LAPLACE TRANSFORMS OF VECTOR-VALUED FUNCTIONS WITH GROWTH ω AND SEMIGROUPS OF OPERATORS

Ralph deLaubenfels, Vũ Quôc Phóng and Shengwang Wang

Abstract

For an arbitrary continuous, increasing function ω : [0, ∞) ↦ C of finite exponential type, we characterize Laplace-Stieltjes transforms of Banach-space-valued functions that are O(ω), and use this to establish a Hille-Yosida type theorem for strongly continuous semigroups that are O(ω). Corollaries include characterizing generators of strongly continuous semigroups that are O(exp(dt^{\frac{1}{d}})), for d > 1, in addition to the already known examples of exponential or polynomial growth.^{1}

2.

11991 mathematics Subject classification: Primary 47D06, Secondary 47B40

2Much of this work was done when the third author was visiting Ohio University, he would like to thank Ohio University and Professors Ralph deLaubenfels and Vũ Quôc Phóng for their invitation and hospitality

LAPLACE TRANSFORMS OF VECTOR-VALUED FUNCTIONS WITH GROWTH ω AND SEMIGROUPS OF OPERATORS

1. Introduction

In this paper we deal with the Laplace-Stieltjes transforms of Banach-space-valued functions in a unified way, that includes the well-known polynomially bounded and exponentially bounded cases (see [7] and [1], respectively), and new cases of non-quasi-analytic growth including O(exp(dt^{\frac{1}{d}})) (d > 1) growth.

In order to deal with our Laplace-Stieltjes transforms in a more general setting, we need a function ω : [0, ∞) ↦ C that satisfies the following conditions.

(a) ω(0) = 1, ω(·) is continuous and increasing on [0, ∞).

(b) The number r_{0} defined by

r_{0} ≡ limsup_{t→∞} \frac{logω(t)}{t}  (*)

is finite. Note that r_{0} ≥ 0.

Once-integrated Laplace transforms of vector-valued functions have been shown to be important for studying abstract Cauchy problems of the first and second order (see [1] and [7]). We restrict ourselves in this paper to the first order abstract Cauchy problem

\frac{d}{dt}u(t, x) = Au(t, x) (t ≥ 0), u(0, x) = x.  (ACP)

As consequences of our Laplace transform results, we will characterize, in Section 3, O(ω(t)) solutions of (ACP) and generators of O(ω(t)) strongly continuous semigroups in terms of the resolvent of the generator. In other words, we will establish a Hille-Yosida type theorem for strongly continuous semigroups that are O(ω(t)).

Our characterization of O(ω(t)) Laplace-Stieltjes transforms is in terms of Laplace transforms of t ↦ t^{n}ω(t), n = 0, 1, 2, ... (Theorem 2.3). For ω(t) = (1 + t^{k}) or e^{rt}, since these Laplace transforms may be easily calculated, this immediately yields those well-known cases (Corollaries 2.4 and 2.5). But other types of growth are possible (see Example 3.6). For a large class of non-quasi-analytic ω; this means that

∫_{0}^{∞} \frac{logω(t)}{1 + t^{2}} dt < ∞;  (**)

we also calculate the required Laplace transforms, to give a new explicit characterization of O(ω) Laplace-Stieltjes transforms (Theorem 2.12). This includes ω(t) = exp(dt^{\frac{1}{d}}), for d > 1 (Corollary 2.14).

For d > 1, we also give a sufficient condition, in terms of only the first power of the resolvent rather than all powers, for an operator A to generate a O(exp(dt^{\frac{1}{d}})) (1 − A)^{−2}-regularized semigroup (Theorem 3.8), that is very close to a necessary condition (see Remark 3.9).

As a slight generalization of [6, Chapter 5], [7, Section 3] and [8, Section 1], we will introduce maximal continuously embedded subspaces where an operator generates a O(ω(t)) strongly continuous semigroup.

In [15], the author studied the asymptotic behaviour of strongly continuous semigroups {T (t)}_{t≥0} that satisfy ||T (t)|| ≤ ω(t). Here, in addition to (**), it is assumed that ω satisfies

ω(t + s) ≤ ω(t)ω(s)  (***)

and that ω_{1}(s) ≡ limsup_{t→∞} \frac{ω(t+s)}{ω(t)} is polynomially bounded. We will also slightly generalize this asymptotic result.

Throughout this paper, X is a Banach space and A is a (possibly unbounded) linear operator on X, with domain D(A) and resolvent set ρ(A). We will denote by B(X) the algebra of all bounded linear operators on X.

2. Properties for Laplace-Stieltjes transform

In this section we present a new approach for Laplace-Stieltjes transforms that seems to be interesting and will be used in Section 3.

Definition 2.1 [1], [7]. Suppose W is a Banach space and r ∈ R. We say that g : (r, ∞) ↦ W is the Laplace-Stieltjes transform of G if G : [0, ∞) ↦ W is continuous, G(0) = 0, and

g(λ) = λ ∫_{0}^{∞} e^{−λt}G(t) dt (λ > r).

We call G the determining function of g.

Let ω be as in Section 1.

Theorem 2.2. Suppose g : (r_{0}, ∞) ↦ C, where r_{0} is the constant defined in (*). Then the following are equivalent.

(a) g is the Laplace-Stieltjes transform of a function G : [0, ∞) ↦ C and there exists a constant K_{1} > 0 so that

limsup_{h→0^{+}} \frac{1}{h}|G(t + h) − G(t)| ≤ K_{1}ω(t), ∀t ≥ 0.

(b) There exists a constant K_{2} > 0 so that g is infinitely differentiable and

|g^{(n)}(λ)| ≤ K_{2} ∫_{0}^{∞} e^{−λt}t^{n}ω(t) dt, ∀λ > r_{0}, n ∈ N ∪ {0}.

