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Technische Universität Dresden

Herausgeber: Der Rektor

Modulus semigroups and perturbation classes for linear delay equations in L_{p}

H. Vogt, J. Voigt Institut für Analysis MATH-AN-07-02

Modulus semigroups and perturbation classes for linear delay equations in L_{p}

Hendrik Vogt and Jürgen Voigt

Fachrichtung Mathematik, Technische Universität Dresden, D-01062 Dresden, Germany

Dedicated to the memory of H. H. Schaefer

Abstract

In this paper we study C_{0}-semigroups on X×L_{p}(−h, 0; X) associated with linear differential equations with delay, where X is a Banach space. In the case that X is a Banach lattice with order continuous norm, we describe the associated modulus semigroup, under minimal assumptions on the delay operator. Moreover, we present a new class of delay operators for which the delay equation is well-posed for p in a subinterval of [1, ∞).

MSC 2000: 47D06, 34K06, 47B60

Keywords: functional differential equation, delay equation, modulus semigroup, perturbation theory, domination, Banach lattice

Introduction

We treat two topics arising in connection with the Cauchy problem for the linear delay equation

\{ \begin{matrix}u′(t) = Au(t) + Lu_{t} (t ⩾ 0), \\ u(0) = x, u_{0} = f, \end{matrix}  (DE)

with initial values x ∈ X, f ∈ L_{p}(−h, 0; X), where X is a Banach space, 1 ⩽ p < ∞, and 0 < h ⩽ ∞. (For a function u: (−h, ∞) → X, we recall the notation

u_{t}(θ) := u(t + θ) (−h < θ < 0),

for t ⩾ 0.) The foundations for treating this problem in the context of C_{0}-semigroups on X × L_{p}(−h, 0; X) have been presented in [2]; we also refer to [3].

One of the topics concerns the question of the kind of operators L that are allowed in (DE). In the previous papers it was assumed that L is associated with a function η: [−h, 0] → L(X) of bounded variation (cf. Example 5.1). Then the problem (DE) could be treated in X × L_{p}(−h, 0; X) for any p ∈ [1, ∞). We present a class of operators L that allows this treatment only for p in a proper subset of [1, ∞): We only require L: W_{p}^{1}(−h, 0; X) → X to be continuous as an operator from L_{r}(μ_{L}; X) to X, for some r ∈ [1, p] and a suitable measure μ_{L} on [−h, 0].

For the other topic we additionally assume that X is a Banach lattice. Then we determine the modulus semigroup in a rather general context. This generalises the results of [5], [13], [10].

In order to put the results of the paper into the proper context, we now describe the C_{0}-semigroup setting of (DE). Let X be a Banach space, A the generator of a C_{0}- semigroup T on X. Let T_{0} be the C_{0}-semigroup on X_{p} := X × L_{p}(−h, 0; X) given by

T_{0}(t) = (\begin{matrix}T (t) & 0 \\ T_{t} & S(t) \end{matrix}) (t ⩾ 0),

where T_{t} ∈ L(X, L_{p}(−h, 0; X)) denotes the operator given by

T_{t}x(θ) := \{ \begin{matrix}0 & for − h < θ ⩽ −t, \\ T (t + θ)x & for − t < θ < 0, \end{matrix}

and S is the C_{0}-semigroup of left translation on L_{p}(−h, 0; X), i.e., S(t)f = f_{t}, where we assume that f ∈ L_{p}(−h, 0; X) is extended by 0 to a function on (−h, ∞). It is known (see [2; Prop. 3.1]) that the generator of T_{0} is given by

A_{0} = (\begin{matrix}A & 0 \\ 0 & \frac{d}{dθ} \end{matrix}) , D(A_{0}) = \{( \begin{matrix}x \\ φ \end{matrix}) ∈ D(A) × W_{p}^{1}(−h, 0; X); x = φ(0)\}.

Let now L ∈ L(W_{p}^{1}(−h, 0; X), X). Then B := ( \begin{matrix}0 & L \\ 0 & 0 \end{matrix} ) ∈ L(D(A_{0}), X_{p}), and we define

A := A_{0} + B = (\begin{matrix}A & L \\ 0 & \frac{d}{dθ} \end{matrix}) , D(A) = D(A_{0}).

Assuming that A is the generator of a C_{0}-semigroup T one knows that, for ( \begin{matrix}x \\ φ \end{matrix}) ∈ D(A), the first component of the function t ↦ T (t)( \begin{matrix}x \\ φ \end{matrix}) is the unique solution of (DE); cf. [2].

Next, assume that X is a Banach lattice with order continuous norm and that T possesses a modulus semigroup (which is the smallest semigroup dominating T), whose generator will be denoted by A^{#}. Assume that L possesses a modulus |L|, that L is massless at 0 (cf. Section 1), and that Ã := ( \begin{matrix}A^{#} & |L| \\ 0 & \frac{d}{dθ} \end{matrix} ), with domain

D(Ã) = \{( \begin{matrix}x \\ φ \end{matrix}) ∈ D(A^{#}) × W_{p}^{1}(−h, 0; X); x = φ(0)\}, is a generator. Then we show that Ã generates the modulus semigroup of T (Theorem 2.7).

The paper is organized as follows. In Section 1 we investigate operators from W_{p}^{1}(−h, 0; X) to Y, where X, Y are Banach lattices. The main objective is establishing relations between masslessness at 0 of L and |L|. In Section 2 we determine the modulus semigroup of T ; see above.

Modulus semigroups and perturbation classes for linear delay equations in L_{p} 3

In Section 3 we discuss an inequality needed for L in order to make B a small Miyadera perturbation of A. In Section 4 we present our class of operators L mentioned above; the conditions are such that the inequality singled out in Section 3 is satisfied. Section 5 presents an example illustrating that the class of operators from Section 4 contains more operators than those associated with functions η: [−h, 0] → L(X) of bounded variation.

Throughout this paper, let 1 ⩽ p < ∞, 0 < h ⩽ ∞.

1 Operators defined on W_{p}^{1}(−h, 0; X) that are massless at 0

In this section we investigate how to describe that operators L: W_{p}^{1}(−h, 0; X) → Y attribute no mass to the point 0, and we present relations between different notions of this kind. We start with a result in a more abstract setting.

Proposition 1.1. Let X be a (real or complex) vector lattice, Y a Banach lattice with order continuous norm. Let L: X → Y be a linear operator, and assume that L has a modulus |L|,

|L|x = sup{|Lz|; z ∈ X, |z| ⩽ x}

for x ∈ X_{+}. Let Q_{n}: X → X be linear operators, 0 ⩽ Q_{n} ⩽ I (n ∈ ℕ).

(a) Assume that the strong limit L_{Q} := s-lim_{n→∞} LQ_{n} exists. Then the linear operators L_{Q} and L − L_{Q} both have a modulus, and |L|Q_{n} → |L_{Q}|, |L|(I − Q_{n}) → |L − L_{Q}| strongly.

(b) If the sequence (Q_{n}) is monotone then s-lim_{n→∞} LQ_{n} exists.

