manuscripta math. © Springer-Verlag Berlin Heidelberg 2013

Bienvenido Barraza Martínez · Robert Denk · Jairo Hernández Monzón

Pseudodifferential operators with non-regularoperator-valued symbols

Received: 31 August 2012 / Revised: 8 October 2013

Abstract. In this paper, we consider pseudodifferential operators with operator-valued sym-bols and their mapping properties, without assumptions on the underlying Banach space E .We show that, under suitable parabolicity assumptions, the W k

p(Rn, E)-realization of the

operator generates an analytic semigroup. Our approach is based on oscillatory integrals andkernel estimates for them. An application to non-autonomous pseudodifferential Cauchyproblems gives the existence and uniqueness of a classical solution. As an example, weinclude a discussion of coagulation–fragmentation processes.

1. Introduction

In this paper, we consider pseudodifferential operators with operator-valued sym-bols, their mapping properties, generation of an analytic semigroup and the ap-plication to non-autonomous vector-valued evolution equations. Operator-valuedsymbols and vector-valued function spaces appear in a natural way in several ap-plications, e.g. in models of coagulation–fragmentation processes where an addi-tional parameter (the cluster size) appears in the model, see [5]. Another exampleis given by boundary value problems in cylindrical domains which can be treatedby a Fourier multiplier approach which leads to operator-valued symbols, too (see[21]). Therefore, during the last decade, the investigation of vector-valued func-tion spaces and differential equations with operator-valued coefficients has gainedincreasing interest.

Our work is motivated by two directions of research: On one hand, operator-valued Fourier multipliers in arbitrary Banach spaces and their mapping propertieshave been considered in [4]. Based on these Fourier multiplier results, the generationof an analytic semigroup for differential operators could be shown in [6]. On theother hand, under an additional geometric assumption on the Banach space (to be aUMD space), Weis could establish in [27] a vector-valued Mikhlin type theorem (see

B. Barraza Martínez · J. Hernández Monzón: Departamento de Matemáticas, Universidaddel Norte, Km 5 Via a Puerto Colombia Barranquilla, Colombia

B. Barraza Martínez: e-mail: bbarraza@uninorte.edu.co

J. Hernández Monzón: e-mail: jahernan@uninorte.edu.co

R. Denk (B): Fachbereich Mathematik, Universität Konstanz, 78457 Konstanz, Germany.e-mail: robert.denk@uni-konstanz.de

Mathematics Subject Classification (2000): 35S05 · 47D06 · 35R20

DOI: 10.1007/s00229-013-0649-3

B. Barraza Martínez et al.

also [22,16]). This was the basis for a large number of results on maximal regularityfor differential and pseudodifferential operators in UMD spaces. As referencesfor operator-valued differential boundary value problems in UMD spaces and theconnection to R-sectoriality and maximal regularity, we mention [13] and [20]. Forvector-valued pseudodifferential operators in UMD spaces, we refer to [14,23] andthe references therein.

The restriction to UMD spaces, however, excludes natural state spaces as L1, andtherefore the present paper deals with pseudodifferential operators with operator-valued symbols in arbitrary Banach spaces. We consider operator-valued symbolsa in the standard Hörmander class with limited smoothness both in the variablex and in the covariable ξ and with positive order m > 0. Similar operators werealso considered in [17,18] with additional assumptions on the symbol (e.g., orderm > 1, existence of a homogeneous principal part, infinite smoothness in ξ ).

One of the main results in the present paper states that, under suitable par-abolicity assumptions, the W k

p(Rn, E)-realization of the symbol a generates an

analytic semigroup in W kp(R

n, E), see Theorem 4.2 and Corollary 4.3 below. Anapplication to non-autonomous pseudodifferential Cauchy problems gives the ex-istence and uniqueness of a classical solution (Theorem 5.3). As an example of aconcrete application, we include a short discussion of coagulation–fragmentationprocesses in Subsection 5.2, where we investigate a model introduced by Amann[5]. Our approach is based on oscillatory integrals and careful kernel estimates forthem, the technical key result being Lemma 3.2. For the case of constant coeffi-cients, results on vector-valued pseudodifferential operators were obtained in ourpaper [9].

2. Vector-valued pseudo-differential operators

In the following, we set |x | := ( ∑nj=1 |x j |2)1/2, |x, μ| := (|x |2 + μ2)1/2, 〈x〉 :=

(1 + |x |2)1/2, and 〈x, μ〉 := (1 + |x |2 + μ2)1/2 for x = (x1, . . . , xn) ∈ Rn and

μ ∈ R. Throughout this paper, (E, ‖ ·‖) denotes an arbitrary Banach space, and forlocally convex spaces X,Y we write L(X,Y ) for the space of all continuous linearoperators from X to Y , and set L(X) := L(X, X). Let C∞

b (Rn, E)be the space of all

u : Rn → E such that ∂αu is bounded and continuous for all α ∈ N

n0 where we use

standard multi-index notation. The space C∞b (R

n, E) is endowed with the locallyconvex topology given by the seminorms ‖u‖k := max|α|≤k supx∈Rn ‖∂αu(x)‖E ,k ∈ N0. The Schwartz space of rapidly decreasing E-valued functions is denotedby S (Rn, E), and for k ∈ N0 and 1 ≤ p < ∞, we write W k

p(Rn, E) for the

E-valued Sobolev space endowed with the norm

‖u‖W kp(R

n ,E) :=( ∑

|α|≤k

‖∂αu‖pL p(Rn ,E)

)1/p.

Here ‖ · ‖L p(Rn ,E) stands for the norm in the Lebesgue-Bochner space L p(Rn, E).We set R

n+ := Rn × [0,∞), Dα := (−i)|α|∂α and d̄ (ξ, y) := (2π)−nd(ξ, y).

Non-regular operator-valued symbols

We start with the definition of the symbol class (which is a non-smooth parameter-dependent version of the standard Hörmander class Sm

1,0) and the related pseudo-differential operators. In the following, for n ∈ N and m ∈ R, we set

ρn :={

n + 1, if n is odd,n + 2, if n is even,

and ρn,m :={ [n + m] + 1, if [n + m] is odd ,

[n + m] + 2, if [n + m] is even .

Here, [s] stands for the largest integer smaller or equal to s.

Definition 2.1. Let m ∈ R, ν ∈ [0,∞], and r ∈ N0.(a) We define Sm,ν,r := Sm,ν,r (Rn

x × Rn+, L(E)) as the set of all functions

a : Rnx × R

n+ → L(E) such that

(i) x → a(x, ξ, μ) ∈ Cr (Rn, L(E)) for all (ξ, μ) ∈ Rn+,

(ii) (ξ, μ) → a(x, ξ, μ) ∈ C [ν](Rn+, L(E)) for all x ∈ Rn , and

(iii) |a|(ν,r)m < ∞ where

|a|(ν,r)m := supx∈Rn

sup(ξ,μ)∈R

n+max|α|+k≤ν max|β|≤r

〈ξ, μ〉−m+|α|+k∥∥∂βx ∂

αξ ∂

kμa(x, ξ, μ)

∥∥L(E).

(b) For a ∈ Sm,ν,r with ν ≥ ρn , the pseudo-differential operator op(a) :=a(x, D, μ) is defined by

[a(x, D, μ)u](x) := Os −∫∫

R2n

eiξ ·ya(x, ξ, μ)u(x − y) d̄ (ξ, y) (x ∈ Rn)

for u ∈ C∞b (R

nx , E).

For the convenience of the reader, we state the definition and some propertiesof the oscillatory integral in the appendix.

In the following, we will also study parameter-dependent double symbols a =a(x, y, ξ, μ) with a ∈ Sm,ν,r (R2n

x,y × Rn+, L(E)) with ν ≥ ρn and r > ρn,m . Here

we set

[a(x, y, D, μ)u](x) := Os −∫∫

R2n

eiξ ·ya(x, x − y, ξ, μ)u(x − y)d̄ (ξ, y) (x ∈ Rn)

(2.1)

for u ∈ C∞b (R

n, E). In the particular case where a(x, y, ξ, μ) does not depend onx (dual symbol), we will write a(y, D, μ).