K_{1}, K_{2} may be chosen so that K_{2} ≤ K_{1} ≤ 2K_{2}.

Proof. (a) → (b), with K_{2} = K_{1}, is an immediate consequence of the definition of G and the equality in (a).

(b) → (a). First assume that g is real-valued. (b) implies that

h_{±}(λ) ≡ K_{2} ∫_{0}^{∞} e^{−λt}ω(t) dt ± g(λ), ∀λ > r_{0},

is completely monotone, that is

(−1)^{n}h_{±}^{(n)}(λ) = K_{2} ∫_{0}^{∞} e^{−λt}t^{n}ω(t) dt ± (−1)^{n}g^{(n)}(λ) ≥ 0,
∀n ∈ N ∪ {0}, λ > r_{0}.

By Bernstein’s theorem [18, p. 156, Corollary 7], there exist nondecreasing functions α_{±} so that α_{±}(0) = 0 and

h_{±}(λ) = ∫_{0}^{∞} e^{−λt} dα_{±}(t) = λ ∫_{0}^{∞} e^{−λt}α_{±}(t) dt, ∀λ > r_{0}.

Therefore

g(λ) = λ ∫_{0}^{∞} e^{−λt}[α_{+}(t) − K_{2} ∫_{0}^{t} ω(s) ds] dt
= λ ∫_{0}^{∞} e^{−λt}[K_{2} ∫_{0}^{t} ω(s) ds − α_{−}(t)] dt.

By the uniqueness of the Laplace transform, we have

α_{+}(t) − K_{2} ∫_{0}^{t} ω(s) ds = K_{2} ∫_{0}^{t} ω(s) ds − α_{−}(t).

Denote the preceding quantity by G(t). Then g is the Laplace-Stieltjes transform of G and

limsup_{h→0^{+}} \frac{1}{h}|G(t + h) − G(t)| ≤ K_{2}ω(t).

Now assume g is complex-valued. Write g(λ) = g_{1}(λ) + ig_{2}(λ) where g_{1}(λ) and g_{2}(λ) are real-valued. Since g_{j}(j = 1, 2) satisfies the conditions in (b), there exists G_{j} : [0, ∞) ↦ C so that

g_{j}(λ) = λ ∫_{0}^{∞} e^{−λt}G_{j}(t) dt;

limsup_{h→0^{+}} \frac{1}{h}|G_{j}(t + h) − G_{j}(t)| ≤ K_{2}ω(t).

Denote G(t) ≡ G_{1}(t) + iG_{2}(t). Then

g(λ) = λ ∫_{0}^{∞} e^{−λt}G(t) dt and

limsup_{h→0^{+}} \frac{1}{h}|G(t + h) − G(t)| ≤ 2K_{2}ω(t). □

The following Theorem 2.3 is a vector-valued version of Theorem 2.2. In Theorem 2.3 and Corollaries 2.4 and 2.5, K_{1}, K_{2} may be also chosen so that K_{2} ≤ K_{1} ≤ 2K_{2}.

Theorem 2.3. Suppose W is a Banach space, r_{0} is the constant defined in (*) and g : (r_{0}, ∞) ↦ W. Then the following are equivalent.

(a) g is the Laplace-Stieltjes transform of a function G : [0, ∞) ↦ W and there is a constant K_{1} > 0 so that

limsup_{h→0^{+}} \frac{1}{h}||G(t + h) − G(t)|| ≤ K_{1}ω(t), ∀t ≥ 0.

(b) g is infinitely differentiable and there is a constant K_{2} > 0 so that

||g^{(n)}(λ)|| ≤ K_{2} ∫_{0}^{∞} e^{−λt}t^{n}ω(t) dt, ∀λ > r_{0}, n ∈ N ∪ {0}.

Proof. (a) → (b) is clear.

(b) → (a). From Theorem 2.2, for any x^{∗} ∈ W^{∗}, there exists a function G_{x^{∗}} : [0, ∞) ↦ C so that λ ↦< g(λ), x^{∗} > is the Laplace-Stieltjes transform of G_{x^{∗}} and

limsup_{h→0^{+}} \frac{1}{h}|G_{x^{∗}}(t + h) − G_{x^{∗}}(t)| ≤ 2K_{2}||x^{∗}||ω(t), ∀t ≥ 0.

From [1, Theorem 1.1] and its proof, there is G : [0, ∞) ↦ W such that

G_{x^{∗}}(t) =< G(t), x^{∗} >, ∀t ≥ 0, x^{∗} ∈ W^{∗}.

G(·) is clearly the desired function. □

Corollary 2.4 [7]. Suppose W is a Banach space, k ∈ N ∪{0} and g : (0, ∞) ↦ W. Then the following are equivalent.

(a) g is the Laplace-Stieltjes transform of a function G : [0, ∞) ↦ W and there is a constant K_{1} > 0 so that

limsup_{h→0_{+}} \frac{1}{h}||G(t + h) − G(t)|| ≤ K_{1}(1 + t^{k}), ∀t ≥ 0.

(b) g is infinitely differentiable and there is a constant K_{2} > 0 so that

||g^{(n)}(λ)|| ≤ K_{2}[\frac{n!}{λ^{n+1}} + \frac{(n + k)!}{λ^{n+k+1}}], ∀λ > 0, n ∈ N ∪ \{0\}.

