Technische Universität Dresden

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1 Als Manuskript gedruckt 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-7-2

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3 Modulus semigroups and perturbation classes for linear delay equations in L p Hendrik Vogt and Jürgen Voigt Fachrichtung Mathematik, Technische Universität Dresden, D-162 Dresden, Germany Dedicated to the memory of H. H. Schaefer Abstract In this paper we study C -semigroups on X L p h,;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 2: 47D6, 34K6, 47B6 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 { u t) = Aut) + Lu t t ), DE) u) = x, u = f, with initial values x X, f L p h, ; X), where X is a Banach space, 1 p <, and < h. For a function u: h, ) X, we recall the notation u t θ) := ut + θ) h < θ < ), for t.) The foundations for treating this problem in the context of C -semigroups on X L p h, ; 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 1

4 2 Hendrik Vogt, Jürgen Voigt η: [ h, ] LX) of bounded variation cf. Example 5.1). Then the problem DE) could be treated in X L p h, ; 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: Wp 1 h, ; X) X to be continuous as an operator from L rµ L ; X) to X, for some r [1, p] and a suitable measure µ L on [ h, ]. 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], [1]. In order to put the results of the paper into the proper context, we now describe the C -semigroup setting of DE). Let X be a Banach space, A the generator of a C - semigroup T on X. Let T be the C -semigroup on X p := X L p h, ; X) given by ) Tt) T t) = t ), St) where T t LX, L p h, ; X)) denotes the operator given by { for h < θ t, T t xθ) := Tt + θ)x for t < θ <, T t and S is the C -semigroup of left translation on L p h, ; X), i.e., St)f = f t, where we assume that f L p h, ; X) is extended by to a function on h, ). It is known see [2; Prop. 3.1]) that the generator of T is given by A A = d dθ ), DA ) = { x ϕ ) DA) W 1 p h, ; X); x = ϕ) }. Let now L LWp 1 h, ; X), X). Then B := L ) LDA ), X p ), and we define ) A L A := A + B = d, DA) = DA ). dθ Assuming that A is the generator of a C -semigroup T one knows that, for ϕ) x DA), the first component of the function t T t) ϕ x ) 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 cf. Section 1), and that à :=, with domain A # L d dθ DÃ) = { x ϕ ) DA # ) W 1 p h, ; X); x = ϕ)}, is a generator. Then we show that à generates the modulus semigroup of T Theorem 2.7). The paper is organized as follows. In Section 1 we investigate operators from Wp 1 h, ; X) to Y, where X, Y are Banach lattices. The main objective is establishing relations between masslessness at of L and L. In Section 2 we determine the modulus semigroup of T ; see above.

5 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, ] LX) of bounded variation. Throughout this paper, let 1 p <, < h. 1 Operators defined on Wp 1 h, ; X) that are massless at In this section we investigate how to describe that operators L: Wp 1 h, ; X) Y attribute no mass to the point, 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, 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 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 >. 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 LQ x y ε, L LQ x z ε. 1.1) The lattice operations in Y are continuous, so we have z n := sup j y n := sup j LQ n x j sup L Q x j = y, j LI Q n )x j sup L L Q )x j = z. j Let now n N such that y n y ε, z n z ε for all n n. 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.

6 4 Hendrik Vogt, Jürgen Voigt For n n 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 [, 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: Wp 1 h, ; X) Y be a bounded linear operator. We say that L is massless at if for all x X and for all sequences χ n ) in W h, 1 ), χ n 1, χ n t) = t 1/n) for all n N, one has Lχ n x) n ). We say that L is strongly massless at if for all ϕ Wp 1 h, ; X) and for all sequences χ n ) as above, one has note that χ n ϕ Wp 1 h, ; X)). Lχ n ϕ) n ) 1.2) Remark 1.2. Obviously, L is massless at if and only if for all x X and for all ε > there exists δ > such that Lχx) ε for all χ W 1 h, ), χ 1, χt) = t δ). Remarks 1.3. Let X, Y be Banach lattices with order continuous norm. a) We recall that Wp 1 h, ; X) is a vector lattice, and that ϕ p,1 ϕ p,1 for all ϕ Wp 1 h, ; X); cf. [13; Thm. 1 and Rem. 5]. b) Assume that L: Wp 1 h, ; 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 in Wp 1 h, ; X) implies Lϕ n 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 1 Lϕ for all n 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: Wp 1 h, ; X) Y is linear and possesses a modulus, and that L is strongly massless at. Proposition 1.1 implies that then L is strongly massless at 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 =, and therefore L Q =. The following lemma shows that it suffices to require property 1.2) for a special sequence in the definition of strongly massless at if L possesses a modulus.