Remark 2.2. (a) For ν ≥ ρn , Lemma A.4 shows that the oscillatory integral in b)exists since in this case [(ξ, y) → a(x, ξ, μ)u(x − y)] ∈ A(m,ν,∞)

0,0 (Rnξ × R

ny, E)

and we can take e.g. = ρn/2 and ′ as any natural number such that m + n < 2′(see Lemma A.4 and Remark A.5). Furthermore, due to Lemma A.6, we have thata(x, D, μ) ∈ L(C∞

b (Rn, E),Cr

b(Rn, E)) for every fixed μ ∈ [0,∞).

(b) For ν ≥ ρn and r ≥ ρn,m , the oscillatory integral in (2.1) exists because inthis case [(ξ, y) → a(x, x − y, ξ, μ)u(x − y)] ∈ A(m,ν,r)

0,0 (Rnξ × R

ny, E) and we

can take e.g. = ρn/2 and ′ = ρn,m/2 (see Lemma A.4 and Remark A.5 again).Note that for n + m < 0 the condition r ≥ ρn,m is satisfied for all r ∈ N.

B. Barraza Martínez et al.

The following result is taken from [18], Remark 2.2.6.

Lemma 2.3. Let ai ∈ Smi ,ν,r , i = 1, 2, with m1, m2 ∈ R, ν ≥ ρn and r ≥ρ2n,m1+m2 . Then a1(x, D, μ)a2(y, D, μ) = op

(a1(x, ξ, μ)a2(y, ξ, μ)

).

Definition 2.4. (a) Let m, ν ∈ (0,∞) and r ∈ N0, and let a ∈ Sm,ν,r . Then a iscalled parameter-elliptic if there are constants κ > 0 and ω ≥ 0 such that for all(x, ξ, μ) ∈ R

n × Rn × [0,∞) with |ξ, μ| ≥ ω we have

(i) a(x, ξ, μ) : E → E is bijective, and(ii) ‖a(x, ξ, μ)−1‖L(E) ≤ κ〈ξ, μ〉−m .

The set of all a ∈ Sm,ν,r satisfying (i)–(ii) will be denoted by Em,ν,rκ,ω := Em,ν,r

κ,ω (Rnx×

Rn+, L(E)).

(b) Let m ∈ (0,∞) and r, ν ∈ N0, and let a ∈ Sm,ν,r (Rnx × R

nξ , L(E)) be

a symbol independent of μ ∈ (0,∞). Then a is called parabolic with constantsκ > 0, ω ≥ 0 if (x, ξ, μ) → a(x, ξ)+ μmeiθ belongs to Em,ν,r

κ,ω for all |θ | ≤ π2 .

For a symbol a ∈ Em,ν,rκ,ω , we define a smooth version of the inverse symbol

in the following way: Let ψ ∈ C∞(Rn+1) with 0 ≤ ψ ≤ 1, ψ(ξ, μ) = 0 for|ξ, μ| ≤ 1

2 and ψ(ξ, μ) = 1 for |ξ, μ| ≥ 1 be a zero extinction function. Forω0 > ω we set

a#(x, ξ, μ) :={ψ(

ξ2ω0,μ

2ω0)a(x, ξ, μ)−1, if x ∈ R

n, |ξ, μ| ≥ ω0,

0, if x ∈ Rn, |ξ, μ| ≤ ω0.

Then a#(x, ξ, μ) = 0 if |ξ, μ| ≤ ω0 and a#(x, ξ, μ) = a(x, ξ, μ)−1 if |ξ, μ| ≥2ω0.

3. Key estimates for parameter-elliptic symbols

In this section, we will prove key estimates for the inverse of parameter-ellipticsymbols and mapping properties for the corresponding operators. We start with aresult which shows that the inverse of a parameter-elliptic symbol is again in thecalculus.

Lemma 3.1. Let m, ν ∈ (0,∞), r ∈ N0, and let A ⊂ Sm,ν,r be bounded, i.e. thereexists a constant K > 0 such that |a|(ν,r)m ≤ K (a ∈ A). Assume moreover thatA ⊂ Em,ν,r

κ,ω . Then there exists a constant C = C(K , κ, ν, r) > 0 such that

‖∂βx ∂αξ ∂kμa#(x, ξ, μ)‖L(E) ≤ C〈ξ, μ〉−m−|α|−k

for all x ∈ Rn, (ξ, μ) ∈ R

n+, |α|+k ≤ ν, |β| ≤ r , and a ∈ A, i.e., {a# : a ∈ A} ⊂S−m,ν,r is bounded.

Non-regular operator-valued symbols

Proof. Note that {〈ξ, μ〉m+|α|+k‖∂βx ∂αξ ∂kμa#(x, ξ, μ)‖L(E) : x ∈ R

n, |ξ, μ| ≤2ω0, |α| + k ≤ ν, |β| ≤ r} is bounded by continuity and compactness. There-fore, it suffices to consider the case |ξ, μ| ≥ 2ω0.

Let x ∈ Rn, |ξ, μ| ≥ 2ω0, α, β ∈ N

n0 \ {0}, k ∈ N0 with |β| ≤ r and

|α| + k ≤ ν. Then ∂βx ∂αξ ∂kμa(x, ξ, μ)−1 is a finite linear combination of terms of

the form

a(x, ξ, μ)−1[∂β(1)

x ∂α(1)

ξ ∂k(1)μ a(x, ξ, μ)

]a(x, ξ, μ)−1 . . .

. . .[∂β

(p)

x ∂α(p)

ξ ∂k(p)μ a(x, ξ, μ)

]a(x, ξ, μ)−1

(3.1)

with 1 ≤ p ≤ |α| + |β| + k and with α(i), β(i) ∈ Nn0, k(i) ∈ N (i = 1, . . . , p)

satisfying∑p

i=1 α(i) = α,

∑pi=1 β

(i) = β,∑p

i=1 k(i) = k. The norm of the operatorin (3.1) can be estimated by

‖a(x, ξ, μ)−1‖p+1L(E)

p∏

i=1

∥∥∂β(i)

x ∂α(i)

ξ ∂k(i)μ a(x, ξ, μ)

∥∥L(E)

≤ κ p+1〈ξ, μ〉−m(p+1)p∏

i=1

|a|(ν,r)m 〈ξ, μ〉m−|α(i)|−k(i)

= κ p+1(|a|(ν,r)m

)p〈ξ, μ〉−m−|α|−k .

Summing up over all terms of the form (3.1), we obtain

‖∂βx ∂αξ ∂kμa−1(x, ξ, μ)‖L(E) ≤ C〈ξ, μ〉−m−|α|−k

with a constant C = C(K , κ, ν, r). �From now on we set χε(ξ, y) := χ(εξ)χ(εy) for ε > 0 and ξ, y ∈ R

n , whereχ ∈ S (Rn)with χ(0) = 1. The following result is the key estimate for parameter-dependent symbols of negative order.

Lemma 3.2. Let b ∈ S−m,ν,r with m > 0, ν ≥ n + 1, r ∈ N0 and ω0 > 0. Forε ∈ (0, 1) define

Kε(x, y, μ) :=∫

Rn

eiξ ·yχε(ξ, y)b(x − y, ξ, μ)dξ((x, y, μ) ∈ R

2n × [ω0,∞)).

(a) There exists a constant C such that

‖Kε(x, y, μ)‖L(E)

≤ C |b|(ν,r)−m μ−m+n |μy|θ0 + |μy|θ1

|μy|n(1 + |μy|)((x, y, μ) ∈ R

2n × [ω0,∞))

where θ0 := 12 min{m, 1} and θ1 := 1

2 . Moreover,

B. Barraza Martínez et al.

‖Kε(x, ·, μ)‖L1(Rn ,L(E)) ≤ C |b|(ν,r)−m μ−m ((x, μ) ∈ R

n × [ω0,∞))

(3.2)

holds with a constant C independent of ε, x and μ.(b) There exists a strongly measurable function K : R

2n ×[ω0,∞)→L(E)withKε(x, y, μ)→K (x, y, μ) (ε ↘ 0) pointwise, and the estimate (3.2) holds with Kεbeing replaced by K .