Proof. Set ω(t) ≡ 1 + t^{k}. From

∫_{0}^{∞} e^{−λt}t^{n}ω(t) dt = ∫_{0}^{∞} e^{−λt}[t^{n} + t^{n+k}] dt

= \frac{n!}{λ^{n+1}} + \frac{(n + k)!}{λ^{n+K+1}}

and Theorem 2.3, the corollary follows. □

Corollary 2.5 [1]. Suppose W is a Banach space, r > 0 and g : (r, ∞) ↦ W. Then the following are equivalent.

(a) g is the Laplace-Stieltjes transform of a function G : [0, ∞) ↦ W and there is a constant K_{1} > 0 so that

limsup_{h→0_{+}} \frac{1}{h}||G(t + h) − G(t)|| ≤ K_{1}e^{rt}, ∀t ≥ 0.

(b) g is infinitely differentiable and there is a constant K_{2} > 0 so that

||g^{(n)}(λ)|| ≤ K_{2}n!(λ − r)^{−(n+1)}, ∀λ > r, n ∈ N ∪ {0}.

Proof. Set ω(t) ≡ e^{rt}. From

∫_{0}^{∞} e^{−λt}t^{n}ω(t) = ∫_{0}^{∞} e^{−(λ−r)}t^{n} dt
= n!(λ − r)^{−(n+1)}, ∀λ > r, n ∈ N ∪ {0},

and Theorem 2.3, the corollary follows. □

To study the third special case, we first make some preparations.

Let {M_{k}} (k ∈ N ∪ {0}) be a sequence of positive numbers with M_{0} = 1. Denote m_{k} ≡ \frac{M_{k}}{M_{k−1}} (k ∈ N). The following two conditions can be found in [4, 14].

(M.1) Logrithmic convexity : M_{k}^{2} ≤ M_{k−1}M_{k+1}, or equivalently, {m_{k}} increases. (M.2) Non-quasi-analyticity : ∑_{k=1}^{∞} \frac{M_{k−1}}{M_{k}} < ∞, or equivalently, ∑_{k=1}^{∞} \frac{1}{m_{k}} < ∞.

A simple example that satisfies (M.1) but not (M.2) is {k!} (k ∈ N ∪ {0}). Another well-known sequence that satisfies (M.1) is N_{0,d} = 1, N_{k,d} = (\frac{k}{e})^{kd} (k ∈ N), where d > 0 (see [3]). This sequence satisfies (M.2) if and only if d > 1. Denote n_{k,d} = \frac{N_{k,d}}{N_{k−1,d}} (k ∈ N). If d = 1, we will write N_{k} = N_{k,d}, n_{k} = n_{k,d} and M_{k}^{∗} = \frac{M_{k}}{N_{k}}. We define the respective associate functions for {M_{k}}, {N_{k}} and {M_{k}^{∗}} on [0, ∞) in the following way.

M(t) ≡ sup\{log \frac{t^{k}}{M_{k}} : k ∈ N ∪ \{0\}\}, ∀t > 0, M(0) = 1;

N(t) ≡ sup\{log \frac{t^{k}}{N_{k}} : k ∈ N ∪ \{0\}\}, ∀t > 0, N(0) = 1;

M^{∗}(t) ≡ sup\{log \frac{t^{k}}{M_{k}^{∗}} : k ∈ N ∪ \{0\}\}, ∀t > 0, M^{∗}(0) = 1.

The following lemma is an immediate consequence of the definition of M(·).

Lemma 2.6 [14]. Assume {M_{k}} satisfies (M.1) and lim_{k→∞}m_{k} = ∞. Then

e^{M(t)} = \frac{t^{k}}{M_{k}} ∀m_{k} ≤ t < m_{k+1}, k ∈ N.

Lemma 2.7. The function N(t) satisfies

e^{N(t)} ≤ e^{t} and lim_{t→∞} \frac{e^{N(t)}}{e^{t}} = 1.

Proof. Regard k as a continuous variable, differentiate \frac{t^{k}}{N_{k}} in k and set the resulting derivative equal to zero to find k = t. Here we note that N_{k} = (\frac{k}{e})^{k}. Thus sup_{k∈[0,∞)} \frac{t^{k}}{N_{k}} = e^{t}. This implies that e^{N(t)} ≤ e^{t} and that

e^{N(t)} = g(t) ≡ max\{\frac{t^{[t]}}{(\frac{[t]}{e})^{[t]}} , \frac{t^{[t]+1}}{(\frac{[t]+1}{e})^{[t]+1}} \}

A direct calculation gives the desired equality. □

It is easy to see that (M.1) and (M.2) imply that \frac{m_{k}}{k} → ∞. From n_{k} < k, we have \frac{m_{k}}{n_{k}} → ∞. In the following, instead of (M.1) and (M.2), we introduce the following condition.

(M.3) \{\frac{m_{k}}{n_{k}} \} increases and lim_{t→∞} \frac{m_{k}}{n_{k}} → ∞. The former is equivalent to \{M_{k}^{∗}\} being logarithmically convex.

When (M.3) is true, M^{∗}(t) is finite for all t ≥ 0 by Lemma 2.6. Clearly, (M.3) implies (M.1). But the following example shows that (M.3) does not imply (M.2). Let

M_{0} = 1; M_{k} ≡ N_{k} Π_{j=1}^{k}log(1 + j), ∀k ∈ N.

Then m_{k} = n_{k}log(1 + k) and hence (M.3) holds. The following calculation

∑_{k=1}^{∞} \frac{1}{m_{k}} = ∑_{k=1}^{∞} \frac{1}{n_{k}log(1 + k)}

≥ ∑_{k=1}^{∞} \frac{1}{klog(1 + k)} = ∞

shows that (M.2) is not true. Here we note that n_{k} < k. Lemma 2.8. Assume (M.3) holds. Then M(·), N(·) and M^{∗}(·) satisfy

e^{M(λt)} ≤ e^{N(αt)}e^{M^{∗}( \frac{λ}{α})}, ∀λ > 0, α > 0, t > 0.