Proof. (a) For x ∈ X we have |L_{Q}x| = lim_{n→∞} |LQ_{n}x| ⩽ |L||x| since 0 ⩽ Q_{n} ⩽ I. Thus L_{Q} has a modulus, by the order completeness of Y (which, in turn, follows from the order continuity of the norm); cf. [9]. Similarly, L − L_{Q} has a modulus.

Since X = lin X_{+} it now suffices to show that |L|Q_{n}x → |L_{Q}|x and |L|(I − Q_{n})x → |L − L_{Q}|x for all x ∈ X_{+}. Let ε > 0. Since Y has order continuous norm, there exist x_{j} ∈ X, |x_{j}| ⩽ x (j = 1, …, m) such that y := sup_{j} |L_{Q}x_{j}| and z := sup_{j} |(L − L_{Q})x_{j}| satisfy

|||L_{Q}|x − y|| ⩽ ε, |||L − L_{Q}|x − z|| ⩽ ε.  (1.1)

The lattice operations in Y are continuous, so we have

y_{n} := sup_{j} |LQ_{n}x_{j}| → sup_{j} |L_{Q}x_{j}| = y,

z_{n} := sup_{j} |L(I − Q_{n})x_{j}| → sup_{j} |(L − L_{Q})x_{j}| = z.

Let now n_{0} ∈ ℕ such that ||y_{n} − y|| ⩽ ε, ||z_{n} − z|| ⩽ ε for all n ⩾ n_{0}. By the definition of y_{n}, z_{n} we obtain, using the estimate |L|x ⩽ |L_{Q}|x + |L − L_{Q}|x,

z_{n} − |L − L_{Q}|x ⩽ |L|(I − Q_{n})x − |L − L_{Q}|x ⩽ |L_{Q}|x − |L|Q_{n}x ⩽ |L_{Q}|x − y_{n}.

For n ⩾ n_{0} the left and right hand sides of this chain of inequalities have norm ⩽ 2ε by (1.1), so |L|Q_{n}x → |L_{Q}|x and |L|(I − Q_{n})x → |L − L_{Q}|x as n → ∞.

(b) It clearly suffices to treat the case that (Q_{n}) is monotone increasing. Let x ∈ X. For n ⩽ m we estimate

|LQ_{m}x − LQ_{n}x| ⩽ |L||Q_{m}x − Q_{n}x| ⩽ |L|(Q_{m} − Q_{n})|x|.

The order continuity of the norm in Y implies that the increasing sequence (|L|Q_{n}|x|) in [0, |L||x|] is convergent, and therefore is a Cauchy sequence. The previous estimate implies that (LQ_{n}x) is a Cauchy sequence as well.

Definition. Let X, Y be Banach spaces, and let L: W_{p}^{1}(−h, 0; X) → Y be a bounded linear operator. We say that L is massless at 0 if for all x ∈ X and for all sequences (χ_{n}) in W_{∞}^{1}(−h, 0), 0 ⩽ χ_{n} ⩽ 1, χ_{n}(t) = 0 (t ⩽ −1/n) for all n ∈ ℕ, one has

L(χ_{n}x) → 0 (n → ∞).

We say that L is strongly massless at 0 if for all φ ∈ W_{p}^{1}(−h, 0; X) and for all sequences (χ_{n}) as above, one has

L(χ_{n}φ) → 0  (n → ∞)  (1.2)

(note that χ_{n}φ ∈ W_{p}^{1}(−h, 0; X)).

Remark 1.2. Obviously, L is massless at 0 if and only if for all x ∈ X and for all ε > 0 there exists δ > 0 such that ||L(χx)|| ⩽ ε for all χ ∈ W_{∞}^{1}(−h, 0), 0 ⩽ χ ⩽ 1, χ(t) = 0 (t ⩽ −δ).

Remarks 1.3. Let X, Y be Banach lattices with order continuous norm.

(a) We recall that W_{p}^{1}(−h, 0; X) is a vector lattice, and that |||φ|||_{p,1} ⩽ ||φ||_{p,1} for all φ ∈ W_{p}^{1}(−h, 0; X); cf. [13; Thm. 1 and Rem. 5].

(b) Assume that L: W_{p}^{1}(−h, 0; X) → Y is linear and possesses a modulus. We show that then L and |L| are continuous. Because of (a) it suffices to prove the assertion for the case that L is positive.

We show that φ_{n} → 0 in W_{p}^{1}(−h, 0; X) implies Lφ_{n} → 0 in Y. Without loss of generality suppose ∑_{n=1}^{∞} n||φ_{n}||_{p,1} < ∞. Defining φ := ∑_{n=1}^{∞} n|φ_{n}| we obtain |Lφ_{n}| ⩽ L|φ_{n}| ⩽ \frac{1}{n}Lφ for all n ∈ ℕ.

We note that the continuity of positive operators between ordered Banach spaces can be obtained in more general situations; cf. [7; Thm. 2.1].

(c) Assume that L: W_{p}^{1}(−h, 0; X) → Y is linear and possesses a modulus, and that L is strongly massless at 0. Proposition 1.1 implies that then |L| is strongly massless at 0 as well. Indeed, if (χ_{n}) is a sequence as in the definition above then with Q_{n} defined by Q_{n}φ := χ_{n}φ we obtain L_{Q} = 0, and therefore |L_{Q}| = 0.

The following lemma shows that it suffices to require property (1.2) for a special sequence in the definition of ‘strongly massless at 0’ if L possesses a modulus.

Lemma 1.4. Let X, Y be Banach lattices with order continuous norm, L: W_{p}^{1}(−h, 0; X) → Y a linear operator. Assume that L possesses a modulus and that there exists a sequence (χ_{n}) in W_{∞}^{1}(−h, 0), 0 ⩽ χ_{n} ⩽ 1, χ_{n}(0) = 1 for all n ∈ ℕ, such that

L(χ_{n}φ) → 0 (n → ∞) (1.3)

for all φ ∈ W_{p}^{1}(−h, 0; X). Then L is strongly massless at 0.

Proof. Since Proposition 1.1 implies |L|(χ_{n}φ) → 0 (n → ∞), it is sufficient to treat the case L ⩾ 0. Let (χ̃_{n}) be a sequence as in the definition of ‘strongly massless at 0’, φ ∈ W_{p}^{1}(−h, 0; X), φ ⩾ 0. Let n ∈ ℕ. Then there exists m_{0} ∈ ℕ such that for all m ⩾ m_{0} one has χ̃_{m} ⩽ 2χ_{n}, and therefore

0 ⩽ L(χ̃_{m}φ) ⩽ 2L(χ_{n}φ).

By (1.3) this implies that lim_{m→∞} L(χ̃_{m}φ) = 0.

We are going to show that in a rather general context L can be decomposed as the sum of two operators, the first only depending on the value of the function at 0, the second being strongly massless at 0.

Proposition 1.5. Let X, Y be Banach lattices with order continuous norm. Assume that L: W_{p}^{1}(−h, 0; X) → Y is linear and possesses a modulus.