Proof. (a) Substituting ξ → μξ , we have

Kε(x, y, μ) = μn∫

Rn

eiμξ ·yχε(μξ, y)b(x − y, μξ, μ)dξ.

We fix γ ∈ Nn0 with |γ | = n + i, i ∈ {0, 1}. By assumption, b(x − y, ·, μ) ∈

C |γ |(Rn, L(E)) and therefore∫

Rn

Dγξ

[χε(μξ, y)b(x − y, μξ, μ)

]dξ = 0.

With this and partial integration, we see

(μy)γ Kε(x, y, μ) = μn∫

Rn

(eiμξ ·y − 1)Dγξ

[χε(μξ, y)b(x − y, μξ, μ)

]dξ.

To estimate the integrand, we apply the Leibniz rule noting that for α + β = γ wehave

∣∣Dαξ χε(μξ, y)

∣∣ = ∣∣Dαξ χ(εμξ)

∣∣ |χ(εy)| ≤ cαμ|α|〈μξ 〉−|α|

by Lemma A.3. This and

‖Dβξ [b(x − y, μξ, μ)]‖L(E) = μ|β|‖(Dβ

ξ b)(x − y, μξ, μ)‖L(E)

≤ μ|β||b|(ν,r)−m 〈μξ,μ〉−m−|β| (3.3)

gives∥∥Dγ

ξ

[χε(μξ, y)b(x − y, μξ, μ)

]∥∥L(E)

≤∑

α+β=γcαβμ

|α|〈μξ 〉−|α||μ||β|〈μξ,μ〉−m−|β||b|(ν,r)−m

≤ cγ μ|γ |〈μξ 〉−m−|γ ||b|(ν,r)−m

where we used 〈μξ,μ〉−m−|β| ≤ 〈μξ 〉−m−|β|. Now we apply the elementary esti-mate |eiμξ y − 1| ≤ 2|μy|θ |ξ |θ valid for all θ ∈ (0, 1) and obtain

‖(μy)γ Kε(x, y, μ)‖L(E) ≤ Cμn+|γ ||μy|θ |b|(ν,r)−m

∫

Rn

|ξ |θ 〈μξ 〉−m−|γ |dξ

= Cμn−m |μy|θ |b|(ν,r)−m I (θ, |γ |)

Non-regular operator-valued symbols

with I (θ, |γ |) := ∫Rn |ξ |θ (μ−2 + |ξ |2)−m/2−|γ |/2dξ .

For |γ | = n + i with i ∈ {0, 1} and θ0 := 12 min{m, 1}, θ1 := 1

2 , we haveθi ∈ (0, 1) and

I (θi , |γ |) =∫

|ξ |≤1

|ξ |θi (μ−2 + |ξ |2)−m/2−|γ |/2dξ

+∫

|ξ |≥1

|ξ |θi (μ−2 + |ξ |2)−m/2−|γ |/2dξ

≤ ω−m−|γ |0

∫

|ξ |≤1

|ξ |θi dξ +∫

|ξ |≥1

|ξ |θi −m−|γ |dξ ≤ C < ∞

due to θi > 0 and θi − m − |γ | < −n. Therefore, for all γ with |γ | = n + i ,i ∈ {0, 1}, we have

‖(μy)γ Kε(x, y, μ)‖L(E) ≤ Cμn−m |μy|θi |b|(ν,r)−m

and consequently

|μy|n+i‖Kε(x, y, μ)‖L(E) ≤ n(n+i)/2∑

|γ |=n+i

‖(μy)γ Kε(x, y, μ)‖L(E)

≤ Cμn−m |μy|θi |b|(ν,r)−m .

Summing up these inequalities for i = 0 and i = 1, we obtain the first assertion ina). Since the function

y → μn |μy|θ0 + |μy|θ1

|μy|n(1 + |μy|)belongs to L1(Rn) and its L1-norm does not depend on μ, we obtain inequality(3.2).

(b) Let ε, ε′ ∈ (0, 1), ξ, y ∈ Rn and μ ∈ [ω0,∞). From the proof of (a) we see

that

(μy)γ (Kε(x, y, μ)− Kε′(x, y, μ)

= μn∫

Rn

(eiμξ ·y − 1)Dγξ

[χε(μξ, y)− χε′(μξ, y)

]b(x − y, μξ, μ)

]dξ. (3.4)

From Lemma A.3 we know that Dαξ

(χε(μξ, y)−χε′(μξ, y)

) → 0 for ε, ε′ ↘ 0 forall α ∈ N

n0 and all μ, ξ, y. Therefore the integrand in (3.4) converges pointwise to

zero for ε, ε′ ↘ 0. By (a), we have Kε(x, ·, μ) ∈ L1(Rn, L(E))with a dominatingfunction independent of ε, and by dominated convergence we see that for fixed(x, y, μ) ∈ R

2n ×[ω0,∞)we get ‖Kε(x, y, μ)− Kε′(x, y, μ)‖L(E) → 0 (ε, ε′ ↘0). Therefore there exists a strongly measurable function K : R

2n × [ω0,∞) →L(E) such that Kε → K (ε ↘ 0) pointwise. By dominated convergence again,inequality (3.2) holds for K (x, ·, μ) instead of Kε(x, ·, μ). �

B. Barraza Martínez et al.

Theorem 3.3. Let m > 0, 1 ≤ p < ∞, r, k, ν ∈ N0 with r ≥ ρn,−m + k + 1,ν ≥ ρn, and ω0 > 0. Let b ∈ S−m,ν,r . Then b(y, D, μ) ∈ L(W k

p(Rn, E)) and we

have

‖b(y, D, μ)‖L(W kp(R

n ,E)) ≤ Cμ−m |b|(ν,r)−m (μ ∈ [ω0,∞))

with a constant C independent of b and μ.

Proof. (i) Consider first the case k = 0. For u ∈ S (Rn, E) we have

(b(y, D, μ)u)(x) = Os −∫∫

R2n

eiξ ·yb(x − y, ξ, μ)u(x − y)d̄ (ξ, y)

= limε↘0

∫

Rn

∫

Rn

eiξ ·yχε(ξ, y)b(x − y, ξ, μ)u(x − y)d̄ ξ d̄ y

= limε↘0

∫

Rn

Kε(x, y, μ)u(x − y)d̄ y

with Kε(x, y, μ) := ∫Rn eiξ ·yχε(ξ, y)b(x−y, ξ, μ)d̄ ξ . By Lemma 3.2, Kε(x, ·, μ)

∈ L1(Rn, L(E))with ε-independent dominating function, and Kε converges point-wise to a strongly measurable function K for ε ↘ 0. By the dominated convergencetheorem, we have

(b(y, D, μ)u)(x) =∫

Rn

K (x, y, μ)u(x − y)d̄ y = [K (x, ·, μ) ∗ u](x)

where ∗ stands for the standard convolution. Because of the L1-estimate of K inLemma 3.2, we obtain for μ ∈ [ω0,∞)

‖b(y, D, μ)u‖L p(Rn ,E) ≤ C‖K (x, ·, μ)‖L1(Rn ,L(E))‖u‖L p(Rn ,E)

≤ Cμ−m |b|(ν,r)−m ‖u‖L p(Rn ,E).

(ii) Now let k ∈ N. By Lemma A.6, for β ∈ Nn0 with |β| ≤ k we write

(∂βx [b(y, D, μ)u])(x) = Os −

∫∫

R2n

eiξ ·y∂βx[b(x − y, ξ, μ)u(x − y)

]d̄ (ξ, y)

=∑

β ′≤βcββ ′ Os −

∫∫

R2n

eiξ ·y(∂β ′x b)(x − y, ξ, μ)(∂β−β ′

x u)(x − y)d̄ (ξ, y).