Proof. The relation follows from

log \frac{(λt)^{k}}{M_{k}} = log \frac{(αt)^{k}}{N_{k}} + log \frac{(\frac{λ}{α})^{k}}{M_{k}^{∗}} , ∀k ∈ N ∪ \{0\},

and the definitions of M(·), N(·) and M^{∗}(·). □

Lemma 2.9. Assume (M.3) holds. Then

lim_{t→∞} \frac{M(t)}{t} = 0.

Proof. Lemmas 2.8 and 2.7 imply that

\frac{M(λt)}{t} ≤ \frac{N(αt)}{t} + \frac{M^{∗}(\frac{λ}{α})}{t} ≤ α + \frac{M^{∗}(\frac{λ}{α})}{t}.

Set λ = 1 to obtain

0 ≤ limsup_{t→∞} \frac{M(t)}{t} ≤ α.

α being arbitrarily fixed, the lemma follows. □

Lemma 2.10. Assume (M.3) holds. Then for any λ > 0, there exists a function α_{λ}(·) on [0, ∞) so that

(a) α_{λ}(·) is positive, continuous and decreasing;

(b) lim_{t→∞}α_{λ}(t) = 0;

(c) there exists K > 0 such that

e^{tα_{λ}(t)}e^{M^{∗}( \frac{λ}{α_{λ}(t)} )} ≤ Ke^{M(λt)}, ∀t ≥ 0.

K may be chosen so that K ≤ e.

Proof. Given λ > 0. Define

α_{λ}(t) = [λ n_{k}m_{k}(\frac{1}{m_{k}} − \frac{1}{m_{k+1}} )][(n_{k+1} − n_{k}) + λt(\frac{n_{k}}{m_{k}} − \frac{n_{k+1}}{m_{k+1}} )]^{−1},
∀m_{k} ≤ λt < m_{k+1}, k ∈ N;
α_{λ}(t) = α_{λ}(m_{1}), ∀0 ≤ λt < m_{1}.

The fact that α_{λ}(t) satisfies (a) and (b) is an immediate consequence of its definition. We now prove (c). For m_{k} ≤ λt < m_{k+1}, a few steps of calculation give us

n_{k} ≤ tα_{λ}(t) ≤ n_{k+1};

\frac{λ}{α_{λ}(t)} = [(\frac{m_{k+1}}{n_{k+1}} − \frac{m_{k}}{n_{k}})(λt − m_{k})](m_{k+1} − m_{k})^{−1} + \frac{m_{k}}{n_{k}} .

The latter implies that

\frac{m_{k}}{n_{k}} ≤ \frac{λ}{α_{λ}(t)} ≤ \frac{m_{k+1}}{n_{k+1}} , ∀m_{k} ≤ λt < m_{k+1}.

This, together with Lemmas 2.6 and 2.7, shows that there is K > 0 so that

e^{tα_{λ}(t)}e^{M^{∗}( \frac{λ}{α_{λ}(t)} )} ≤ Ke^{N(tα_{λ}(t))}e^{M^{∗}( \frac{λ}{α_{λ}(t)} )}

= K \frac{[tα_{λ}(t)]^{k}}{N_{k}} \frac{[\frac{λ}{α_{λ}(t)}]^{k}}{M^{∗}} = Ke^{M(λt)}.

We now show that K may be chosen so that K ≤ e. Assume k ≤ t < k + 1, then from the proof of Lemma 2.7,

ee^{N(t)} ≥ e\frac{t^{k}}{(\frac{k}{e})^{k}} = \frac{e^{k+1}t^{k}}{k^{k}} ≥ e^{k+1} ≥ e^{t}.

The lemma follows. □

We now state and prove our third special case.

Theorem 2.11. Assume (M.3) holds and g : (0, ∞) ↦ C. Then the following are equivalent.

(a) g is the Laplace-Stieltjes transform of a function G : [0, ∞) ↦ C and there is a constant K_{1} > 0 so that

limsup_{h→0^{+}} \frac{1}{h}|G(t + h) − G(t)| ≤ K_{1}e^{M(t)}, ∀t > 0.

(b) g is infinitely differentiable and there is a constant K_{2} > 0 so that, for every α > 0,

|g^{(n)}(λ)| ≤ K_{2}n!(λ − α)^{−(n+1)}e^{M^{∗}( \frac{1}{α})}, ∀λ > α, n ∈ N ∪ \{0\}.

K_{1}, K_{2} may be chosen so that K_{2} ≤ K_{1} ≤ 2eK_{2}.

Proof. (a) → (b). Set ω(t) ≡ e^{M(t)}. Then r_{0} = lim_{t→∞} \frac{logω(t)}{t} = 0 by Lemma 2.9. Theorem 2.2 implies that (a) is equivalent to

(a’) g is infinitely differentiable and

|g^{(n)}(λ)| ≤ K_{2} ∫_{0}^{∞} e^{−λt}t^{n}e^{M(t)} dt, ∀λ > 0.

where K_{2} may be chosen so that K_{2} ≤ K_{1} ≤ 2K_{2} by Theorem 2.2. Given α > 0, let λ > α. From Lemmas 2.7 and 2.8,

∫_{0}^{∞}e^{−λt}t^{n}e^{M(t)} dt
= ∫_{0}^{∞} e^{−(λ−α)t}t^{n}e^{−αt}e^{M(t)} dt
≤ ∫_{0}^{∞} e^{−(λ−α)t}t^{n}e^{M^{∗}( \frac{1}{α})} dt
= n!(λ − α)^{−(n+1)}e^{M^{∗}( \frac{1}{α})}.