7 Modulus semigroups and perturbation classes for linear delay equations in L p 5 Lemma 1.4. Let X, Y be Banach lattices with order continuous norm, L: Wp 1 h, ; X) Y a linear operator. Assume that L possesses a modulus and that there exists a sequence χ n ) in W h, 1 ), χ n 1, χ n ) = 1 for all n N, such that Lχ n ϕ) n ) 1.3) for all ϕ Wp 1 h, ; X). Then L is strongly massless at. Proof. Since Proposition 1.1 implies L χ n ϕ) n ), it is sufficient to treat the case L. Let χ n ) be a sequence as in the definition of strongly massless at, ϕ W 1 p h, ; X), ϕ. Let n N. Then there exists m N such that for all m m one has χ m 2χ n, and therefore L χ m ϕ) 2Lχ n ϕ). By 1.3) this implies that lim m L χ m ϕ) =. 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, the second being strongly massless at. Proposition 1.5. Let X, Y be Banach lattices with order continuous norm. Assume that L: Wp 1 h, ; X) Y is linear and possesses a modulus. a) Let χ n ) be a sequence in W 1 h, ), χ n 1, χ n t) = t 1/n), χ n ) = 1 for all n N. Then L ϕ := lim n Lχ n ϕ) exists for all ϕ Wp 1 h, ; X), and the limit does not depend on the sequence χ n ). The linear operator L thus defined satisfies L ϕ = if ϕ Wp 1 h, ; X), ϕ) = ; the operator L L is strongly massless at. b) The operator L has the modulus L, and L L has the modulus L L. Proof. a) We first suppose that χ n ) is a monotone decreasing sequence satisfying the assumptions. Then L ϕ := lim n Lχ n ϕ) exists for all ϕ Wp 1 h, ; X), and L as well as L L possess a modulus, by Proposition 1.1. Obviously, L ϕ = if ϕ = in a neighbourhood of. Since the set of those ϕ is dense in {ϕ Wp 1 h, ; X); ϕ) = } and L is continuous cf. Remark 1.3b)), we obtain that L ϕ = if ϕ) =. From this we conclude that L L )χ n ϕ) = Lχ n ϕ) L χ n ϕ) = Lχ n ϕ) L ϕ n ). Thus, by Lemma 1.4, L L is strongly massless at. Let now χ n ) be any sequence satisfying the assumptions. Then with L obtained as above we have Lχ n ϕ) = L L )χ n ϕ) + L ϕ L ϕ n ) by the definition of strongly massless at. This completes the proof of a). b) follows from Proposition 1.1.

8 6 Hendrik Vogt, Jürgen Voigt The following result is another version of Lemma 1.4. It contains as a consequence that, for operators possessing a modulus, massless at implies strongly massless at. Proposition 1.6. Let X, Y be Banach lattices with order continuous norm, L: W 1 p h, ; X) Y a linear operator. Assume that L possesses a modulus and that there exists a sequence χ n ) in W 1 h, ), χ n 1, χ n t) = t 1/n), χ n ) = 1 for all n N such that Lχ n x) n ) for all x X. Then L is strongly massless at. Proof. We use the decomposition obtained in Proposition 1.5. Let χ Wp 1 h, ), χ = 1 in a neighbourhood of. Then L ϕ = L χϕ)) = lim n Lχ n χϕ)) = for all ϕ W 1 p h, ; X), i.e., L =. Therefore L = L L is strongly massless at. 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 -semigroup T on X, and that L LWp 1 h, ; X), X). Let A, T and B be as defined in the introduction. For later use we note that ) ) ) BT s) ϕ x L Ts)x LTs x + Ss)ϕ) ) = = 2.1) T s x + Ss)ϕ for x ϕ) DA). Lemma 2.1. The spectral radius of Bλ A ) 1 tends to as λ. Proof. For c > we endow X p = X L p h, ; X) with the norm x f ) c := x + c f p and also denote the corresponding norm on LX p ) by c. Let x f ) X p, λ larger than the type of T. Denoting by P 2 the projection of X p onto its second component, we obtain that Bλ A ) 1 x f ) c = LP 2 λ A ) 1 x f ) L P 2 λ A ) 1 x f ) p,1 and P 2 λ A ) 1 x f ) = e λt P 2 T t) x f ) dt = e λt T t x + St)f ) dt. A straightforward computation shows that e λt T t xdt = ε λ λ A) 1 x, where ε λ θ) := e λθ for θ h, ) cf. [6; Prop. VI.6.7]). This implies P 2 λ A ) 1 x f ) = ε λ λ A) 1 x + λ d dθ ) 1 f,