From ∂β ′x b ∈ Sm,ν,r−|β ′| and part (i) of the proof we obtain

‖∂βx [b(y, D, μ)u]‖L p(Rn ,E) ≤ Cμ−m |b|(ν,r)−m

∑

|γ |≤k

‖∂γ u‖L p(Rn ,E)

≤ Cμ−m |b|(ν,r)−m ‖u‖W kp(R

n ,E).

�

Non-regular operator-valued symbols

Remark 3.4. The last result shows that the mapping b → b(y, D, μ), S−m,ν,r →L(W k

p(Rn, E)) is continuous with norm not greater than a constant times μ−m for

μ ∈ [ω0,∞).

Corollary 3.5. In the situation of Lemma 3.1, we have a#(y, D, μ) ∈ L(W kp(R

n, E))and

supa∈A, μ≥0

‖a#(y, D, μ)‖L(W kp(R

n ,E)) < ∞

for all k ∈ N0 with r ≥ ρn,−m + k + 1.

Proof. This follows for μ ≥ ω0 immediately from Lemma 3.1 and Remark 3.4while forμ ≤ ω0 we have a#(y, D, μ) = 0 for all a ∈ A. �

4. Generation of an analytic semigroup

We will need the following slight generalization of Lemma 3.2.

Lemma 4.1. Let m > 0, m1,m2 ∈ R with m1 + m2 = −m, ν, r ∈ N0 withν ≥ n + 1, ω0 > 0. Let b1 ∈ Sm1,ν,r and b2 ∈ Sm2,ν,r . For ε ∈ (0, 1) define

Kε(x, y, μ) :=∫

Rn

eiξ ·yχε(ξ, y)b(x, y, ξ, μ)dξ

with b(x, y, ξ, μ) := ∫ 10 b1(x − τ y, ξ, μ)b2(x − y, ξ, μ)dτ . Then the assertions

of Lemma 3.2 hold with |b|(ν,r)−m being replaced by |b1|(ν,r)m1 |b2|(ν,r)m2 .

Proof. In the proof of Lemma 3.2, we replace inequality (3.3) by

‖Dβξ [b(x, y, μξ, μ)]‖L(E) ≤ μ|β||b1|(ν,r)m1

|b2|(ν,r)m2〈μξ,μ〉−m−|β|. (4.1)

This can be seen by

‖Dβξ [b(x, y, μξ, μ)]‖L(E)

≤ μ|β| ∑

β ′≤βcββ ′

∥∥∥

1∫

0

(Dβ ′ξ b1)(x − τ y, μξ, μ)(Dβ−β ′

ξ b2)(x − y, μξ, μ)dτ∥∥∥

L(E)

≤ μ|β| ∑

β ′≤βcββ ′ |b1|(ν,r)m1

〈μξ,μ〉m1−|β ′||b2|(ν,r)m2〈μξ,μ〉m2−(|β|−|β ′|)

≤ μ|β||b1|(ν,r)m1|b2|(ν,r)m2

〈μξ,μ〉−m−|β|.

With (4.1) instead of (3.3) we can follow the proof of Lemma 3.2 almostliterally. �

B. Barraza Martínez et al.

Throughout the remainder of this section, we fix k ∈ N0, m > 0, 1 ≤ p < ∞,r, ν ∈ N0 with r ≥ ρn,m + k + 1 and ν ≥ n + 2. We will consider the W k

p(Rn, E)-

realization Ak of a(x, D, μ), i.e. the restriction of a(x, D, μ) to Dmax(Ak) := {u ∈W k

p(Rn, E) : a(x, D, μ)u ∈ W k

p(Rn, E)}.

Theorem 4.2. Let a ∈ Sm,ν,r be parameter-elliptic, i.e., a ∈ Em,ν,rκ,ω for some

κ > 0 and ω ≥ 0. Then there exist constants M1 > 0 and μ1 > 0 such that theW k

p-realization Ak of a(x, D, μ) is invertible for μ ≥ μ1 and

‖A−1k ‖L(W k

p(Rn ,E)) ≤ M1

1 + μm(μ ≥ μ1).

Proof. Let ω0 > ω, μ ≥ 2ω0, and u ∈ S (Rn, E). From Lemma 2.3 we know that

a(x, D, μ)a#(y, D, μ)u = [a(x, ξ, μ)a#(y, ξ, μ)](D)u = (I + p(x, y, D, μ)

)u

with the double symbol

p(x, y, ξ, μ) := (a(x, ξ, μ)− a(y, ξ, μ)

)a(y, ξ, μ)−1.

(i) Let x ∈ Rn . By a Taylor expansion we obtain

(p(x, y, D, μ)u

)(x)

= Os −∫∫

R2n

eiξ ·y(a(x, ξ, μ)− a(x − y, ξ, μ)

)a(x − y, ξ, μ)−1u(x − y)d̄ (ξ, y)

= Os −∫∫

R2n

eiξ ·yn∑

j=1

y j

1∫

0

(∂x j a)(x − τ y, ξ, μ)a(x − y, ξ, μ)−1dτu(x − y)d̄ (ξ, y)

=n∑

j=1

Os −∫∫

R2n

eiξ ·y1∫

0

Dξ j

[(∂x j a)(x − τ y, ξ, μ)a(x − y, ξ, μ)−1]

dτu(x − y)d̄ (ξ, y)

=n∑

j=1

limε↘0

∫

Rn

[K ( j,1)ε (x, y, μ)+ K ( j,2)

ε (x, y, μ)]u(x − y)d̄ y

with

K ( j,)ε (x, y, μ) :=

∫

Rn

eiξ ·yχε(ξ, y)b( j,)(x, y, ξ, μ)d̄ ξ ( = 1, 2),

b( j,1)(x, y, ξ, μ) :=1∫

0

(Dξ j ∂x j a)(x − τ y, ξ, μ)a(x − y, ξ, μ)−1dτ,

b( j,2)(x, y, ξ, μ) :=1∫

0

(∂x j a)(x − τ y, ξ, μ)Dξ j [a(x − y, ξ, μ)−1]dτ.

Non-regular operator-valued symbols

We have Dξ j ∂x j a ∈ Sm−1,r−1,ν−1 and Dξ j a−1 ∈ S−m−1,r,ν−1. By Lemma 4.1

we obtain for j = 1, . . . , n and = 1, 2 the existence of a strongly measurablefunction K ( j,) with K ( j,)

ε → K ( j,) (ε ↘ 0) pointwise and

‖K ( j,)(x, ·, μ)‖L1(Rn ,L(E)) ≤ Cμ−m |a|(ν,r)m |a#|(ν,r)−m

for μ ≥ 2ω0. Therefore,

‖p(x, y, D, μ)u‖L p(Rn ,E) ≤n∑

j=1

2∑

=1

‖[K ( j,)(x, ·, μ) ∗ u]‖L p(Rn ,E)

≤ Cμ−m |a|(ν,r)m |a#|(ν,r)−m ‖u‖L p(Rn ,E).

(ii) Now let β ∈ Nn0 with |β| ≤ k. In the same way as in the proof of part (ii) of

Theorem 3.3, we write

(∂βx p(x, y, D, μ)u)(x)

=∑

β ′≤βcββ ′ Os −

∫∫

R2n

eiξ ·y(∂β ′x p)(x, y, ξ, μ)(∂β−β ′

x u)(x − y)d̄ (ξ, y).