(b) follows.

(b) → (a). Given α > 0, denote

h_{±,α}(λ) = K_{2}(λ − α)^{−1}e^{M^{∗}( \frac{1}{α})} ± g(λ) ∀λ > α.

Assume g is real-valued. From the inequality in (b), h_{±,α}(·) are completely monotone. By Bernstein’s theorem again, there exist increasing functions β_{±,α}(t) so that

h_{±,α}(λ) = ∫_{0}^{∞} e^{−λt} dβ_{±,α}(t) = λ ∫_{0}^{∞} e^{−λt}β_{±,α}(t) dt.

Then

g(λ) = λ ∫_{0}^{∞} e^{−λt}[β_{+,α}(t) − \frac{K_{2}}{α} e^{M^{∗}( \frac{1}{α})}(e^{αt} − 1)] dt

= λ ∫_{0}^{∞} e^{−λt}[\frac{K_{2}}{α} e^{M^{∗}( \frac{1}{α})}(e^{αt} − 1) − β_{−,α}(t)] dt.

By the uniqueness of the Laplace transform, we have

β_{+,α}(t) − \frac{K_{2}}{α} e^{M^{∗}( \frac{1}{α})}(e^{αt} − 1)

= \frac{K_{2}}{α} e^{M^{∗}( \frac{1}{α})}(e^{αt} − 1) − β_{−,α}(t),

which is independent of α, since g is. Denote it by G(t). Then

g(λ) = λ ∫_{0}^{∞} e^{−λt}G(t) dt;

limsup_{h→0^{+}} \frac{1}{h}|G(t + h) − G(t)| ≤ K_{2}e^{M^{∗}( \frac{1}{α})}e^{αt}, ∀α > 0.

α being independent of t, we may choose α = α_{1}(t), where α_{1}(t) is the function defined in Lemma 2.10 corresponding to λ = 1. The application of Lemma 2.10 gives (a) and from the process of the arguments of Theorem 2.2, Lemma 2.10, K_{1} may be chosen so that K_{1} ≤ 2eK_{2}. □

The following is a vector-valued version of Theorem 2.11.

Theorem 2.12. Assume {M_{k}} satisfies (M.3), W is a Banach space, and g : (0, ∞) ↦ W. Then the following are equivalent.

(a) g is the Laplace-Stieltjes transform of a function G : [0, ∞) ↦ W and there is a constant K_{1} > 0 so that

limsup_{h→0^{+}} \frac{1}{h}||G(t + h) − G(t)|| ≤ K_{1}e^{M(t)}, ∀t > 0.

(b) g is infinitely differentiable and there is a constant K_{2} > 0 so that, for every α > 0,

||g^{(n)}(λ)|| ≤ K_{2}n!(λ − α)^{−(n+1)}e^{M^{∗}( \frac{1}{α})},
∀λ > α, n ∈ N ∪ {0}.

K_{1}, K_{2} may be chosen so that K_{1} ≤ 2eK_{2}.

We now consider a special case of Theorem 2.12. Let {N_{d,k}} be the sequence given previously and define

N_{d}(t) ≡ sup_{k∈N∪\{0\}}[log \frac{t^{k}}{N_{d,k}}].

When d > 1, write N_{d,k}^{∗} = \frac{N_{d,k}}{N_{k}}. Then N_{d,k}^{∗} = N_{d−1,k}. Define

N_{d}^{∗}(t) ≡ sup_{k∈N∪\{0\}}[log \frac{t^{k}}{N_{d,k}^{∗}}] = N_{d−1}(t).

The following lemma is an immediate generalization of Lemma 2.7.

Lemma 2.13. Given d > 0. Then

e^{N_{d}(t)} ≤ exp(dt^{\frac{1}{d}}) and lim_{t→∞}e^{N_{d}(t)}exp(−dt^{\frac{1}{d}}) = 1.

With the help of Lemma 2.13, we may prove

Corollary 2.14. Suppose W is a Banach space, d > 1 and g : (0, ∞) ↦ W. Then the following are equivalent.

g is the Laplace-Stieltjes transform of a function G : [0, ∞) ↦ W and there is a constant K_{1} > 0 so that

limsup_{h→0^{+}} \frac{1}{h}||G(t + h) − G(t)|| ≤ K_{1}exp(dt^{\frac{1}{d}}).

(b) g is infinitely differentiable and there exists K_{2} > 0 so that, for every α > 0,

||g^{(n)}(λ)|| ≤ K_{2}n!(λ − α)^{−(n+1)}exp((d − 1)α^{− \frac{1}{d−1}} ),
∀λ > α, n ∈ N ∪ {0}.

Recall that ω(·) is a continuous and increasing function defined on [0, ∞) satisfying ω(0) = 1 (see Section 1).

Theorem 2.15. Assume ω(·) is the function given above. Then the following are equivalent.

(a) ω(·) satisfies (**).

(b) There exist a sequence {m_{k}}_{k∈N} of positive numbers satisfying

1 = m_{1} ≤ \frac{m_{2}}{n_{2}} ≤ \frac{m_{3}}{n_{3}} ≤ ⋯; ∑_{k=1}^{∞} \frac{1}{m_{k}} < ∞;

ω(t) ≤ e^{M(t)},

where M(·) is the function associated with {M_{k}}_{k∈N∪{0}}.

Proof. (a) → (b). We may assume that ω(t) is strictly increasing and ω(t) → ∞, as t → ∞. Otherwise, instead of ω(t), we may choose (1 + t)ω(t). For every k ∈ N, define a_{k} ∈ [1, ∞) and l_{k} ∈ N so that

ω(a_{k}) = e^{l_{k}}; \frac{a_{k}}{n_{k}} ≥ \frac{a_{k−1}}{n_{k−1}} ; l_{k} ≥ k.