9 Modulus semigroups and perturbation classes for linear delay equations in L p 7 where d denotes the generator of S. There exist λ dθ >, M 1 such that λ + 1) λ A) 1 M and λ d dθ ) 1 : L p h, ; X) Wp 1 h, ; X) M for all λ λ. In the following let λ λ. Then P 2 λ A ) 1 x f ) p,1 λ + 1) ε λ Lp h,) λ A) 1 x + M f p Mδ λ x + M f p, with δ λ := ε λ Lp h,) as λ. For c := 1 δ λ we conclude that Bλ A ) 1 x f ) c L M δ λ x + f p ) = δλ L M x f ) c, so Bλ A ) 1 c δ λ L M, and this implies that the spectral radius of Bλ A ) 1 is less or equal δ λ L M. Lemma 2.2. Assume that L is massless at and that A = A + B is the generator of a C -semigroup T on X p. Then for all x DA). 1 t T t) T t) ) x ) t ) Proof. Let t >, x DA). For n N define ϕ n Wp 1 h, ; X) by ϕ nθ) := 1 + nθ) + x. Then ϕ n in L p h, ; X) as n and hence T t) T t) ) x ) = lim T t) T t) ) ϕ x n ). 2.2) n For s [, t] let now χ n,s W 1 h, ), χ n,sθ) := 1 + nθ + s) )+ 1 and ψ s Wp 1 h, ; X), ψ s θ) := for θ s, ψ s θ) := Ts + θ)x x for θ > s. Then T s x + Ss)ϕ n = χ n,s x + ψ s. Defining M := sup s t T s) and using the Duhamel formula and 2.1), we obtain T t) T t) ) ϕ x n ) = M T t s)bt s) ϕ x n ) ds LT s x + Ss)ϕ n ) ds M Together with 2.2) this implies the assertion: We have ψ s p p,1 = ψ s p p + s Lχn,s x) + L ψ s p,1 ) ds. Ts + θ)ax p dθ s ), and Lχ n,s x) as n and s since L is massless at. 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, B LX), where X is a Banach lattice, B is dominated by B if Bx B x for all x X. If T, T are C -semigroups on X then T is dominated by T if Tt) is dominated by Tt) for all t.

10 8 Hendrik Vogt, Jürgen Voigt Proposition 2.3. Let X be a Banach lattice. Let T, T be C -semigroups on X, with generators A, à respectively, and assume that T is dominated by T. Let B: DA) X, B: D Ã) X be linear operators, B, Bλ A) 1 dominated by Bλ Ã) 1 λ λ ) for some λ R. Assume that à + B generates a C -semigroup T eb on X and that the spectral radius of Bλ Ã) 1 is less that 1 for all λ λ. Then T B e is positive, and A + B generates a C -semigroup T B on X that is dominated by T B e. Proof. Let λ λ. The assumptions imply that λ à B) 1 = λ Bλ Ã) 1 Ã) 1)k cf. [1; Thm. 3.1]). Therefore, the C -semigroup T B e is positive. Since Bλ A) 1 is dominated by Bλ Ã) 1, we obtain in the same way that k= λ A B) 1 = λ A) 1 Bλ A) 1 )k. We conclude that λ A B) 1 is dominated by λ à B) 1. Hence λ A B) n λ à B) n for all n N, and the assertion follows from the Hille-Yosida generation theorem. The following domination lemma is a generalisation of [1; 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 -semigroups on X, S positive, and let A be the generator of T. Let k N, R: [, 1] LD A k, X) where D A k is the domain of A k endowed with the graph norm of A k ), and assume that for all x DA k ). Further assume that k= 1Rt)x t ) t Tt)x St) x + Rt)x for all x DA k ), t 1. Then T is dominated by S. Proof. Let x DA k ), t >. As in [1; proof of Lemma 1.1, eqn. 1.2)] we obtain Tt)x St) x + n m=1 S n mt) n R t n )Tm 1t)x 2.3) n