Replacing in (i) p by ∂β′

x p and u by ∂β−β ′x u, we obtain

‖∂βx [p(x, y, D, μ)u]‖L p(Rn ,E) ≤ C∑

β ′≤β‖(∂β ′

x p)(x, y, D, μ)(∂β−β ′x u)‖L p(Rn ,E)

≤ C∑

β ′≤βμ−m |a|(ν,r)m |a#|(ν,r)−m

∑

|γ |≤k

‖∂γx u‖L p(Rn ,E)

and therefore ‖p(x, y, D, μ)u‖W kp(R

n ,E) ≤ Cμ−m |a|(ν,r)m |a#|(ν,r)−m ‖u‖W kp(R

n ,E) forμ ≥ 2ω0 with a constant C independent of a and μ. Taking μ ≥ μ1 with μ1sufficiently large, we obtain ‖p(x, y, D, μ)‖L(W k

p (Rn ,E)) ≤ 1

2 . By a Neumann series

argument, we see that I+p(x, y, D, μ) ∈ L(W kp(R

n, E)) is invertible and thereforea(x, D, μ)a#(y, D, μ)(I + p(x, y, D, μ))−1 = idW k

p(Rn ,E) for μ ≥ μ1. This

shows that Ak is surjective.(iii) In the same way, we see that a#(x, D, μ)a(y, D, μ) = I + p̃(x, y, D, μ)

withp̃(x, y, ξ, μ) := (a(x, ξ, μ)−1 − a(y, ξ, μ)−1)a(y, ξ, μ),

and that ‖ p̃(x, y, D, μ)‖L(W kp(R

n ,E)) ≤ 12 for sufficiently large μ. Therefore, (1 +

p̃(x, y, D, μ))−1a#(x, D, μ)a(y, D, μ) = idDmax(Ak ). This shows that Ak is injec-tive, and we obtain the invertibility of Ak . Moreover, we have for sufficiently largeμ

‖A−1k ‖L(W k

p(Rn ,E))≤‖a#(y, D, μ)‖L(W p

k (Rn ,E))‖(I+p(x, y, D, μ))−1‖L(W k

p(Rn ,E))

≤ 2M0

1 + μm

forμ ≥ μ1 withμ1 sufficiently large, where M0 does not depend onμ. �

B. Barraza Martínez et al.

In the following, we setΣϑ,R := {λ ∈ C : |λ| ≥ R, | arg λ| ≤ ϑ} forϑ ∈ [0, π)and R > 0.

Corollary 4.3. Let a ∈ Sm,ν,r (Rnx × R

nξ , L(E)) be parabolic with constants κ > 0

andω ≥ 0, and let Ak denote the W kp(R

n, E)-realization of a(x, D). Then there areconstants M2 > 0, ϑ ∈ (π2 , π) and R > 0 such that for the resolvent set ρ(−Ak)

of −Ak we have Σϑ,R ⊂ ρ(−Ak),

‖(λ+ Ak)−1‖L(W k

p(Rn ,E)) ≤ M2

1 + |λ| (λ ∈ Σϑ,R). (4.2)

Therefore, −Ak : W kp(R

n, E) ⊃ D(Ak) → W kp(R

n, E) generates an analytic

semigroup on W kp(R

n, E).

Proof. By Definition 2.4 b), the symbol a(x, ξ)+μmeiθ is parameter-elliptic for allθ with θ ∈ Σπ

2 ,R. By a standard continuity argument, we see that the set of angles θ

where parameter-ellipticity holds is open. Therefore, we may assume that the abovesymbol is parameter-elliptic for all θ ∈ Σϑ,R with ϑ > π

2 . Now the invertibilityof λ + Ak , λ = μmeiθ and the estimate (4.2) follows directly from Theorem 4.2.

�Remark 4.4. The proof of Theorem 4.2 shows that the constants μ1 and M1 inTheorem 4.2 can be chosen independently of a for all a ∈ A where A ⊂ Sm,ν,r

is bounded and A ⊂ Em,ν,rκ,ω for fixed κ > 0 and ω ≥ 0. In the same way, the

constants ϑ , M2, and R in Corollary 4.3 can be chosen independently of a for alla ∈ A′ where A′ ⊂ Sm,ν,r (Rn

x × Rnξ , L(E)) is bounded and where all a ∈ A′ are

parabolic with the same constants κ > 0 and ω ≥ 0.

5. Applications

5.1. Application to abstract non-autonomous Cauchy problems

Let T > 0, J := [0, T ] be a closed interval in R and t ∈ J . In the following, t ina (t, ·) ∈ Sm,ν,r

(R

2n, L (E))

and in a(t, x, D) denotes only a parameter. Moreoverwe will consider in this section a family A := {a (t, ·) : t ∈ J } ⊂ Em,ν,r

κ,ω for fixedκ > 0 and ω ≥ 0, such that

J � t −→ a (t, ·) ∈ Sm,ν,r(R

2n, L (E))

is a Hölder continuous function relative to the topology in the space of the symbols.We will use the results of the previous sections to study the existence and uniquenessof solutions for the Cauchy problem

{∂t u + Ak(t)u = f (t) , t ∈ J\ {0} ,

u (0) = u0,(5.1)

Non-regular operator-valued symbols

in W kp (R

n, E). There, Ak (t), t ∈ J , is the W kp(R

n, E)-realization of a(t, x, D)

and f : [0, T ] → W kp (R

n, E) is a given Hölder continuous function. A function

u ∈ C1 ((0, T ], E) ∩ C ([0, T ] , E)

is called a classical solution of (5.1) in [0, T ], if u(t) ∈ D(Ak(t)) and u′ (t) +Ak (t) u (t) = f (t) for all t ∈ (0, T ], and u(0) = u0. More precisely, we willuse Corollary 4.3, Theorem 2.5.1 of Chapter IV in [3] and Lemma 5.2 to obtainresults on the existence and uniqueness of solutions of the Cauchy problem (5.1)in W k

p (Rn, E) if 1 ≤ p < ∞.

In the following, B̊sp,q(R

n, E), s ∈ R, p, q ∈ [1,∞], denotes the E-valuedhomogeneous Besov space of order s and parameters p and q (see [4] for theirdefinition and properties).

Lemma 5.1. Let m ∈ R, ν, r ∈ N with ν ≥ 2n + 1 and a ∈ Sm,ν,r (R2n, L(E)).Then

a(x, D) : B̊s+mp,q (R

n, E) → B̊sp,q(R

n, E)

is a bounded and linear map for all p, q ∈ [1,∞] and all s ∈ R with 0 ≤ s < r .Moreover, the following estimate holds:

‖a(x, D)‖L(

B̊s+mp,q (Rn ,E),B̊s

p,q (Rn ,E)

) ≤ c|a|(2n+1,r)m .

Proof. See [17], Theorem 3.5. �Lemma 5.2. Let m ∈ R, ν ≥ 2n + 1, r ≥ ρn,m + k + 1, p ∈ [1,∞) andA := {a(t, ·) : t ∈ J } ⊂ Sm,ν,r . Let Ak(t) be the W k

p(Rn, E)-realization of

a(t, x, D). Then

B̊k+mp,1 (Rn, E) ↪→ D (Ak(t))

d↪→ W k

p(Rn, E). (5.2)

Proof. It is clear that D (Ak(t)) ↪→ W kp(R

n, E) (with D(Ak(t)) being endowedwith the graph norm) and it is also known that

S (Rn, E) ↪→ B̊k+mp,1 (Rn, E) ↪→ B̊k

p,1(Rn, E) ↪→ W k

p(Rn, E), 1 ≤ p < ∞.

(5.3)

From this and Lemma 5.1 it follows that u, a(t, x, D)u ∈ W kp(R

n, E) when-

ever u ∈ S (Rn, E). Therefore S (Rn, E) ⊂ D (Ak(t)). But S (Rn, E)d↪→

W kp(R

n, E) for 1 ≤ p < ∞, and we have D (Ak(t))d↪→ W k

p(Rn, E).

On the other hand, (5.3) and Lemma 5.1 imply that B̊k+mp,1 (Rn, E) ⊂ D (Ak(t))

and

‖u‖D(Ak (t)) = ‖u‖W kp(R

n ,E) + ‖a(t, x, D)u‖W kp(R

n ,E)

≤ c1‖u‖B̊k+mp,1 (Rn ,E) + c2‖a(t, x, D)u‖B̊k

p,1(Rn ,E)

B. Barraza Martínez et al.

≤ c1‖u‖B̊k+mp,1 (Rn ,E) + c3‖u‖B̊k+m

p,1 (Rn ,E)

= c‖u‖B̊k+mp,1 (Rn ,E).