Then we have

\frac{k}{a_{k}} ≤ \frac{l_{k}}{a_{k}} = l_{k} ∫_{a_{k}}^{∞} \frac{1}{t^{2}} dt ≤ ∫_{a_{k}}^{∞} \frac{logω(t)}{t^{2}} dt.

Since a_{k} → ∞, it follows that \frac{k}{a_{k}} → 0. Thus the following calculation

∑_{j=1}^{k} \frac{1}{a_{j}} − \frac{k}{a_{k+1}} = ∑_{j=1}^{k} j(\frac{1}{a_{j}} − \frac{1}{a_{j+1}} )

≤ ∑_{j=1}^{k} l_{j} ∫_{a_{j}}^{a_{j+1}} \frac{1}{t^{2}} dt ≤ ∫_{1}^{∞} \frac{logω(t)}{t^{2}} dt

implies that

∑_{j=1}^{∞} \frac{1}{a_{j}} ≤ ∫_{1}^{∞} \frac{logω(t)}{t^{2}} dt < ∞.

Let m_{k} ≡ \frac{a_{k}}{e} (k ∈ N). Then

m_{1} ≤ \frac{m_{2}}{n_{2}} ≤ \frac{m_{3}}{n_{3}} ≤ ⋯; ∑_{k=1}^{∞} \frac{1}{m_{k}} < ∞.

Define M_{0} ≡ 1, M_{k} ≡ Π_{j=1}^{k}m_{j} for k ∈ N. Given t ∈ [0, ∞), there exists a unique k ∈ N such that a_{k} ≤ t < a_{k+1}. From

e^{M(t)} = sup_{j≥0} \frac{t^{j}}{M_{j}} ≥ \frac{t^{l_{k+1}}}{M_{l_{k+1}}} ≥ \frac{a_{1} ⋯ a_{l_{k+1}}}{M_{l_{k+1}}} ,

we have

\frac{ω(t)}{e^{M(t)}} ≤ Π_{j=1}^{l_{k+1}} \frac{em_{j}}{a_{j}} = 1.

(b) → (a) is clear, since ∑_{k=1}^{∞} \frac{1}{m_{k}} < ∞ is equivalent to ∫_{0}^{∞} \frac{M(t)}{1+t^{2}} dt < ∞. □

3. A Hille-Yosida type theorem for strongly continuous semigroups with growth ω

In this section we apply the results established in the previous section to produce Hille-Yosida type theorems for once integrated semigroups, when the generating operators are not densely defined, and strongly continuous semigroups, when the generating operators are densely defined (Theorem 3.2 and its corollaries). In particular, we will produce a Hille-Yosida type theorem for strongly continuous semigroups with non-quasi-analytic growth, which seems to be first appearing here. This includes O(exp(dt^{\frac{1}{d}})) growth (Corollary 3.5). Theorem 3.8 involves only the resolvent, rather than all powers of the resolvent.

Definition 3.1 [2]. Assume A is closed. The exponentially bounded strongly continuous family of operators {S(t)}_{t≥0} on X is a once integrated semigroup generated by A if S(0) = 0 and there is r ∈ R such that (r, ∞) ⊆ ρ(A) and

(λ − A)^{−1}x = λ ∫_{0}^{∞} e^{−λt}S(t)x dt (x ∈ X, λ > r).

Recall that ω(·) is the function defined in Section 1. The following theorem is a direct consequence of Theorem 2.3.

Theorem 3.2. Suppose A is closed. Then the following are equivalent.

(a) A is the generator of a once integrated semigroup {S(t)}_{t≥0} and there is a constant K_{1} > 0 so that

limsup_{h→0^{+}} \frac{1}{h}||S(t + h) − S(t)|| ≤ K_{1}ω(t) ∀t ≥ 0.

(b) (r_{0}, ∞) ⊆ ρ(A) and there is a constant K_{2} > 0 so that

||(λ − A)^{−n}|| ≤ \frac{K_{2}}{(n − 1)!} ∫_{0}^{∞} e^{−λt}t^{n}ω(t) dt, ∀λ > r_{0}, n ∈ N.

If A is densely defined, then (a) and hence (b) are equivalent to

(c) A is the generator of a strongly continuous semigroup {T (t)}_{t≥0} so that ||T (t)|| ≤ K_{1}ω(t), ∀t ≥ 0.

Proof. To use Theorem 2.3 to prove the theorem, it suffices to note that

(\frac{d}{dλ})^{n−1}[(λ − A)^{−1}] = (−1)^{n−1}(n − 1)!(λ − A)^{−n}, ∀λ > r_{0}, n ∈ N. □

Setting ω(t) = (1 + t^{k}) and e^{rt} gives [6, Theorem 2.9] and [1, Theorem 4.1], respectively. The following theorem and its corollaries characterize strongly continuous semigroups of other types of growth, and follow directly from Theorem 2.12.

Theorem 3.3. Suppose A is closed and {M_{k}} satisfies (M.3). Then the following are equivalent.

(a) A is the generator of a once-integrated semigroup {S(t)}_{t≥0} and there is a constant K_{1} > 0 so that

limsup_{h→0^{+}} \frac{1}{h}||S(t + h) − S(t)|| ≤ K_{1}e^{M(t)}, ∀t ≥ 0.

(b) (0, ∞) ⊆ ρ(A) and there is a constant K_{2} > 0 so that, for any α > 0,

||(λ − α)^{n}(λ − A)^{−n}|| ≤ K_{2}e^{M^{∗}( \frac{1}{α})}, ∀λ > α, n ∈ N.