11 Modulus semigroups and perturbation classes for linear delay equations in L p 9 for all n N, n t. With c t := sup s t Ss) we estimate n S n mt) n R t n )Tm 1t)x n n c t R t n )Tm 1t)x n m=1 m=1 tc t max { n t R t n )Tm 1 n t)x ; 1 m n }. The hypothesis on R implies that nr t ) strongly in LD t n Ak, X) as n. We recall that strong convergence is uniform on compact sets and that T )x: [, ) D A k is continuous. Therefore the right hand side of the last inequality converges to as n. Letting n in 2.3) we obtain Tt)x St) x for all x DA k ). This implies the assertion since DA k ) is dense in X. The following proposition generalises [1; Prop. 2.1]. Proposition 2.5. Let X be a Banach lattice. Assume that L is massless at, and that A is the generator of a C -semigroup T on X p. If T is dominated by a C -semigroup S on X p then T is dominated by S as well. Proof. For t > we define R 1 t) := T t) T t), Rt) := R 1 t) 1 ). Let x f ) DA ). Estimating as in [1; proof of Prop. 2.1] we obtain that T t) x f ) St) x f ) + Rt) x f ). Lemma 2.2 implies that 1 t Rt) x f ) as t. 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. Indeed, let χ Wp 1 h, ), χ) = 1. Define E LX, Wp 1 h, ; X)) by Ex := χx. Let L LWp 1 h, ; X), X) be as in Proposition 1.5. Since ϕ) = x for x ϕ) DA), we obtain A = ) A L d dθ = ) A + L E L L d. dθ Thus, we have written A with a generater A + L E and an off-diagonal term L L that is strongly) massless at. 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 -semigroup dominating T ; see [4] for results on modulus semigroups.) It was shown in [1; Prop. 3.2b)] that then the C -semigroup T possesses ) a modulus semigroup, T #, whose generator is given by A # = DA # ) W 1 p h, ; X); x = ϕ)}. A # d dθ, with domain DA # ) = { x ϕ )

12 1 Hendrik Vogt, Jürgen Voigt Assume additionally that L is massless at, that L possesses a modulus and that à := A # + ) ) L A = # L, with domain DÃ) = DA# ), is the generator of a d dθ C -semigroup T. Taking into account Lemma 2.1, we then obtain by Proposition 2.3 that A is the generator of a C -semigroup T on X p, and that T is dominated by T. 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 # t) T # t) T t) for all t ; cf. [1; Prop. 3.2a)]. Theorem 2.7. Let X, T, and T be as introduced above. Then T # = T. The proof of this result is the same as for [1; Theorem 3.1]. Therefore we are not going to reproduce it; cf. [1; pp ]. We only mention that in [1; bottom of p. 397], for ϕ W 1 p h, ; X), the existence of a sequence ϕ k) in W 1 p h, ; X), ϕ k ) =, 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. Note that the hypotheses and Proposition 1.6 imply that L is strongly massless at, and recall Remark 1.3c).) 3 Miyadera perturbations of delay semigroups We assume that X is a Banach space, T a C -semigroup on X, L LWp 1 h, ; 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 [1; 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, ; X). Theorem 3.1. Assume that there exist t, c > and γ < 1 such that LT s x + Ss)ϕ) ds γ x + c ϕ p x ϕ ) DA ) ). 3.1) Then A = A + B is the generator of a C -semigroup T on X p. More precisely, if X p is endowed with the norm x f ) c/γ := x + c γ f p then B is a small Miyadera perturbation of A, i.e., BT s) x ϕ) c/γ ds γ x ϕ) c/γ x ϕ ) DA ) ). For Miyadera perturbations and the Miyadera perturbation theorem we refer to [12], [6; III.3.c].) Proof. By 2.1) we obtain that BT s) x ϕ) c/γ ds = LT s x + Ss)ϕ) ds γ x + c ϕ p = γ x ϕ ) c/γ.

13 Modulus semigroups and perturbation classes for linear delay equations in L p 11 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 >. 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 [1; Sec. 1.2]. b) Condition 3.1) is in particular satisfied if there exist t, c > such that Lψ s ds c ψ p ψ W 1 p h, t; X) ) 3.2) recall that ψ s θ) = ψθ+s)). Indeed, let ϕ x ) DA ) and define ψ Wp 1 h, t; X) by ψ := ϕ on h, ), ψθ) := Tθ)x for θ t. Then ψ p p = ϕ p p + where c t := sup θ t Tθ). Hence 3.2) implies LT s x + Ss)ϕ) ds = Tθ)x p dθ ϕ p p + tc t x ) p, It remains to choose t > such that γ := ct 1/p c t < 1. Lψ s ds ct 1/p c t x + ϕ p ). 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 LWp 1 h, ; X), Y ). For a Borel measure µ on R and t we define v µ,t : R [, ) by v µ,t s) := µs t, s]) s R). Theorem 4.1. Assume that there exist r [1, p] and a Borel measure µ L on [ h, ], ] in case h = ) such that v µl,1 L p h, 1) and p r Lϕ ϕ Lrµ L ;X) ϕ W 1 p h, ; X) ). 4.1) Then for all < t 1, ψ Wp 1 h, t; X). Lψ s ds t 1/p v µl,1 1/r p ψ p p r This theorem is an immediate consequence of Proposition 4.3 below.