(a) Let (χ_{n}) be a sequence in W_{∞}^{1}(−h, 0), 0 ⩽ χ_{n} ⩽ 1, χ_{n}(t) = 0 (t ⩽ −1/n), χ_{n}(0) = 1 for all n ∈ ℕ. Then

L_{0}φ := lim_{n→∞} L(χ_{n}φ)

exists for all φ ∈ W_{p}^{1}(−h, 0; X), and the limit does not depend on the sequence (χ_{n}). The linear operator L_{0} thus defined satisfies L_{0}φ = 0 if φ ∈ W_{p}^{1}(−h, 0; X), φ(0) = 0; the operator L − L_{0} is strongly massless at 0.

(b) The operator L_{0} has the modulus |L|_{0}, and L − L_{0} has the modulus |L| − |L|_{0}.

Proof. (a) We first suppose that (χ_{n}) is a monotone decreasing sequence satisfying the assumptions. Then L_{0}φ := lim_{n→∞} L(χ_{n}φ) exists for all φ ∈ W_{p}^{1}(−h, 0; X), and L_{0} as well as L − L_{0} possess a modulus, by Proposition 1.1. Obviously, L_{0}φ = 0 if φ = 0 in a neighbourhood of 0. Since the set of those φ is dense in {φ ∈ W_{p}^{1}(−h, 0; X); φ(0) = 0} and L_{0} is continuous (cf. Remark 1.3(b)), we obtain that L_{0}φ = 0 if φ(0) = 0. From this we conclude that

(L − L_{0})(χ_{n}φ) = L(χ_{n}φ) − L_{0}(χ_{n}φ) = L(χ_{n}φ) − L_{0}φ → 0 (n → ∞).

Thus, by Lemma 1.4, L − L_{0} is strongly massless at 0.

Let now (χ_{n}) be any sequence satisfying the assumptions. Then with L_{0} obtained as above we have

L(χ_{n}φ) = (L − L_{0})(χ_{n}φ) + L_{0}φ → L_{0}φ (n → ∞)

by the definition of ‘strongly massless at 0’. This completes the proof of (a). (b) follows from Proposition 1.1.

The following result is another version of Lemma 1.4. It contains as a consequence that, for operators possessing a modulus, ‘massless at 0’ implies ‘strongly massless at 0’.

Proposition 1.6. Let X, Y be Banach lattices with order continuous norm, L: W_{p}^{1}(−h, 0; X) → Y a linear operator. Assume that L possesses a modulus and that there exists a sequence (χ_{n}) in W_{∞}^{1}(−h, 0), 0 ⩽ χ_{n} ⩽ 1, χ_{n}(t) = 0 (t ⩽ −1/n), χ_{n}(0) = 1 for all n ∈ ℕ such that L(χ_{n}x) → 0 (n → ∞) for all x ∈ X. Then L is strongly massless at 0.

Proof. We use the decomposition obtained in Proposition 1.5. Let χ ∈ W_{p}^{1}(−h, 0), χ = 1 in a neighbourhood of 0. Then

L_{0}φ = L_{0}(χφ(0)) = lim_{n→∞} L(χ_{n}χφ(0)) = 0

for all φ ∈ W_{p}^{1}(−h, 0; X), i.e., L_{0} = 0. Therefore L = L − L_{0} is strongly massless at 0.

2 The modulus semigroup

The following two lemmas are technical results which will be needed later. We assume that X is a Banach space, that A is the generator of a C_{0}-semigroup T on X, and that L ∈ L(W_{p}^{1}(−h, 0; X), X). Let A_{0}, T_{0} and B be as defined in the introduction. For later use we note that

BT_{0}(s)( \begin{matrix}x \\ φ \end{matrix}) = (\begin{matrix}0 & L \\ 0 & 0 \end{matrix}) (\begin{matrix}T (s)x \\ T_{s}x + S(s)φ \end{matrix}) = (\begin{matrix}L(T_{s}x + S(s)φ) \\ 0 \end{matrix})  (2.1)

for ( \begin{matrix}x \\ φ \end{matrix}) ∈ D(A)_{0}.

Lemma 2.1. The spectral radius of B(λ − A_{0})^{−1} tends to 0 as λ → ∞.

Proof. For c > 0 we endow X_{p} = X × L_{p}(−h, 0; X) with the norm ||( \begin{matrix}x \\ f \end{matrix})||_{c} := ||x|| + c||f||_{p} and also denote the corresponding norm on L(X_{p}) by || · ||_{c}.

Let ( \begin{matrix}x \\ f \end{matrix}) ∈ X_{p}, λ larger than the type of T_{0}. Denoting by P_{2} the projection of X_{p} onto its second component, we obtain that

||B(λ − A_{0})^{−1}( \begin{matrix}x \\ f \end{matrix})||_{c} = ||LP_{2}(λ − A_{0})^{−1}( \begin{matrix}x \\ f \end{matrix})|| ⩽ ||L||||P_{2}(λ − A_{0})^{−1}( \begin{matrix}x \\ f \end{matrix})||_{p,1}

and

P_{2}(λ − A_{0})^{−1}( \begin{matrix}x \\ f \end{matrix}) = ∫_{0}^{∞} e^{−λt}P_{2}T_{0}(t)( \begin{matrix}x \\ f \end{matrix}) dt = ∫_{0}^{∞} e^{−λt}(T_{t}x + S(t)f) dt.

A straightforward computation shows that ∫_{0}^{∞} e^{−λt}T_{t}x dt = ε_{λ}(λ − A)^{−1}x, where ε_{λ}(θ) := e^{λθ} for θ ∈ (−h, 0) (cf. [6; Prop. VI.6.7]). This implies

P_{2}(λ − A_{0})^{−1}( \begin{matrix}x \\ f \end{matrix}) = ε_{λ}(λ − A)^{−1}x + (λ − \frac{d}{dθ})^{−1}f,

where \frac{d}{dθ} denotes the generator of S. There exist λ_{0} > 0, M ⩾ 1 such that (λ + 1)||(λ − A)^{−1}|| ⩽ M and ||(λ − \frac{d}{dθ})^{−1}: L_{p}(−h, 0; X) → W_{p}^{1}(−h, 0; X)|| ⩽ M for all λ ⩾ λ_{0}.

In the following let λ ⩾ λ_{0}. Then

||P_{2}(λ − A_{0})^{−1}( \begin{matrix}x \\ f \end{matrix})||_{p,1} ⩽ (λ + 1)||ε_{λ}||_{L_{p}(−h,0)}||(λ − A)^{−1}x|| + M||f||_{p}
⩽ Mδ_{λ}||x|| + M||f||_{p},

with δ_{λ} := ||ε_{λ}||_{L_{p}(−h,0)} → 0 as λ → ∞. For c := \frac{1}{δ_{λ}} we conclude that

||B(λ − A_{0})^{−1}( \begin{matrix}x \\ f \end{matrix})||_{c} ⩽ ||L||M(δ_{λ}||x|| + ||f||_{p}) = δ_{λ}||L||M||( \begin{matrix}x \\ f \end{matrix})||_{c},

so ||B(λ−A_{0})^{−1}||_{c} ⩽ δ_{λ}||L||M, and this implies that the spectral radius of B(λ−A_{0})^{−1} is less or equal δ_{λ}||L||M.