If A is densely defined, then (a) and hence (b) are equivalent to (c) A is the generator of a strongly continuous semigroup {T (t)}_{t≥0} so that

||T (t)|| ≤ K_{1}e^{M(t)}, ∀t ≥ 0.

K_{1}, K_{2} may be chosen so that K_{2} ≤ K_{1} ≤ 2eK_{2}.

The following corollary offers a Hille-Yosida type theorem for once integrated semigroups and strongly continuous semigroups with non-quasi-analytic growth.

Corollary 3.4. Suppose A is closed. Then the following are equivalent.

(a) A is the generator of a once-integrated semigroup {S(t)}_{t≥0} and there exists a continuous increasing function ω(·) defined on [0, ∞) that satisfies ω(0) = 1 and (**), such that

limsup_{h→0^{+}} \frac{1}{h}||S(t + h) − S(t)|| ≤ ω(t),

(b) There is a sequence {M_{k}} (k ∈ N ∪ {0}) satisfying (M.2) and (M.3) such that (a) and hence (b) of Theorem 3.3 is true.

If A is densely defined, then (a) and hence (b) are equivalent to

(c) A is the generator of a strongly continuous semigroup {T (t)}_{t≥0} and there exists a continuous increasing function ω(·) defined on [0, ∞) that satisfies ω(0) = 1 and (**), such that

||T (t)|| ≤ ω(t), ∀t ≥ 0.

Proof. (a) → (b) is an immediate consequence of Theorems 2.15 and 3.3. (b) → (a) follows by setting ω(t) ≡ e^{M(t)} for all t ≥ 0. □

Similarly, Corollary 2.14 gives us the following.

Corollary 3.5. Suppose A is closed and d > 1. Then the following are equivalent.

(a) A is the generator of a once-integrated semigroup {S(t)}_{t≥0} and there is a constant K_{1} > 0 so that

limsup_{h→0^{+}} \frac{1}{h}||S(t + h) − S(t)|| ≤ K_{1}exp(dt^{\frac{1}{d}}), ∀t ≥ 0.

(b) (0, ∞) ⊆ ρ(A) and there is a constant K_{2} > 0 so that, for any α > 0,

||(λ − α)^{n}(λ − A)^{−n}|| ≤ K_{2}exp((d − 1)α^{− \frac{1}{d−1}} ), ∀λ > α, n ∈ N.

If A is densely defined, then (a) and hence (b) are equivalent to (c) A is the generator of a strongly continuous semigroup {T (t)}_{t≥0} such that

||T (t)|| ≤ K_{1}exp(dt^{\frac{1}{d}}), ∀t ≥ 0.

Example 3.6. Weighted translation provides examples of strongly continuous semigroups with growth as in Corollary 3.5 that grow more rapidly than any polynomial.

Let g : [0, ∞) → (0, ∞), 1 ≤ p < ∞, and define

L^{p}([0, ∞), g ds) ≡ {f : [0, ∞) → C | ∫_{0}^{∞} |f(s)|^{p}g(s) ds < ∞}

with norm

∥f∥^{p} ≡ ∫_{0}^{∞} |f(s)|^{p}g(s) ds.

Let A be the generator of right translation on L^{p}([0, ∞), g dt); that is,

(e^{tA}f) (s) ≡ f(s − t) (s, t ≥ 0),

where f(s) ≡ 0 for s < 0.

Then it is not hard to show that

∥e^{tA}∥^{p} = sup_{s≥0} [\frac{g(t + s)}{g(s)}] (t ≥ 0).

If we choose, for 0 < r < 1, g(s) ≡ exp(s^{r}), then, for any positive t, [\frac{g(t+s)}{g(s)}] is decreasing as a function of s, thus

∥e^{tA}∥ = exp (\frac{t^{r}}{p} ) (t ≥ 0).

Definition 3.7. If C ∈ B(X) is injective, then a C-regularized semigroup generated by A is a strongly continuous family {W (t)}_{t≥0} ⊆ B(X) such that W (0) = C, W (t)W (s) = CW (t + s), for all s, t ≥ 0, and

Ax = C^{−1} [lim_{t→0} \frac{1}{t}(W (t)x − Cx)] ,

with maximal domain. Note that, when C = I, a C-regularized semigroup is a strongly continuous semigroup. See [6] for basic information.

Here is a sufficient condition for generating a O(exp(dt^{\frac{1}{d}})) (1 − A)^{−2}-regularized semigroup, that involves only the resolvent rather than all its powers.

Theorem 3.8. Suppose d > 1 and

{z ∈ C | Re(z) > 0} ⊆ ρ(A)

with

∥(z − A)^{−1}∥ = O ((Re(z) − α)^{−1} exp ((d − 1)α^{− \frac{1}{d−1}} )) ,  (1)

whenever Re(z) > α > 0. Then A generates a norm-continuous (1 − A)^{−2}- regularized semigroup {W (t)}_{t≥0} such that

∥W (t)∥ = O (t^{1− \frac{1}{d}} exp ((d + 1)t^{\frac{1}{d}} )) .  (2)

Proof: As in the proof of [7, Theorem 4.10], define, for 0 < ϵ < \frac{1}{2},

W (t) ≡ ∫_{2ϵ−i∞}^{2ϵ+i∞} e^{zt}(z − A)^{−1} \frac{dz}{2πi(1 − z)^{2}} .