So we have B̊k+mp,1 (Rn, E) ↪→ D (Ak(t)). �

Theorem 5.3. Let k ∈ N0, α ∈ (0, 1), m ∈ (0,∞), p ∈ [1,∞), ν ≥ 2n + 1 andr ≥ ρn,m + k + 1. Furthermore for fixed κ > 0 and ω ≥ 0, let A := {a(t, ·) ; t ∈J } ⊂ Em,ν,r

κ,ω be a family of parabolic symbols such that

t −→ a(t, ·) ∈ Cα(J, Sm,ν,r (R2n, L(E))

)(5.4)

and such that there is a constant M with

‖(λ+ Ak(t))−1‖L(B̊k

p,∞(Rn ,E),B̊k+mp,∞ (Rn ,E)) ≤ M, (5.5)

for all t ∈ J and λ in the sector Σϑ,R of Corollary 4.3. There Ak (t) is theW k

p(Rn, E)-realization of a(t, x, D).

If u0 ∈ W kp (R

n, E) and f ∈ Cσ(

J,W kp (R

n, E))

for some σ ∈ (0, 1), then

the Cauchy problem (5.1) has a unique solution

u ∈ C([0, T ],W kp(R

n, E)) ∩ C1((0, T ],W kp(R

n, E)).

Proof. First we claim that D(Ak(t)) ↪→ B̊k+mp,∞ (Rn, E). In fact, if u ∈ D(Ak(t)),

then u, a(t, x, D)u ∈ W kp(R

n, E) ⊂ B̊kp,∞(Rn, E). For λ0 ∈ Σϑ,R fixed and

v := (λ0 + Ak(t))u ∈ B̊kp,∞(Rn, E), (5.5) implies that u ∈ B̊k+m

p,∞ (Rn, E) and

‖u‖B̊k+mp,∞ (Rn ,E) ≤ M‖v‖B̊k

p,∞(Rn ,E)

≤ c̃‖v‖W kp(R

n ,E)

≤ c̃(|λ0|‖u‖W k

p(Rn ,E) + ‖Ak(t)u‖W k

p(Rn ,E)

)

≤ c‖u‖D(Ak (t)),

i.e. D(Ak(t)) ↪→ B̊k+mp,∞ (Rn, E). From this, (5.2), Corollary 4.3 and Theorem 2.5.1

of Chapter IV in [3], we obtain the desired result in similar way to the proof of The-orem 4.3 in [9]. �

Remark 5.4. The condition (5.5) holds, e.g., if Ak(t) is the realization of a normalelliptic differential operator (see [6], Theorem 4.1).

Non-regular operator-valued symbols

5.2. Application to coagulation–fragmentation processes

In this section, we will consider a coagulation–fragmentation problem with diffu-sion for a many particle system. This model was introduced by H. Amann in [5]in order to describe clusters of particles which can merge to form larger clusters orbreak apart into smaller ones, and which move under the influence of diffusion andtransport processes. The aim is to describe the particle size distribution u(x, t, y)depending on the position x ∈ R

n , the time t > 0, and the cluster size (i.e., volume)y ∈ Y .

In [5], the dynamic behaviour of the particle size distribution is given in theform of an initial-value problem of reaction-diffusion type:

{∂t u + A (x, t, y) u = R (x, t, y, u) (t ∈ (0, T ], x ∈ R

n),u (x, 0, y) = u0 (x, y) (x ∈ R

n),(5.6)

where n ∈ {1, 2, 3}. The diffusion-convection operator A is given by

A (x, t, y) u : = −∇x · [a (x, t, y)∇x u + −→a (t, y) u

]

+−→b (x, t, y) · ∇x u + a0 (x, t, y) u, (5.7)

with the symmetric diffusion matrix a, the drift vectors −→a and−→b , and the absorp-

tion rate a0. To deal with discrete and continuous cluster volumes simultaneously,the measure space (Y, dy) stands either for the set of positive real numbers and theLebesgue measure or for the positive integers and the counting measure. Again, weset J := [0, T ].

The reaction term R in (5.6) is of the form

R (x, t, y, u) := c (x, t, y, u)+ f (x, t, y, u)+ h (x, t, y) . (5.8)

Here, the coagulation term c given by

c (x, t, y, u) := 1

2

y∫

0

η(x, t, y − y′, y′) u

(y − y′) u

(y′) dy′

− u (y)

∞∫

0

η(x, t, y, y′) u

(y′) dy′,

(5.9)

where the coagulation kernel η(x, t, y, y′) describes the rate of coalescences of

clusters of volumes y and y′ at time t and position x . The kernel η is supposed tobe nonnegative and symmetric with respect to y, y′.

The fragmentation term f in (5.8) is given by

f (x, t, y, u) :=∞∫

y

ϕ(x, t, y′, y

)u

(y′) dy′ − u (y)

y

y∫

0

ϕ(x, t, y, y′) y′dy′,

(5.10)

B. Barraza Martínez et al.

where the fragmentation kernel ϕ satisfies

ϕ(x, t, y, y′) ≥ 0 (y, y′ ∈ Y, y > y′).

The nonnegative source term h stands for the generation of clusters of volumey at time t and position x due to particle input.

As in [5], we assume that there exist α, r > 0 such that the following assertionshold.

(i) Setting [[a (t)] (x)] (y) := a (x, t, y) := [ai j (x, t, y)

]i, j=1,...,n , we have

[t −→ a(t)] ∈ Cα(J, BUC1+r (Rn, L∞(Y, dy; R

n×nsym ))

), (5.11)

where Rn×nsym stands for the space of all symmetric real n × n-matrices. In

addition, there exists δ > 0 such that for all x ∈ Rn , t ∈ J and y ∈ Y the

inequality

n∑

i, j=1

ai j (x, t, y) ξiξ j ≥ δ |ξ |2 (ξ ∈ Rn) (5.12)

holds. For simplicity of presentation, in the present paper we additionallyassume that ai j (x, t, y) = c(t, y)a0

i j (x) with c(t, y) > 0 (problems with thiscondition can be found e.g. in [11,15,24,25] and [28]).

(ii) Let Kcoag be the closed subspace of L∞ (Y 2, dy ⊗ dy; R

)defined by

Kcoag := {η ∈ L∞(Y 2, dy ⊗ dy; R) : η(y, y′) = η(y′, y) for almost all y, y′ ∈ Y

}.

(5.13)

Then we suppose that

[t −→ η (·, t, ·, ·)] ∈ Cα(J, BUCr (

Rn, Kcoag

))(5.14)

holds.(iii) Set Y 2

Δ := {(y, y′) ∈ Y 2 : 0 ≤ y ≤ y′}, and for ϕ ∈ L∞(Y 2Δ, dy ⊗ dy) define

Φϕ(y) := 1

y

y∫

0

ϕ(y, y′)dy′ (y ∈ Y ).

The space Kfrag := {ϕ ∈ L∞(Y 2Δ, dy ⊗ dy) : Φϕ ∈ L∞(Y, dy)} is a Banach

space with the norm ϕ → ‖ϕ‖∞ + ‖Φϕ‖∞. Then we suppose that

[t −→ ϕ (·, t, ·, ·)] ∈ Cα(J, BUCr (

Rn, Kfrag

))(5.15)

(iv) For the drift vectors −→a and−→b and the absorption rate a0, we assume

[t −→ −→a (t)] ∈ Cα(J, BUC1+r (Rn, L∞(Y, dy; Rn))),

[t −→ −→b (t)] ∈ Cα(J, BUC1+r (Rn, L∞(Y, dy; R

n))),

[t −→ a0(t)] ∈ Cα(J, BUCr (Rn, L1(Y, dy; Rn))).

Non-regular operator-valued symbols

(v) Finally, it is assumed that [t −→ h(·, t, ·)] ∈ Cα(J,W τ1 (R

n,E)) holds forsome 0 < τ < r , with E := L1(Y, (1 + y)dy).