Lemma 2.2. Assume that L is massless at 0 and that A = A_{0} + B is the generator of a C_{0}-semigroup T on X_{p}. Then

\frac{1}{t}(T (t) − T_{0}(t))( \begin{matrix}x \\ 0 \end{matrix} ) → 0 (t → 0)

for all x ∈ D(A).

Proof. Let t > 0, x ∈ D(A). For n ∈ ℕ define φ_{n} ∈ W_{p}^{1}(−h, 0; X) by φ_{n}(θ) := (1 + nθ)^{+}x. Then φ_{n} → 0 in L_{p}(−h, 0; X) as n → ∞ and hence

||(T (t) − T_{0}(t))( \begin{matrix}x \\ 0 \end{matrix} )|| = lim_{n→∞} ||(T (t) − T_{0}(t))( \begin{matrix}x \\ φ_{n} \end{matrix})||.  (2.2)

For s ∈ [0, t] let now χ_{n,s} ∈ W_{∞}^{1}(−h, 0), χ_{n,s}(θ) := (1 + n(θ + s))^{+} ∧ 1 and ψ_{s} ∈ W_{p}^{1}(−h, 0; X), ψ_{s}(θ) := 0 for θ ⩽ −s, ψ_{s}(θ) := T (s + θ)x − x for θ > −s. Then T_{s}x + S(s)φ_{n} = χ_{n,s}x + ψ_{s}. Defining M := sup_{0⩽s⩽t} ||T (s)|| and using the Duhamel formula and (2.1), we obtain

||(T (t) − T_{0}(t))( \begin{matrix}x \\ φ_{n} \end{matrix})|| = || ∫_{0}^{t} T (t − s)BT_{0}(s)( \begin{matrix}x \\ φ_{n} \end{matrix}) ds||

⩽ M ∫_{0}^{t} ||L(T_{s}x + S(s)φ_{n})|| ds ⩽ M ∫_{0}^{t}(||L(χ_{n,s}x)|| + ||L||||ψ_{s}||_{p,1}) ds.

Together with (2.2) this implies the assertion: We have

||ψ_{s}||_{p,1}^{p} = ||ψ_{s}||_{p}^{p} + ∫_{−s}^{0} ||T (s + θ)Ax||^{p} dθ → 0 (s → 0),

and L(χ_{n,s}x) → 0 as n → ∞ and s → 0 since L is massless at 0.

The following proposition is an abstract result on generation, positivity and domination of perturbed semigroups. It will be needed for the application to delay semigroups.

We recall that for operators B, B̃ ∈ L(X), where X is a Banach lattice, B is dominated by B̃ if |Bx| ⩽ B̃|x| for all x ∈ X. If T, T̃ are C_{0}-semigroups on X then T is dominated by T̃ if T (t) is dominated by T̃(t) for all t ⩾ 0.

Proposition 2.3. Let X be a Banach lattice. Let T, T̃ be C_{0}-semigroups on X, with generators A, Ã respectively, and assume that T is dominated by T̃. Let B: D(A) → X, B̃: D(Ã) → X be linear operators, B̃ ⩾ 0,

B(λ − A)^{−1} dominated by B̃(λ − Ã)^{−1} (λ ⩾ λ_{0})

for some λ_{0} ∈ ℝ. Assume that Ã+ B̃ generates a C_{0}-semigroup T̃B̃ on X and that the spectral radius of B̃(λ − Ã)^{−1} is less that 1 for all λ ⩾ λ_{0}. Then T̃B̃ is positive, and A + B generates a C_{0}-semigroup T_{B} on X that is dominated by T̃B̃.

Proof. Let λ ⩾ λ_{0}. The assumptions imply that

(λ − Ã − B̃)^{−1} = (λ − Ã)^{−1} ∑_{k=0}^{∞}(B̃(λ − Ã)^{−1})^{k} ⩾ 0

(cf. [1; Thm. 3.1]). Therefore, the C_{0}-semigroup T̃B̃ is positive.

Since B(λ − A)^{−1} is dominated by B̃(λ − Ã)^{−1}, we obtain in the same way that

(λ − A − B)^{−1} = (λ − A)^{−1} ∑_{k=0}^{∞}(B(λ − A)^{−1})^{k}.

We conclude that (λ − A − B)^{−1} is dominated by (λ − Ã − B̃)^{−1}. Hence ||(λ − A − B)^{−n}|| ⩽ ||(λ − Ã − B̃)^{−n}|| for all n ∈ ℕ, and the assertion follows from the Hille-Yosida generation theorem.

The following ‘domination lemma’ is a generalisation of [10; Lemma 1.1]; a more restricted form has already been used in [11; proof of Prop. 1.2].

Lemma 2.4. Let X be a Banach lattice. Let T, S be C_{0}-semigroups on X, S positive, and let A be the generator of T. Let k ∈ ℕ_{0}, R: [0, 1] → L(D_{A^{k}}, X) (where D_{A^{k}} is the domain of A^{k} endowed with the graph norm of A^{k}), and assume that

\frac{1}{t}R(t)x → 0 (t → 0)

for all x ∈ D(A^{k}). Further assume that

|T (t)x| ⩽ S(t)|x| + |R(t)x|

for all x ∈ D(A^{k}), 0 ⩽ t ⩽ 1. Then T is dominated by S.

Proof. Let x ∈ D(A^{k}), t > 0. As in [10; proof of Lemma 1.1, eqn. (1.2)] we obtain

|T (t)x| ⩽ S(t)|x| + ∑_{m=1}^{n} S(\frac{n−m}{n}t) |R(\frac{t}{n})T (\frac{m−1}{n}t)x|  (2.3)

for all n ∈ ℕ, n ⩾ t. With c_{t} := sup_{0⩽s⩽t} ||S(s)|| we estimate

|| ∑_{m=1}^{n} S(\frac{n−m}{n}t) |R(\frac{t}{n})T (\frac{m−1}{n}t)x| || ⩽ c_{t} ∑_{m=1}^{n} ||R(\frac{t}{n})T (\frac{m−1}{n}t)x||

⩽ tc_{t} max\{||\frac{n}{t}R(\frac{t}{n})T (\frac{m−1}{n}t)x||; 1 ⩽ m ⩽ n\}.

The hypothesis on R implies that \frac{n}{t}R(\frac{t}{n}) → 0 strongly in L(D_{A^{k}}, X) as n → ∞. We recall that strong convergence is uniform on compact sets and that T (·)x: [0, ∞) → D_{A^{k}} is continuous. Therefore the right hand side of the last inequality converges to 0 as n → ∞. Letting n → ∞ in (2.3) we obtain |T (t)x| ⩽ S(t)|x| for all x ∈ D(A^{k}). This implies the assertion since D(A^{k}) is dense in X.

The following proposition generalises [10; Prop. 2.1].

Proposition 2.5. Let X be a Banach lattice. Assume that L is massless at 0, and that A is the generator of a C_{0}-semigroup T on X_{p}.