As in that proof, this is a (1 − A)^{−2}-regularized semigroup generated by A. For 0 < ϵ < \frac{1}{4}, there is a constant M so that (choosing α = ϵ, z = 2ϵ + iy), by (1),

∥W (t)∥ ≤ M ∫_{R} e^{2ϵt}ϵ^{−1} exp ((d − 1)ϵ^{− \frac{1}{d−1}} ) \frac{dy}{(\frac{1}{2} + |y|)^{2}} ;  (3)

Define

K ≡ M ∫_{R} \frac{dy}{(\frac{1}{2} + |y|)^{2}} ,

and in (3), for t > 4^{\frac{d}{d−1}} , choose

ϵ ≡ t^{− \frac{(d−1)}{d}} ,

then we get from (3)

∥W (t)∥ ≤ Kt^{\frac{d−1}{d}} exp ((d + 1)t^{\frac{1}{d}} ) .

□

Remark 3.9. If A generated a (1 − A)^{−2}-regularized semigroup such that

∥W (t)∥ = O (exp (dt^{\frac{1}{d}} )) ,

then we would have

∥(z − A)^{−1}(1 − A)^{−2}∥ = O ((Re(z) − α)^{−1} exp ((d − 1)α^{− \frac{1}{d−1}} )) ,

the same as (1) with (z − A)^{−1} replaced by (z − A)^{−1}(1 − A)^{−2}.

Thus Theorem 3.8 is almost an equivalence involving only the resolvent (in the right half plane) rather than all powers of the resolvent.

In [6, Chapter 5], [7], [8], [12] and [13], maximal continuously embedded subspaces, where a closed operator generates a bounded strongly continuous group, semigroup, or a Lipschitz continuous once-integrated semigroup, were constructed. In the following, we establish analogous constructions for strongly continuous semigroups and locally Lipschitz continuous once-integrated semigroups with growth ω.

The following concepts can be found in [6].

A mild solution of (ACP) is u so that t ↦ u(t, x) ∈ C([0, ∞), X), ∫_{0}^{t} u(s, x) ds ∈ D(A), for all t ≥ 0, and

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

A weak mild solution of (ACP) is v so that t ↦ v(t, x) is locally Lipschitz continuous, ∫_{0}^{t} v(s, x) ds ∈ D(A), for all t ≥ 0, and

v(t, x) = A ∫_{0}^{t} v(s, x) ds + tx (t ≥ 0).

Definition 3.10. Let Z(A, ω) be the set of all x ∈ X for which (ACP) has a mild solution u(·, x) so that

t → [ω(t)]^{−1}||u(t, x)||

is bounded and uniformly continuous on [0, ∞). Define

||x||_{Z(A,ω)} = sup_{t≥0}[ω(t)]^{−1}||u(t, x)||.

Let Y (A, ω) be the set of all x ∈ X for which (ACP) has a mild solution v(·, x) so that

esssup\{ω(t)^{−1}|\frac{d}{dt} < v(t, x), x^{∗} > | : t ≥ 0, x^{∗} ∈ X^{∗}, ||x^{∗}|| ≤ 1\} < ∞.

We recall that ω_{1}(s) ≡ limsup_{t→∞} \frac{ω(t+s)}{ω(t)} is polynomially bounded (see Section 1).

The following theorem is an analogue of [6, Theorem 5.5], [7, Theorem 3.6], [8, Proposition 1.8] and [15, Theorem 7]. We omit its proof.

Theorem 3.11. Suppose m ∈ N ∪ {0} and λ − A is injective for λ > r. Then the following are true.

(a) Y (A, ω) and Z(A, ω) are Banach spaces continuously embedded in X.

(b) A|_{Z(A,ω)} generates a strongly continuous semigroup {T (t)}_{t≥0} such that

||T (t)|| ≤ ω(t), ∀t ≥ 0.

(c) A|_{Z(A,ω)} generates a once-integrated semigroup {S(t)}_{t≥0} such that

limsup_{h→0^{+}} \frac{1}{h}||S(t + h) − S(t)|| ≤ ω(t), ∀t ≥ 0.

(d) Z(A, ω) equals the closure of D(A|_{Z(A,ω)}) in Y (A, ω).

(e) x ∈ Y (A, ω) if and only if x ∈ Im((λ − A)^{−n}) for all n ∈ N, λ > 0 and there is a constant K > 0 such that

||(λ − A)^{−n}x|| ≤ \frac{K}{(n − 1)!} ∫_{0}^{∞} e^{−λt}t^{n}ω(t) dt.

(f) Z(A, ω) is maximal unique; that is, if W is a Banach space continuously embedded in X so that A|_{W} generates a strongly continuous semigroup that is O(ω(t)), then W is continuously embedded in Z(A, ω).

(g) Assume ω(·) satisfies (∗∗), (∗ ∗ ∗) and ω_{1}(s) is polynomially bounded and

P σ((A|_{Z(A,ω)})^{∗}) ∩ iR = ∅.

Then for every x ∈ Z(A, ω), such that σ(A, x) ∩ iR is countable, we have

lim_{t→∞} \frac{1}{ω_{1}(t)}||T (t)x|| = 0.

Letting ω(t) = 1+t^{k}, e^{rt} and e^{M(t)}, respectively, we will have three special cases of (a)—(f) of Theorem 3.11.

Remark 3.12. It is surprising that if ω(·) satisfies (***), (b) of Theorem 3.11 can be improved as

||T (t)||_{Z(A,ω)} ≤ ω(t), ∀t ≥ 0.

Given x ∈ Z(A, ω). We have

||[ω(s)]^{−1}T (s)T (t)x|| = ||[ω(s)]^{−1}T (s + t)x||
≤ [ω(s)]^{−1}ω(s + t)||x||_{Z(A,ω)} ≤ ω(t)||x||_{Z(A,ω)}, ∀s ≥ 0,

as desired. □

The results of this section are easily extended to regularized semigroups (see [6]). We omit the details.

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