Let C(t, ·), F(t, ·) and H(t) be the Nemytskii operators induced by c (·, t, ·, ·),f (·, t, ·, ·) and h (·, t, ·), respectively. Then we can write R as g(t, u) = C(t, u)+F (t, u)+ H(t). With the assumptions (i)–(v) above, it was shown in [5] that

[t −→ g (t, ·)] ∈ Cα(J,C∞

b

(W σ

1

(R

n,E),W τ

1

(R

n,E)))

, (5.16)

where

σ ∈ (0,∞) \ N, τ ∈ (0, r)� N and

{τ + n < 2σ < 2n, if τ < n,σ := τ , if τ > n.

(5.17)

The principal part of A in (5.6) is A0(t)(y) := −∑ni, j=1 ai j (x, t, y)∂xi x j with

symbol p0(x, t, y, ξ) = c (t, y)n∑

i, j=1a0

i j (x) ξiξ j , where t and y are considered as

parameters. Let y0 ∈ Y be fixed. Since c (0, y0) �= 0, we have

A0(t)(y) = p0(x, t, y, D) = −c (t, y)n∑

i, j=1

a0i j (x) ∂xi x j = c (t, y)

c (0, y0)A0(0)(y0).

Therefore Dmax(A0k(t)) = Dmax(A0

k(0)) for all t and y, where A0k(t), k ∈ N0 is the

W k1 (R

n,E)-Realization of A0(t). Since the symbol of A − A0 has order 1, thenDmax(Ak(t)) = Dmax(A0

k(t)). Therefore Dmax(Ak(t)) = Dmax(Ak(0)) for all tand y.

On the other hand it follows from (5.12) that p0 is parabolic uniformly in t andy. Then with

μmeiθ I + p = [I + (p − p0)(μ

meiθ I + p0)−1](μmeiθ I + p0)

for all |ξ, μ| ≥ ω1, θ ∈ [−π2 ,

π2

], x ∈ R

n and y ∈ Y , we have that p (the symbolof A) is uniformly parabolic. Therefore

− Ak(t) : D (Ak(0)) ⊂ W k1

(R

n,E) −→ W k

1

(R

n,E)

(5.18)

generates an analytic semigroup on W k1 (R

n,E) for all t ≥ 0 and y ∈ Y due toCorollary 4.3. Furthermore −Ak(t) generates an strongly semigroup on W k

1 (Rn,E)

since D (Ak(t))d↪→ W k

1 (Rn,E) (see Lemma 5.2).

With the notations and assumptions given above, we can write the problem(5.6) as the following semilinear parabolic evolution equation:

{u̇ + Ak(t)u = g(t, u), t ∈ (0, T ),

u (0) = u0,(5.19)

in the Banach space W k1 (R

n,E), where Ak (given in (5.18)) satisfies

[t −→ Ak (t)] ∈ Cα(

J, L(

D (Ak(0)) ,W k1

(R

n,E)))

. (5.20)

B. Barraza Martínez et al.

and

[t −→ g (t, ·)] ∈ Cα(J,C∞

b

(W σ

1

(R

n,E),W τ

1

(R

n,E)))

. (5.21)

Now we can solve the system (5.6).

Proposition 5.5. With the notations and assumptions above, let

0 ≤ k < τ < r , τ /∈ N, (5.22)k + n

2< σ < min {n, k + 2} , if k < n (5.23)

and

k < σ < k + 2, if k > n (5.24)

for some σ /∈ N. If u0 ∈ W σ1 (R

n,E), then the coagulation–fragmentation system(5.6), i.e. the initial value problem (5.19), has a unique solution

u ∈ C([0, T0),W σ1 (R

n,E)) ∩ C1((0, T0),W k1 (R

n,E)) (5.25)

for some T0 ≤ T .

Proof. First assume that k < n. In this case, k+n2 < σ and k < τ . Therefore, we

can choose τ suitably to have k < τ < n and

τ + n

2< σ < min {n, k + 2} . (5.26)

Now we choose σ1 /∈ N with

max

{τ,τ + n

2

}< σ1 < σ < min {n, k + 2} . (5.27)

Then it follows that 0 < ρ < γ < β < 1 holds for

ρ := τ − k

2, γ := σ1 − k

2, β := σ − k

2. (5.28)

In the case k > n, we can choose τ /∈ N with k < τ < σ and σ1 /∈ N withk < τ ≤ σ1 < σ . From this, together with (5.24), it follows that 0 < ρ < γ <

β < 1 holds, where ρ, γ and τ are again defined by (5.28). Furthermore, becauseof (5.22), we have in this case that τ > n.

Now, let E1 := D (Ak(0)), E0 := W k1 (R

n,E) and Eθ := (E0,E1)θ,1 forθ ∈ {ρ, γ, β}. From

Bkp,1

(R

n,E)↪→ W k

p

(R

n,E)↪→ Bk

p,∞(R

n,E),

Bk+2p,1

(R

n,E)↪→ D (Ak(0)) ↪→ Bk+2

p,∞(R

n,E)

it follows that

Eβ = Bk+2β1,1

(R

n,E) = W σ

1

(R

n,E)

,

Eγ = Bk+2γ1,1

(R

n,E) = W σ1

1

(R

n,E)

,

Eρ = Bk+2ρ1,1

(R

n,E) = W τ

1

(R

n,E)

.

From this, (5.18), (5.20), (5.21) and Theorem 5.1 in [5], the assertion of the proposi-tion follows. �

Non-regular operator-valued symbols

Remark 5.6. The result of Proposition 5.5 complements the results of Theorem 5.2in [5], since there k /∈ N0. It is important to remark that there are several mathe-matical models of phenomena which include coagulation fragmentation processes,for example: the formation of aerosols, colloidal aggregates, polymerization anddepolimerization phenomena in chemical engineering, celestial bodies on astro-nomical scale (see [26,28] and the literature therein), evolution of phytoplanktonaggregates (see [12,24]), the development of solid tumours (see [2]), the diffusionand agglomeration of β−amyloid in Brain affected by Alzheimer’s disease (see[1]) and in studying of neural systems in neuroscience (see [10]). Our results canbe applied to the problems (1.1)–(1.2) in [28], (26)–(30) in [24] and could be ap-plied to problems in [7,10,15,24] and [25] too. In most of the references previouslymentioned E in the solution space is of the type L1 and therefore not a reflexivespace.

A. Appendix: Oscillatory integrals

For the definition of pseudo-differential operators we needed the theory of (vector-valued) oscillatory integrals. Therefore, we summarize below some definitions andresults which can be found in [8] and [19].

Definition A.1. Let a : R2n → E be a strongly measurable function. We say that

a is integrable in the oscillatory sense if for each χ ∈ S (R2n) with χ(0, 0) = 1the limit

Os −∫∫

R2n

eiξ ·ya(ξ, y)d̄ (ξ, y) := limε↘0

∫∫

R2n

eiξ ·yχ(εξ, εy)a(ξ, y)d̄ (ξ, y)

exists and does not depend on the choice of χ .

Definition A.2. Let m ∈ R, τ ∈ [0,∞), 0 ≤ δ < 1 and ν, ρ ∈ N0 ∪{∞}. Then thespace A(m,ν,ρ)

δ,τ (Rnξ×R

ny, E) of amplitude functions consists of all a : R

nξ×R

ny → E

for which all derivatives ∂αξ ∂βy a with |α| ≤ ν, |β| ≤ ρ are continuous and for which

all norms |a|m,,′ with , ′ ∈ N0, ≤ ν, ′ ≤ ρ are finite. Here

|a|m,,′ := max|α|≤ max|β|≤′

supξ∈Rn

supy∈Rn

〈ξ 〉−m−δ|β|〈y〉−τ‖∂αξ ∂βy a(ξ, y)‖E .

Lemma A.3. For χ ∈ S (Rn) with χ(0) = 1 the following assertions hold.

(i) χ(εx) → 1 (ε ↘ 0) uniformly on all compact subsets of Rn,

(ii) ∂αx χ(εx) → 0 (ε ↘ 0) uniformly on Rn for all α ∈ N

n0 \ {0},

(iii) for all α ∈ Nn0 there exists cα > 0 such that for all ε ∈ (0, 1) we have

|∂αx χ(εx)| ≤ cαεσ 〈x〉−(|α|−σ) (σ ∈ [0, |α|], x ∈ R

n).