If T is dominated by a C_{0}-semigroup S on X_{p} then T_{0} is dominated by S as well.

Proof. For t > 0 we define R_{1}(t) := T (t) − T_{0}(t), R(t) := R_{1}(t) ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ). Let ( \begin{matrix}x \\ f \end{matrix}) ∈ D(A_{0}). Estimating as in [10; proof of Prop. 2.1] we obtain that

|T_{0}(t)( \begin{matrix}x \\ f \end{matrix})| ⩽ S(t)|( \begin{matrix}x \\ f \end{matrix})| + |R(t)( \begin{matrix}x \\ f \end{matrix})|.

Lemma 2.2 implies that \frac{1}{t}R(t)( \begin{matrix}x \\ f \end{matrix}) → 0 as t → 0. Now applying Lemma 2.4, with k = 1, we obtain the assertion.

Remark 2.6. Assume that X is a Banach lattice with order continuous norm and that L possesses a modulus. Then one may rearrange the setting in such a way that L is (strongly) massless at 0.

Indeed, let χ ∈ W_{p}^{1}(−h, 0), χ(0) = 1. Define E ∈ L(X, W_{p}^{1}(−h, 0; X)) by Ex := χx. Let L_{0} ∈ L(W_{p}^{1}(−h, 0; X), X) be as in Proposition 1.5. Since φ(0) = x for ( \begin{matrix}x \\ φ \end{matrix}) ∈ D(A), we obtain

A = (\begin{matrix}A & L \\ 0 & \frac{d}{dθ} \end{matrix}) = (\begin{matrix}A + L_{0}E & L − L_{0} \\ 0 & \frac{d}{dθ} \end{matrix}) .

Thus, we have written A with a generater A + L_{0}E and an off-diagonal term L − L_{0} that is (strongly) massless at 0.

For the remainder of this section we assume that X is a Banach lattice with order continuous norm and that T possesses a modulus semigroup, T^{#}, with generator A^{#}. (We recall that the modulus semigroup of T is the smallest C_{0}-semigroup dominating T; see [4] for results on modulus semigroups.) It was shown in [10; Prop. 3.2(b)] that then the C_{0}-semigroup T_{0} possesses a modulus semigroup, T_{0}^{#}, whose generator is given by A_{0}^{#} = ( \begin{matrix}A^{#} & 0 \\ 0 & \frac{d}{dθ} \end{matrix} ), with domain D(A_{0}^{#}) = \{( \begin{matrix}x \\ φ \end{matrix}) ∈ D(A^{#}) × W_{p}^{1}(−h, 0; X); x = φ(0)}.

Assume additionally that L is massless at 0, that L possesses a modulus and that Ã:= A_{0}^{#} + ( \begin{matrix}0 & |L| \\ 0 & 0 \end{matrix} ) = ( \begin{matrix}A^{#} & |L| \\ 0 & \frac{d}{dθ} \end{matrix} ), with domain D(Ã) = D(A_{0}^{#}), is the generator of a

C_{0}-semigroup T̃.

Taking into account Lemma 2.1, we then obtain by Proposition 2.3 that A is the generator of a C_{0}-semigroup T on X_{p}, and that T is dominated by T̃. From the order completeness of X_{p} we conclude that T possesses a modulus semigroup, T^{#}, with generator A^{#}, and it is a consequence of Proposition 2.5 that then T_{0}^{#}(t) ⩽ T^{#}(t) ⩽ T̃(t) for all t ⩾ 0; cf. [10; Prop. 3.2(a)].

Theorem 2.7. Let X, T , and T̃ be as introduced above. Then T^{#} = T̃.

The proof of this result is the same as for [10; Theorem 3.1]. Therefore we are not going to reproduce it; cf. [10; pp. 397–398]. We only mention that in [10; bottom of p. 397], for 0 ⩽ φ ∈ W_{p}^{1}(−h, 0; X), the existence of a sequence (φ_{k}) in W_{p}^{1}(−h, 0; X), φ_{k}(0) = 0, 0 ⩽ φ_{k} ⩽ φ (k ∈ ℕ) such that |L|φ_{k} → |L|φ (k → ∞) was needed. In the present context, the existence of such a sequence follows from the fact that |L| is strongly massless at 0. (Note that the hypotheses and Proposition 1.6 imply that L is strongly massless at 0, and recall Remark 1.3(c).)

3 Miyadera perturbations of delay semigroups

We assume that X is a Banach space, T a C_{0}-semigroup on X, L ∈ L(W_{p}^{1}(−h, 0; X), X), and we use the notation of the introduction.

The following theorem is a slight generalisation of [2; Thm. 3.2]; see also [3; Thm. 3.26]. Our contribution—already used in [10; Sec. 1.2] as well as in the proof of Lemma 2.1 above—is the use of a suitable norm on the product space X_{p} = X × L_{p}(−h, 0; X).

Theorem 3.1. Assume that there exist t, c > 0 and γ < 1 such that

∫_{0}^{t} ||L(T_{s}x + S(s)φ)|| ds ⩽ γ||x|| + c||φ||_{p}  (( \begin{matrix}x \\ φ \end{matrix}) ∈ D(A_{0})).  (3.1)

Then A = A_{0} + B is the generator of a C_{0}-semigroup T on X_{p}.

More precisely, if X_{p} is endowed with the norm ||( \begin{matrix}x \\ f \end{matrix})||_{c/γ} := ||x|| + \frac{c}{γ} ||f||_{p} then B is a small Miyadera perturbation of A_{0}, i.e.,

∫_{0}^{t} ||BT_{0}(s)( \begin{matrix}x \\ φ \end{matrix})||_{c/γ} ds ⩽ γ||( \begin{matrix}x \\ φ \end{matrix})||_{c/γ} (( \begin{matrix}x \\ φ \end{matrix}) ∈ D(A_{0})).

(For Miyadera perturbations and the Miyadera perturbation theorem we refer to [12], [6; III.3.c].)

Proof. By (2.1) we obtain that

∫_{0}^{t} ||BT_{0}(s)( \begin{matrix}x \\ φ \end{matrix})||_{c/γ} ds = ∫_{0}^{t} ||L(T_{s}x + S(s)φ)|| ds ⩽ γ||x|| + c||φ||_{p} = γ||( \begin{matrix}x \\ φ \end{matrix})||_{c/γ}.

Remarks 3.2. (a) The only difference between condition (3.1) in Theorem 3.1 and [2; Thm. 3.2, condition (M)] is that in the latter c = γ < 1 is assumed, whereas we allow arbitrary c > 0. This refinement is important for the application to more general perturbations than studied in [2], and it avoids the separate study of the case p = 1 given in [8; Sec. 2]; see also [10; Sec. 1.2].