14 12 Hendrik Vogt, Jürgen Voigt 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 + 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 µl,1 L p h, 1) p r in Theorem 4.1 is automatically satisfied. If µ L is not finite then this condition is responsible for the inclusion Wp 1 h, ; X) L r µ L ; X). Indeed, for ϕ Wp 1 h, ; X) we compute ϕ r L rµ L ;X) = θ+1 θ dt ϕθ) r dµ L θ) = 1 t 1 ϕθ) r dµ L θ) dt. Now t 1 ϕθ) r dµ L θ) v µl,1t) ϕ t 1,t 1) r, so by Hölder s inequality we conclude that ϕ r L rµ L ;X) v µl,1 t p ϕ p r t 1,t ) r p = v µl,1 t p ϕ r p r t 1,t ) r Using the Sobolev embedding Wp 1, 1; X) L, 1; X), we deduce that the latter can be estimated by c ϕ r p,1, with c not depending on ϕ, so the asserted inclusion follows. c) Assume that 4.1) holds. Then the operator L extends to a bounded operator L: L r µ L ; X) Y. d) Assume additionally that µ L {}) =. Then it is obvious that L is strongly massless at, in the sense of Section 1. Conversely, assume that L is massless at. We show that then µ L can be chosen such that µ L {}) =. Indeed, define µ L := µ L µ L {})δ, where δ is the Dirac measure at. Let ϕ Wp 1 h, ; X), ϕ constant in a neighbourhood of. Let χ n) be a sequence as in the definition of massless at, χ n ) = 1 n N). Then p. L1 χ n )ϕ) 1 χ n )ϕ Lrµ L ;X) = 1 χ n )ϕ Lreµ L ;X) for all n N. From L1 χ n )ϕ) Lϕ in Y, 1 χ n )ϕ ϕ in L r µ L ; X) n ) we obtain that Lϕ ϕ Lreµ L ;X). Thus, 4.1) holds with µ L replaced by µ 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 p p r R). Let < t 1, ψ L pr). Then ψ s Lrµ) ds t 1/r v µ,t spt ψ 1/r p p r ψ p t 1/p v µ,1 1/r p ψ p. p r Proof. We only show the first inequality, the second one being a consequence of

15 Modulus semigroups and perturbation classes for linear delay equations in L p 13 Lemma 4.4 below. We have ψθ + s) r dµθ) ds = R = = R R R R R 1 [,t) s) ψθ + s) r ds dµθ) 1 [,t) s θ) dµθ) ψs) r ds Since ψ r p r = ψ r p, we obtain by Hölder s inequality that ψ s Lrµ) ds t 1/r v µ,t s) ψs) r ds v µ,t spt ψ p p r ψ r p r. ) 1/r ψ s r L rµ) ds t 1/r v µ,t spt ψ 1/r p ψ p. p r In view of Remark 4.2a) 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 [, 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 >, t [, 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 = 1+t, too. Since 4.2) trivially holds for s =, 2 by induction we obtain 4.2) for s = t n := 1 2 n n N) and α >. Again by induction, this shows the assertion of the lemma for t D := {t k n ; k, n N}. Since D is dense in [, 1], and t v µ,t q is increasing, the asserted inequality follows for all t [, 1]. Now let α >, t [, 1), and assume that 4.2) holds for s = t. Let t := 1+t. 2 Observe that, since t [ 1, 1), we have θ α < θ 2 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 : 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 [ σ 2, σ] t tq + σ t) q is monotone increasing. Together with 4.3) we obtain v µ,t α α + t α) q + v q µ,t α vq µ,α + v µ,tα α + t α) q.

