(T(u)f) = T(u)(D^{β}f).  (2.18)

Therefore, by virtue of Lemma 2.1, we obtain that (2.18) holds for every β ∈ N_{0}^{l} and f ∈ E_{l}. This implies

T(u)E_{l} ⊂ E_{l}.

Now define T_{l}(u) to be the part of T(u) in E_{l}. Then for each q_{β} ∈ Γ, f ∈ E_{l},

q_{β}(T_{l}(u)f) ≤ q_{β}(f) \{ \begin{matrix}∥u∥_{M_{p}}, & if X = L^{p}(R^{n}) (1 < p < ∞) \\ ∥u∥_{FL^{1}}, & otherwise, \end{matrix}

where ∥u∥_{FL^{1}} := ∥F^{−1}u∥_{L^{1}} . Consequently

T_{l}(u) ∈ B_{Γ}(E_{l})

and

∥T_{l}(u)∥_{Γ} ≤ \{ \begin{matrix}∥u∥_{M_{p}}, & if X = L^{p}(R^{n}) (1 < p < ∞) \\ ∥u∥_{FL^{1}}, & otherwise. \end{matrix}

Let k ∈ N, m ∈ N_{0}, a_{α} ∈ C for each α ∈ N_{0}^{n} with |α| ≤ k, and b_{α} ∈ C for each α ∈ N_{0}^{n} with |α| ≤ m. In the sequel, we fix l = 0, 1, …, n and consider the differential operators A and B in E_{l}, given by

Af  := ∑_{|α|≤k} a_{α}D^{α}f,
D(A) := {f ∈ E_{l}; ∑_{|α|≤k} a_{α}D^{α}f ∈ E_{l} distributionally} .

and

Bf  := ∑_{|α|≤m} b_{α}D^{α}f,
D(B) := {f ∈ E_{l}; ∑_{|α|≤m} b_{α}D^{α}f ∈ E_{l} distributionally} .

Clearly, A and B are closed linear operators in E_{l}. The symbol of A, B will be denoted respectively by

a(ξ) := ∑_{|α|≤k} i^{|α|}a_{α}ξ^{α} (ξ ∈ R^{n}) , b(ξ) := ∑_{|α|≤m} i^{|α|}b_{α}ξ^{α} (ξ ∈ R^{n}) .

Theorem 2.2. Let X be one of L^{p}(R^{n}) (1 < p < ∞). Set

C_{r,l} = T_{l} ((1 + |ξ|^{2})^{− \frac{(k+m)r}{2}} ) .

Assume that b(ξ) ≠ 0 for each ξ ∈ R^{n} and ω := sup_{ξ∈R^{n}} Re (a(ξ)b^{−1}(ξ)) < ∞. Then for

any r ≥ n |\frac{1}{2} − \frac{1}{p} |, (DCP) has a C_{r,l}-propagation family \{W_{r,l}(t)\}_{t≥0} satisfying

∥W_{r,l}(t)∥_{Γ} ≤ const (1 + t)^{n|\frac{1}{2} − \frac{1}{p} |}e^{ωt}, t ≥ 0.

Furthermore, hypothesis (ii) of Theorem 1.6 is also satisfied.

Proof. First, we see easily (cf. the proof of Theorem 1.5.10 of [14]) that C_{r,l} is an injective and continuous operator on E_{l} and

C_{r,l}A ⊂ AC_{r,l}, C_{r,l}B ⊂ BC_{r,l}.

\underline{Step 1.} Since 0 ∉ b(R^{n}), we have that B is injective and

\{  \begin{matrix}B^{−1}f = F^{−1} (b^{−1}(ξ)f̂) , \\ f ∈ D(B^{−1}) := \{f ∈ E_{l}; F^{−1} (b^{−1}(ξ)f̂) ∈ E_{l}\} . \end{matrix}  (2.19)

Define operator Q as follows

\{  \begin{matrix}Qf := F^{−1} (a(ξ)b^{−1}(ξ)f̂) , \\ f ∈ D(Q) := \{f ∈ E_{l}; F^{−1} (a(ξ)b^{−1}(ξ)f̂) ∈ E_{l}\} . \end{matrix}

It is clear that Q is a closed linear operator in E_{l} and

(λ − Q)Bf = (λB − A)f, f ∈ D(B), λ ∈ R.  (2.20)

\underline{Step 2.} For each t ≥ 0, ξ ∈ R^{n}, put

v_{t,r}(ξ) = (1 + |ξ|^{2})^{− \frac{(k+m)r}{2}} e^{ta(ξ)b^{−1}(ξ)}.