(b) Condition (3.1) is in particular satisfied if there exist t, c > 0 such that

∫_{0}^{t} ||Lψ_{s}|| ds ⩽ c||ψ||_{p}  (ψ ∈ W_{p}^{1}(−h, t; X))  (3.2)

(recall that ψ_{s}(θ) = ψ(θ+s)). Indeed, let ( \begin{matrix}x \\ φ \end{matrix}) ∈ D(A_{0}) and define ψ ∈ W_{p}^{1}(−h, t; X) by ψ := φ on (−h, 0), ψ(θ) := T (θ)x for 0 ⩽ θ ⩽ t. Then

||ψ||_{p}^{p} = ||φ||_{p}^{p} + ∫_{0}^{t} ||T (θ)x||^{p} dθ ⩽ ||φ||_{p}^{p} + t(c_{t}||x||)^{p},

where c_{t} := sup_{0⩽θ⩽t} ||T (θ)||. Hence (3.2) implies

∫_{0}^{t} ||L(T_{s}x + S(s)φ)|| ds = ∫_{0}^{t} ||Lψ_{s}|| ds ⩽ c(t^{1/p}c_{t}||x|| + ||φ||_{p}).

It remains to choose t > 0 such that γ := ct^{1/p}c_{t} < 1.

4 Perturbation classes

In this section we present a condition on L sufficient for (3.2). Here, as in Section 1, we assume more generally that X, Y are Banach spaces and that L ∈ L(W_{p}^{1}(−h, 0; X), Y). For a Borel measure μ on ℝ and t ⩾ 0 we define v_{μ,t}: ℝ → [0, ∞) by

v_{μ,t}(s) := μ((s − t, s]) (s ∈ ℝ).

Theorem 4.1. Assume that there exist r ∈ [1, p] and a Borel measure μ_{L} on [−h, 0] ((−∞, 0] in case h = ∞) such that v_{μ_{L},1} ∈ L_{\frac{p}{p−r}} (−h, 1) and

||Lφ|| ⩽ ||φ||_{L_{r}(μ_{L};X)}  (φ ∈ W_{p}^{1}(−h, 0; X)).  (4.1)

Then

∫_{0}^{t} ||Lψ_{s}|| ds ⩽ t^{1/p′}||v_{μ_{L},1}||1/r_{\frac{p}{p−r}} ||ψ||_{p}

for all 0 < t ⩽ 1, ψ ∈ W_{p}^{1}(−h, t; X).

This theorem is an immediate consequence of Proposition 4.3 below.

Remarks 4.2. (a) Assume that X = Y. If (4.1) holds then L satisfies condition (3.2) and hence condition (3.1). Thus, (4.1)is a condition for A = A_{0} +B to be a generator. Observe that this condition does not depend on the generator A.

(b) If μ_{L} is finite (in particular if h < ∞) then the condition v_{μ_{L},1} ∈ L_{\frac{p}{p−r}} (−h, 1) in Theorem 4.1 is automatically satisfied. If μ_{L} is not finite then this condition is responsible for the inclusion W_{p}^{1}(−h, 0; X) ⊆ L_{r}(μ_{L}; X).

Indeed, for φ ∈ W_{p}^{1}(−h, 0; X) we compute

||φ||_{L_{r}(μ_{L};X)}^{r} = ∫_{−∞}^{0} ∫_{θ}^{θ+1} dt ||φ(θ)||^{r} dμ_{L}(θ) = ∫_{−∞}^{1} ∫_{t−1}^{t∧0} ||φ(θ)||^{r} dμ_{L}(θ) dt.

Now ∫_{t−1}^{t∧0} ||φ(θ)||^{r} dμ_{L}(θ) ⩽ v_{μ_{L},1}(t)||φ|_{(t−1,t∧1)}||_{∞}^{r}, so by Hölder’s inequality we conclude that

||φ||_{L_{r}(μ_{L};X)}^{r} ⩽ ||v_{μ_{L},1}||_{\frac{p}{p−r}} ||t ↦ ||φ|_{(t−1,t∧0)}||_{∞}^{r}||_{\frac{p}{r}} = ||v_{μ_{L},1}||_{\frac{p}{p−r}} ||t ↦ ||φ|_{(t−1,t∧0)}||_{∞}||_{p}^{r}.

Using the Sobolev embedding W_{p}^{1}(0, 1; X) ⊆ L_{∞}(0, 1; X), we deduce that the latter can be estimated by c||φ||_{p,1}^{r}, with c ⩾ 0 not depending on φ, so the asserted inclusion follows.

(c) Assume that (4.1) holds. Then the operator L extends to a bounded operator L̂: L_{r}(μ_{L}; X) → Y.

(d) Assume additionally that μ_{L}({0}) = 0. Then it is obvious that L is strongly massless at 0, in the sense of Section 1.

Conversely, assume that L is massless at 0. We show that then μ_{L} can be chosen such that μ_{L}({0}) = 0. Indeed, define μ̃_{L} := μ_{L} − μ_{L}({0})δ_{0}, where δ_{0} is the Dirac measure at 0. Let φ ∈ W_{p}^{1}(−h, 0; X), φ constant in a neighbourhood of 0. Let (χ_{n}) be a sequence as in the definition of ‘massless at 0’, χ_{n}(0) = 1 (n ∈ ℕ). Then

||L((1 − χ_{n})φ)|| ⩽ ||(1 − χ_{n})φ||_{L_{r}(μ_{L};X)} = ||(1 − χ_{n})φ||_{L_{r}(μ̃_{L};X)}

for all n ∈ ℕ. From L((1 − χ_{n})φ) → Lφ in Y, (1 − χ_{n})φ → φ in L_{r}(μ̃_{L}; X) (n → ∞) we obtain that ||Lφ|| ⩽ ||φ||_{L_{r}(μ̃_{L};X)}. Thus, (4.1) holds with μ_{L} replaced by μ̃_{L} since the set of φ under consideration in dense in L_{r}(μ_{L}; X).

Proposition 4.3. Let r ∈ [1, p], and let μ be a Borel measure on ℝ with v_{μ,1} ∈ L_{\frac{p}{p−r}} (ℝ). Let 0 < t ⩽ 1, ψ ∈ L_{p}(ℝ). Then

∫_{0}^{t} ||ψ_{s}||_{L_{r}(μ)} ds ⩽ t^{1/r′}||v_{μ,t} _{spt ψ}||1/r_{\frac{p}{p−r}} ||ψ||_{p} ⩽ t^{1/p′}||v_{μ,1}||1/r_{\frac{p}{p−r}} ||ψ||_{p}.

Proof. We only show the first inequality, the second one being a consequence of Lemma 4.4 below. We have

∫_{0}^{t} ∫_{ℝ} |ψ(θ + s)|^{r} dμ(θ) ds = ∫_{ℝ} ∫_{ℝ} 1_{[0,t)}(s) |ψ(θ + s)|^{r} ds dμ(θ)
= ∫_{ℝ} ∫_{ℝ} 1_{[0,t)}(s − θ) dμ(θ) |ψ(s)|^{r} ds
= ∫_{ℝ} v_{μ,t}(s) |ψ(s)|^{r} ds ⩽ ||v_{μ,t} _{spt ψ}||_{\frac{p}{p−r}} |||ψ|^{r}||_{\frac{p}{r}} .