16 14 Hendrik Vogt, Jürgen Voigt Integrating both sides we infer, using 4.2) for s = t, that i.e., 4.2) holds for s = t. 2 v µ,t α q q v µ,α q q + v µ,tα q q 1 + t) v µ,α q q, 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 f L q R) and Lebesgue measure λ on R, we obtain 1 [,t] f q t 1/q 1 [,1] f q 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 LWp 1 h, ; 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 LWp 1 h, ; 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 L defined in Remark 4.2c) possesses a modulus. Then L possesses a modulus, L is the restriction of L to Wp 1 h, ; X), and L satisfies the hypotheses of Theorem 4.1, with µ L := L µl. Proof. It is sufficient to show that the restriction of L to W 1 p h, ; X) is the modulus of L. This, however, is a consequence of [13; Thm. 1, Rem. 2]; see also Remark 4.7a) 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 then τ g ft) = τ gt) ft) 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 τy1x 1 τ y2x 2 x1 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. b) We point out that, in Proposition 4.7, it is much more restrictive to require the existence of a modulus for L than for L.

17 Modulus semigroups and perturbation classes for linear delay equations in L p 15 5 Examples Example 5.1. a) Let X, Y be Banach spaces. Let η: [ h, ] LX, Y ) be a function of bounded variation. Then a continuous linear operator L η : C[ h, ]; X) Y is defined by L η ϕ := dηθ)ϕθ); cf. [13; Sec. 3]. Due to the embedding Wp 1 h, ; X) C[ h, ]; X), the operator L η can be restricted to Wp 1 h, ; 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 4.2b). b) Let µ L be a finite Borel measure on [ h, ], ] 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 4.2d) we obtain that L η,p is strongly) massless at if and only if η {}) =. The latter holds if and only if η) = η ) := lim θ ηθ) i.e., η does not give rise to mass at ). Proposition 5.2. Let X, Y be Banach spaces, L LW 1 p h, ; X), Y ), and assume that 4.1) holds for r = 1 and a finite Borel measure µ L. Then there exists η: [ h, ] LX, Y ) of bounded variation such that η µ L and L = L η,p. Proof. Let L be as in Remark 4.2c). For a bounded interval I [ h, ] we define ηi)x := L1 I x) x X); moreover, ηθ) := ηθ, ]) θ h, ]), η h) := η[ h, ]) if h, ). 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 >. Then there exist x j X, x j = 1 j = 1,...,n) such that n ηi j ) 1 + ε) j=1 n ηi j )x j. Since 4.1) implies ηi j )x j = L1 Ij x j ) 1 Ij x j L1 µ L ;X) = µ L I j ) for j = 1,...,n, we conclude that j=1 n n ) ηi j ) 1 + ε)µ L I j = 1 + ε)µl I) j=1 and hence η I) µ L I). j=1

18 16 Hendrik Vogt, Jürgen Voigt Remark 5.3. In Example 5.1, let X, Y be Banach lattices, Y order complete. Assume that the regular variation η of η exists, and that η 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 = L eη. b) Assuming additionally that X, Y have order continuous norm we show that η) = η ) implies η) = η ). This reproduces the result of [13; Lemma 1]. Indeed, part a) and [13; Thm. 1 and Rem. 2] show that the modulus of L η,p exists and is given by L eη,p the restriction of L eη to W 1 p h, ; X)). Remark 5.1c) shows that L η,p is massless at. Therefore L eη,p = L η,p ) is massless at, by Proposition 1.1. Applying again Remark 5.1c) we obtain η) = η ). 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 ) Y n ; y n ) q := y n q < }. n=1 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, ] LX, Y n ) be a function of bounded variation. Then µ n := η n is a finite measure on [ h, ]. Let r 1, q] and assume that n=1 µ n r <. Then n=1 L ηn q <, so we can define L: C[ h, ]; X) Y by Lf := L ηn f) n N. Assume without loss of generality that η n ) = for all n N. Then n=1 η n θ) q n=1 µ n q < for all θ [ h, ], and hence we can define η: [ h, ] LX, Y ) by ηθ)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, ]; X). Recall from Example 5.1a) that L ηn ϕ ϕ L1 µ n;x) for all n N. Since r q, we infer that Lϕ r = n=1 L ηn ϕ q ) r q n=1 ϕ q L 1 µ n;x) ) r q n=1 ϕ r L 1 µ. n;x) By Hölder s inequality we have ϕ L1 µ n;x) µ n 1 r ϕ Lrµ n;x) for all n N. We conclude that Lϕ r µ n r 1 ϕθ) r dµ n θ) = ϕθ) r dµ L θ), n=1 by the definition of µ L. This proves Lϕ ϕ Lrµ L ;X). We note that the operator L defined above is massless at if and only if the operators L ηn are massless at. n=1

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