Notice sup_{ξ∈R^{n}} |b(ξ)|^{−1} < ∞ by hypothesis. A direct calculation then shows that for each multiindex α with |α| ≤ [\frac{n}{2}] + 1,

\begin{matrix}|D^{α}v_{t,r}(ξ)| , |D^{α} (b^{−1}(ξ)v_{t,r}(ξ))| \\ ≤ const (1 + t)^{|α|}e^{ωt}(1 + |ξ|)^{(k+m−1)|α|−(k+m)r}. \end{matrix}  (2.21)

Hence we obtain by Lemma 1.5.4 of [14] that v_{t,r}(ξ), b^{−1}(ξ)v_{t,r}(ξ) ∈ M_{p} and

∥v_{t,r}(ξ)∥_{M_{p}}, ∥b^{−1}(ξ)v_{t,r}(ξ)∥_{M_{p}} ≤ const (1 + t)^{n|\frac{1}{2} − \frac{1}{p} |}e^{ωt}, t ≥ 0.

Therefore, defining

V_{r,l}(t) = T_{l} (v_{t,r}(ξ)) for each t ≥ 0,

we deduce by (2.19) that V_{r,l}(t)E_{l} ⊂ D(B^{−1}) and

∥V_{r,l}(t)∥_{Γ}, ∥B^{−1}V_{r,l}(t)∥_{Γ} ≤ const (1 + t)^{n|\frac{1}{2} − \frac{1}{p} |}e^{ωt}, t ≥ 0.  (2.22)

\underline{Step 3.} (2.21) also implies (by Lemma 1.5.2 of [14]) that

V_{r+1}(·), B^{−1}V_{r+1}(·) ∈ C (R^{+}, B_{Γ}(E_{l})) .  (2.23)

Whence for any f ∈ E_{l},

V_{r,l}(·)C_{1,l}f, B^{−1}V_{r,l}(·)C_{1,l}f ∈ C (R^{+}, E_{l}) .

But R(C_{1,l}) ⊃ S(R^{n}) and S(R^{n}) is dense in E_{l} by Lemma 2.1. We therefore justify by (2.22) that for all f ∈ E_{l},

V_{r,l}(·)f, B^{−1}V_{r,l}(·)f ∈ C (R^{+}, E_{l}) .  (2.24)

\underline{Step 4.} Write

L_{r,l}(λ)f = ∫_{0}^{∞} e^{−λt}V_{r,l}(t)fdt, f ∈ E_{l}, λ > ω + 1.

We obtain by Fubini’s theorem that for each f ∈ S(R^{n}), λ > ω + 1,

\begin{aligned}L_{r,l}(λ)f & = & F^{−1} [(∫_{0}^{∞} e^{−λt}v_{t,r}(ξ)dt) f̂] \\ & = & F^{−1} [(λ − a(ξ)b^{−1}(ξ))^{−1} F (C_{r,l}f)] , \end{aligned}  (2.25)

and hence

(λ − Q)L_{r,l}(λ)f = C_{r,l}f.  (2.26)

Since Q is closed and S(R^{n}) is dense in E_{l} by Lemma 2.1, (2.26) is true for all f ∈ E_{l}.

\underline{Step 5.} (2.21) also implies by Lemma 1.5.3 of [14] that

v_{t,r+n}(ξ) ∈ FL^{1} for each t ≥ 0

and so

V_{r+n,l}(t)f = F^{−1} (v_{t,r+n}f̂) for all f ∈ E_{l}.

In view of this fact, we have that for λ > ω + 1,

L_{r+n,l}(λ)(Af) = AL_{r+n,l}(λ)f, f ∈ D(A),
L_{r+n,l}(λ)(Bf) = BL_{r+n,l}(λ)f, f ∈ D(B).