Since |||ψ|^{r}||_{\frac{p}{r}} = ||ψ||_{p}^{r}, we obtain by Hölder’s inequality that

∫_{0}^{t} ||ψ_{s}||_{L_{r}(μ)} ds ⩽ t^{1/r′}( ∫_{0}^{t} ||ψ_{s}||_{L_{r}(μ)}^{r} ds)^{1/r} ⩽ t^{1/r′}||v_{μ,t} _{spt ψ}||1/r_{\frac{p}{p−r}} ||ψ||_{p}.

In view of Remark 4.2(a) one could be content with the first inequality given in Proposition 4.3. Note, however, that the second inequality yields a better t-exponent if p > r.

Lemma 4.4. Let μ be a Borel measure on ℝ, t ∈ [0, 1], q ∈ [1, ∞]. Then ||v_{μ,t}||_{q} ⩽ t^{1/q}||v_{μ,1}||_{q}.

Proof. For q = ∞ the assertion is clear since v_{μ,t} ⩽ v_{μ,1}, so let q < ∞. Below we will show: If α > 0, t ∈ [0, 1) and

||v_{μ,sα}||_{q}^{q} ⩽ s||v_{μ,α}||_{q}^{q}  (4.2)

holds for s = t then (4.2) holds for s = \frac{1+t}{2}, too. Since (4.2) trivially holds for s = 0, by induction we obtain (4.2) for s = t_{n} := 1 − 2^{−n} (n ∈ ℕ) and α > 0. Again by induction, this shows the assertion of the lemma for t ∈ D := {t_{n}^{k}; k, n ∈ ℕ}. Since D is dense in [0, 1], and t ↦ ||v_{μ,t}||_{q} is increasing, the asserted inequality follows for all t ∈ [0, 1].

Now let α > 0, t ∈ [0, 1), and assume that (4.2) holds for s = t. Let t′ := \frac{1+t}{2}. Observe that, since t′ ∈ [\frac{1}{2}, 1), we have θ − α < θ − t′α ⩽ θ − α + t′α < θ and hence

1_{(θ−α,θ−α+t′α]} + 1_{(θ−t′α,θ]} = 1_{(θ−α,θ]} + 1_{(θ−t′α,θ−α+t′α]} (θ ∈ ℝ).

Since t = 2t′ − 1, this implies

v_{μ,t′α}(· − α + t′α) + v_{μ,t′α} = v_{μ,α} + v_{μ,tα}(· − α + t′α),  (4.3)

where both terms on the left hand side are bounded by v_{μ,α} since (θ − α, θ − α + t′α], (θ − t′α, θ] are subsets of (θ − α, θ].

We now need the following inequality for numbers a, b, c, d ⩾ 0: If a+b = c+d =: σ and a, b ⩽ c then a^{q} + b^{q} ⩽ c^{q} + d^{q}. This is a direct consequence of the fact that the function [\frac{σ}{2}, σ] ∋ t ↦ t^{q} + (σ − t)^{q} is monotone increasing. Together with (4.3) we obtain

v_{μ,t′α}(· − α + t′α)^{q} + v_{μ,t′α}^{q} ⩽ v_{μ,α}^{q} + v_{μ,tα}(· − α + t′α)^{q}.

Integrating both sides we infer, using (4.2) for s = t, that

2||v_{μ,t′α}||_{q}^{q} ⩽ ||v_{μ,α}||_{q}^{q} + ||v_{μ,tα}||_{q}^{q} ⩽ (1 + t)||v_{μ,α}||_{q}^{q},

i.e., (4.2) holds for s = t′.

Remarks 4.5. (a) It is easy to see that for q = 1 the inequality in Lemma 4.4 is in fact an equality. Since for q = ∞ the inequality is clear, one is tempted to use interpolation to prove Lemma 4.4. The authors did not succeed in realising this idea.

(b) Applying Lemma 4.4 to μ = fλ with 0 ⩽ f ∈ L_{q}(ℝ) and Lebesgue measure λ on ℝ, we obtain

||1_{[0,t]} ∗ f||_{q} ⩽ t^{1/q}||1_{[0,1]} ∗ f||_{q} (0 ⩽ t ⩽ 1).

We did not find a straightforward proof for this inequality.

We close this section by some observations concerning the modulus of operators L admitting a measure μ_{L} as in Theorem 4.1.

Let X, Y be Banach lattices with order continuous norm. Assume that L ∈ L(W_{p}^{1}(−h, 0; X), Y) possesses a modulus, and that for |L| there exists a measure μ_{|L|} such that the hypotheses of Theorem 4.1 are satisfies for |L| and μ_{|L|}. Then clearly the hypotheses of Theorem 4.1 are also satisfied for L, with μ_{L} := μ_{|L|}.

Proposition 4.6. Let X, Y be Banach lattices with order continuous norm. Let L ∈ L(W_{p}^{1}(−h, 0; X), Y), and assume that there exist r ∈ [1, p] and μ_{L} such that the hypotheses of Theorem 4.1 are satisfied. Assume further that the operator L̂ defined in Remark 4.2(c) possesses a modulus. Then L possesses a modulus, |L| is the restriction of |L̂| to W_{p}^{1}(−h, 0; X), and |L| satisfies the hypotheses of Theorem 4.1, with μ_{|L|} := |||L̂|||μ_{L}.

Proof. It is sufficient to show that the restriction of |L̂| to W_{p}^{1}(−h, 0; X) is the modulus of L. This, however, is a consequence of [13; Thm. 1, Rem. 2]; see also Remark 4.7(a) below.

Remark 4.7. (a) In order to apply [13; Rem. 2]) in the present context we have to convince ourselves of the following fact: If (Ω, A, μ) is a measure space, X a Banach lattice, and f, g ∈ L_{p}(μ; X), g ⩾ 0 then

τ_{g}f(t) = τ_{g(t)}f(t)  (t ∈ Ω).  (4.4)

(We refer to [13; Sec. 2] for the truncation τ). If f, g are simple functions then the right hand side of (4.4) defines a measurable function enjoying the properties of the truncation; therefore (4.4) holds. Approximating general f, g by simple functions and applying

|τ_{y_{1}}x_{1} − τ_{y_{2}}x_{2}| ⩽ |x_{1} − x_{2}| + |y_{1} − y_{2}|

(cf. [13; Eqn. (1)]) simultaneously to elements x_{1}, y_{1}, x_{2}, y_{2} in X and L_{p}(μ; X), one obtains (4.4) for f, g ∈ L_{p}(μ; X), g ⩾ 0.

(b) We point out that, in Proposition 4.7, it is much more restrictive to require the existence of a modulus for L̂ than for L.

5 Examples

Example 5.1. (a) Let X, Y be Banach spaces. Let η: [−h, 0] → L(X, Y) be a function of bounded variation. Then a continuous linear operator L_{η}: C([−h, 0]; X) → Y is defined by

L_{η}φ := ∫ dη(θ)φ(θ);

cf. [13; Sec. 3]. Due to the embedding W_{p}^{1}(−h, 0; X) ⊆ C([−h, 0]; X), the operator L_{η} can be restricted to W_{p}^{1}(−h, 0; X); the restriction will be denoted by L_{η,p}.