But

L_{r+n,l}(λ) = C_{n,l}L_{r,l},

we get by the injectivity of C_{n,l} that

L_{r,l}(λ)(Af) = AL_{r,l}(λ)f (f ∈ D(A)), L_{r,l}(λ)Bf = BL_{r,l}(λ)f (f ∈ D(B)).

Accordingly, it follows from (2.26) that for any f ∈ D(B), λ > ω + 1,

\begin{aligned}C_{r,l}Bf & = & (λ − Q)L_{r,l}(λ)Bf \\ & = & (λ − Q)BL_{r,l}(λ)f \\ & = & (λB − A)L_{r,l}(λ)f; \end{aligned}  (2.27)

and that for f ∈ D(A) ∩ D(B), λ > ω + 1,

L_{r,l}(λ)(λB − A)f = (λB − A)L_{r,l}(λ)f = C_{r,l}Bf,  (2.28)

by (2.27). It is implied by (2.28) that (λB − A) is injective for each λ > ω + 1, since C_{r,l}B is injective. Therefore (2.27) gives

L_{r,l}(λ)f = (λB − A)^{−1}C_{r,l}Bf, f ∈ D(B).

For each t ≥ 0, let W_{r,l}(t) be the restriction of V_{r,l}(t) on D(B). Then {W_{r,l}(t)}_{t≥0} is the C_{r,l}-propagation family for (DCP).

\underline{Step 6.} By (2.22) and (2.24) together with the closedness of B^{−1}, we know that for all λ > ω + 1, L_{r,l}(λ)E_{l} ⊂ D (B^{−1}), and therefore by (2.20) and (2.26),

C_{r,l}f = (λ − Q)BB^{−1}L_{r,l}(λ)f = (λB − A)B^{−1}L_{r,l}(λ)f, f ∈ E_{l}.

Consequently

B(λB − A)^{−1}C_{r,l}f = L_{r,l}(λ)f, f ∈ E_{l}.

This ends the proof.

Theorem 2.3. Let X be one of L^{1}(R^{n}), L^{∞}(R^{n}), C_{0}(R^{n}), UC_{b}(R^{n}) or C_{b}(R^{n}). Assume that C_{r,l}, a(ξ) and b(ξ) are as in Theorem 2.2. Then for any r > \frac{n}{2}, (DCP) has a C_{r,l}-propagation family {W_{r,l}(t)}_{t≥0}, which belongs to C (R^{+}, B_{Γ} (D(B), E_{l})), satisfying

∥W_{r,l}(t)∥_{Γ} ≤ const (1 + t)^{\frac{n}{2}} e^{ωt}, t ≥ 0,

and hypothesis (ii) of Theorem 1.6 is satisfied.

Proof. Copy the proof of Theorem 2.2, with several changes as follows.

In Step 2, replace M_{p} with FL^{1}, n |\frac{1}{2} − \frac{1}{p} | with \frac{n}{2}.

In Step 3, replace r + 1 with r in (2.23) and delete the remaining part.

In Step 4, for each f ∈ E_{l} and each test function ϕ ∈ C_{c}^{∞}(R^{n}), λ > ω + 1,

⟨L_{r,l}(λ)f, ϕ⟩ = ∫_{0}^{∞} ⟨e^{−λt}F^{−1} (v_{t,r}f̂) , ϕ⟩ dt
= ∫_{0}^{∞} ⟨e^{−λt} (v_{t,r}f̂) , F^{−1}(ϕ)⟩ dt
= ⟨(∫_{0}^{∞} e^{−λt}v_{t,r}dt) f̂, F^{−1}(ϕ)⟩
= ⟨F^{−1} (∫_{0}^{∞} e^{−λt}v_{t,r}dt) f̂, ϕ⟩ ,

and so

L_{r,l}(λ)f = F^{−1} (∫_{0}^{∞} e^{−λt}v_{t,r}dt) f̂, f ∈ E_{l}, λ > ω + 1.

From this we justify that (2.25) and (2.26) hold for all f ∈ E_{l} in this case.

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