It it easy to see that L = L_{η,p} satisfies (4.1)for r = 1 if the variation |η| (a measure; cf. [13; Sec. 3, p. 198]) of η is used as μ_{L}. Therefore, L_{η,p} satisfies the hypotheses of Theorem 4.1; cf. Remark 4.2(b).

(b) Let μ_{L} be a finite Borel measure on [−h, 0] ((−∞, 0] in case h = ∞). Let L_{η,p} be as above and observe that η is uniquely determined by L_{η,p}. It thus follows from part (a) and Proposition 5.2 below that L = L_{η,p} satisfies (4.1) for r = 1 if and only if |η| ⩽ μ_{L}.

(c) From part (b) and Remark 4.2(d) we obtain that L_{η,p} is (strongly) massless at 0 if and only if |η|({0}) = 0. The latter holds if and only if η(0) = η(0−) := lim_{θ→0−} η(θ) (i.e., η does not give rise to mass at 0).

Proposition 5.2. Let X, Y be Banach spaces, L ∈ L(W_{p}^{1}(−h, 0; X), Y), and assume that (4.1)holds for r = 1 and a finite Borel measure μ_{L}. Then there exists η: [−h, 0] → L(X, Y) of bounded variation such that |η| ⩽ μ_{L} and L = L_{η,p}.

Proof. Let L̂ be as in Remark 4.2(c). For a bounded interval I ⊆ [−h, 0] we define

η̂(I)x := L̂(1_{I}x) (x ∈ X);

moreover, η(θ) := η̂((θ, 0]) (θ ∈ (−h, 0]), η(−h) := η̂([−h, 0]) if h ∈ (0, ∞). We show that the variation of η on I is bounded by μ_{L}(I); then L = L_{η,p} follows from the definition of L_{η}.

Let (I_{1}, …, I_{n}) be a partition of I into subintervals. Let ε > 0. Then there exist x_{j} ∈ X, ||x_{j}|| = 1 (j = 1, …, n) such that

∑_{j=1}^{n} ||η̂(I_{j})|| ⩽ (1 + ε) ∑_{j=1}^{n} ||η̂(I_{j})x_{j}||.

Since (4.1) implies ||η̂(I_{j})x_{j}|| = ||L̂(1_{I_{j}}x_{j})|| ⩽ ||1_{I_{j}}x_{j}||_{L_{1}(μ_{L};X)} = μ_{L}(I_{j}) for j = 1, …, n, we conclude that

∑_{j=1}^{n} ||η̂(I_{j})|| ⩽ (1 + ε)μ_{L}( ⋃_{j=1}^{n}I_{j}) = (1 + ε)μ_{L}(I)

and hence |η|(I) ⩽ μ_{L}(I).

Remark 5.3. In Example 5.1, let X, Y be Banach lattices, Y order complete. Assume that the regular variation η̃ of η exists, and that η̃ is of bounded variation; cf. [13; Sec. 3].

(a) It was shown in [13; Prop. 9] that then the modulus of L_{η} exists, and that |L_{η}| = Lη̃.

(b) Assuming additionally that X, Y have order continuous norm we show that η(0) = η(0−) implies η̃(0) = η̃(0−). This reproduces the result of [13; Lemma 10].

Indeed, part (a) and [13; Thm. 1 and Rem. 2] show that the modulus of L_{η,p} exists and is given by L_{η̃,p} (the restriction of Lη̃ to W_{p}^{1}(−h, 0; X)). Remark 5.1(c) shows that L_{η,p} is massless at 0. Therefore L_{η̃,p} ( = |L_{η,p}|) is massless at 0, by Proposition 1.1. Applying again Remark 5.1(c) we obtain η̃(0) = η̃(0−). Example 5.4. Let X be a Banach space, (Y_{n}) a sequence of Banach spaces, q ∈ (1, ∞),

Y := ℓ_{q}((Y_{n})_{n∈ℕ}) := {(y_{n}) ∈ ∏_{n=1}^{∞} Y_{n}; ||(y_{n})||^{q} := ∑_{n=1}^{∞} ||y_{n}||^{q} < ∞}.

(The above is an abstraction of the case that Y = L_{q}(Ω, ν), where (Ω, A, ν) is a measure space, and Y_{n} = L_{q}(Ω_{n}, ν), with a sequence (Ω_{n}) of pairwise disjoint measurable subsets of Ω such that ⋃_{n∈ℕ} Ω_{n} = Ω.) For n ∈ ℕ let η_{n}: [−h, 0] → L(X, Y_{n}) be a function of bounded variation. Then μ_{n} := |η_{n}| is a finite measure on [−h, 0].

Let r ∈ (1, q] and assume that ∑_{n=1}^{∞} ||μ_{n}||^{r} < ∞. Then ∑_{n=1}^{∞} ||L_{η_{n}}||^{q} < ∞, so we can define L: C([−h, 0]; X) → Y by

Lf := (L_{η_{n}}f)_{n∈ℕ}.

Assume without loss of generality that η_{n}(0) = 0 for all n ∈ ℕ. Then ∑_{n=1}^{∞} ||η_{n}(θ)||^{q} ⩽ ∑_{n=1}^{∞} ||μ_{n}||^{q} < ∞ for all θ ∈ [−h, 0], and hence we can define η: [−h, 0] → L(X, Y) by η(θ)f := (η_{n}(θ)f)_{n∈ℕ}. Then formally L = L_{η}, but η is not of bounded variation if the μ_{n} have pairwise disjoint support and ∑_{n=1}^{∞} ||μ_{n}|| = ∞. Nevertheless, we will show that L satisfies the hypotheses of Theorem 4.1 with the finite measure μ_{L} := ∑_{n=1}^{∞} ||μ_{n}||^{r−1}μ_{n}.

Let φ ∈ C([−h, 0]; X). Recall from Example 5.1(a) that ||L_{η_{n}}φ|| ⩽ ||φ||_{L_{1}(μ_{n};X)} for all n ∈ ℕ. Since r ⩽ q, we infer that

||Lφ||^{r} = (∑_{n=1}^{∞} ||L_{η_{n}}φ||^{q})^{\frac{r}{q}} ⩽ (∑_{n=1}^{∞} ||φ||_{L_{1}(μ_{n};X)}^{q})^{\frac{r}{q}} ⩽ ∑_{n=1}^{∞} ||φ||_{L_{1}(μ_{n};X)}^{r}.

By Hölder’s inequality we have ||φ||_{L_{1}(μ_{n};X)} ⩽ ||μ_{n}||^{\frac{1}{r′}} ||φ||_{L_{r}(μ_{n};X)} for all n ∈ ℕ. We conclude that

||Lφ||^{r} ⩽ ∑_{n=1}^{∞} ||μ_{n}||^{r−1} ∫ ||φ(θ)||^{r} dμ_{n}(θ) = ∫ ||φ(θ)||^{r} dμ_{L}(θ),

by the definition of μ_{L}. This proves ||Lφ|| ⩽ ||φ||_{L_{r}(μ_{L};X)}.

We note that the operator L defined above is massless at 0 if and only if the operators L_{η_{n}} are massless at 0.

References

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