Proof. See [19], Lemma 6.3. �

B. Barraza Martínez et al.

Lemma A.4. Let m ∈ R, τ ∈ [0,∞), 0 ≤ δ < 1, ν, ρ ∈ N0 ∪ {∞} and , ′ ∈ N

with n + τ < 2 ≤ ν and m+n1−δ < 2′ ≤ ρ. Then for a ∈ A(m,ν,ρ)

δ,τ (Rnξ × R

ny, E) the

oscillatory integral Os −∫∫

R2n

eiξ ·ya(ξ, y)d̄ (ξ, y) exists, and

∥∥∥ Os −

∫∫

R2n

eiξ ·ya(ξ, y)d̄ (ξ, y)∥∥∥

E≤ c|a|m,2,2′

holds with a constant c not depending on a.

Proof. With 〈Dη〉 := (1 − Δη)1/2, where Δη denotes the Laplacian with respect

to variable η, we have

eiξ ·y = 〈ξ 〉−2′ 〈Dy〉2′(〈y〉−2〈Dξ 〉2eiξ ·y). (A.1)

Let χ ∈ S (R2n) be arbitrary but such that χ(0, 0) = 1 and 0 < ε < 1. Using(A.1) and integration by parts we obtain

∫∫

R2n

eiξ ·yχ(εξ, εy)a(ξ, y) d̄ (ξ, y)

=∫∫

R2n

eiξ ·y〈y〉−2〈Dξ 〉2[〈ξ 〉−2′ 〈Dy〉2′

(χ(εξ, εy)a(ξ, y)

)]d̄ (ξ, y).

(A.2)

Now we put

Jε(ξ, y) : = 〈y〉−2〈Dξ 〉2[〈ξ 〉−2′ 〈Dy〉2′

(χ(εξ, εy)a(ξ, y)

)],

J (ξ, y) : = 〈y〉−2〈Dξ 〉2[〈ξ 〉−2′ 〈Dy〉2′a(ξ, y)

].

Then it holds

Jε(ξ, y)− J (ξ, y)

= 〈y〉−2〈Dξ 〉2{〈ξ 〉−2′ 〈Dy〉2′

[(χ(εξ, εy)− 1

)a(ξ, y)

]}

= 〈y〉−2∑

|γ |≤2θ≤γ, α≤θ

∑

|λ|≤2′β≤λ

Cz(∂γ−θξ 〈ξ 〉−2′)

[∂θ−αξ ∂λ−βy

(χ(εξ, εy)−1

)]∂αξ ∂

βy a(ξ, y)

(A.3)

where Cz are constants depending on z := (α, β, γ, θ, λ). Due to the last equalityin (A.3) and Lemma A.3 we have that Jε(ξ, y) → J (ξ, y) (ε ↘ 0) in E for all(ξ, y) ∈ R

2n .A similar calculation to (A.3) leads to the following estimate

‖Jε(ξ, y)‖E ≤ C,′ |a|m,2,2′ 〈y〉−(2−τ)〈ξ 〉−((1−δ)2′−m)

Non-regular operator-valued symbols

for all (ξ, y) ∈ R2n and 0 < ε < 1. Since

[(ξ, y) → 〈y〉−(2−τ)〈ξ 〉−((1−δ)2′−m)

]∈ L1(R2n

ξ,y)

it follows from (A.2) and the dominated convergence theorem that

limε↘0

∫∫

R2n

eiξ ·yχ(εξ, εy)a(ξ, y) d̄ (ξ, y) = limε↘0

∫∫

R2n

eiξ ·y Jε(ξ, y) d̄ (ξ, y)

=∫∫

R2n

eiξ ·y J (ξ, y) d̄ (ξ, y).

This shows that the oscillatory integral Os −∫∫

R2n

eiξ ·ya(ξ, y) d̄ (ξ, y) exists because

the limit above is independent on χ and we have

Os −∫∫

R2n

eiξ ·ya(ξ, y) d̄ (ξ, y) =∫∫

R2n

eiξ ·y J (ξ, y) d̄ (ξ, y).

On the other hand, since

J (ξ, y) = 〈y〉−2∑

|λ|≤2α≤λ

∑

|β|≤2′Cα,β,λ

(∂λ−αξ 〈ξ 〉−2′)∂αξ ∂

βy a(ξ, y)

we have

‖J (ξ, y)‖E ≤ c,′ |a|m,2,2′ 〈ξ 〉−((1−δ)2′−m)〈y〉−(2−τ).Therefore

∥∥∥ Os −∫∫

R2n

eiξ ·ya(ξ, y)d̄ (ξ, y)∥∥∥

E

=∥∥∥

∫∫

R2n

eiξ ·y J (ξ, y) d̄ (ξ, y)∥∥∥

E

≤∫∫

R2n

‖J (ξ, y)‖E d̄ (ξ, y)

≤ c,′ |a|m,2,2′∫∫

R2n

〈ξ 〉−((1−δ)2′−m)〈y〉−(2−τ) d̄ (ξ, y)

= c|a|m,2,2′

with c := c,′∫∫

R2n〈ξ 〉−((1−δ)2′−m)〈y〉−(2−τ) d̄ (ξ, y) < ∞ a constant which is

independent on a. �

B. Barraza Martínez et al.

Remark A.5. From the proof of Lemma A.4 it can be seen that the existence and

value of the oscillatory integral Os −∫∫

R2n

eiξ ·ya(ξ, y)d̄ (ξ, y) is independent on the

choice of and ′ whenever n + τ < 2 ≤ ν and m+n1−δ < 2′ ≤ ρ.

Lemma A.6. With m, τ, ν, ρ, δ, , and ′ as in Lemma A.4, let N ∈ N0 and leta : R

nx × R

nξ × R

ny → E be a function with ∂γx a(x, ·, ·) ∈ A(m,ν,ρ)δ,τ (Rn

ξ × Rny, E)

for all x ∈ Rn and |γ | ≤ N. Suppose further that for every x ∈ R

n there is aneighbourhood Ux of x such that the set {(∂γx a)(x ′, ·, ·) : x ′ ∈ Ux } is bounded inA(m,ν,ρ)δ,τ (Rn

ξ × Rny, E). Then for all |γ | ≤ N we have

∂γx

[Os −

∫∫

R2n

eiξ ·ya(x, ξ, y)d̄ (ξ, y)]

= Os −∫∫

R2n

eiξ ·y∂γx a(x, ξ, y)d̄ (ξ, y).

Proof. From the proof of Lemma A.4 we have in this case that

∂γx

[Os −

∫∫

R2n

eiξ ·ya(x, ξ, y)d̄ (ξ, y)]

= ∂γx

∫∫

R2n

eiξ ·y J (x, ξ, y)d̄ (ξ, y)

with

J (x, ξ, y) = 〈y〉−2〈Dξ 〉2[〈ξ 〉−2′ 〈Dy〉2′a(x, ξ, y)

]

= 〈y〉−2∑

|λ|≤2α≤λ

∑

|β|≤2′Cα,β,λ

(∂λ−αξ 〈ξ 〉−2′)∂αξ ∂

βy a(x, ξ, y).

Then

∂γx J (x, ξ, y) = 〈y〉−2〈Dξ 〉2

[〈ξ 〉−2′ 〈Dy〉2′∂γx a(x, ξ, y)

]

= 〈y〉−2∑

|λ|≤2α≤λ

∑

|β|≤2′Cα,β,λ

(∂λ−αξ 〈ξ 〉−2′)∂αξ ∂

βy ∂

γx a(x, ξ, y)

and therefore

‖∂γx J (x, ξ, y)‖E

≤ 〈y〉−2∑

|λ|≤2α≤λ

∑

|β|≤2′|Cα,β,λ|〈ξ 〉−2′+m+δ|β|〈y〉τ |(∂λx a)(x, ·, ·)|m,2,2′ .

Because of the conditions on ∂γx a we have that for each x ∈ Rn there is a neigh-

bourhood Ux of x such that |(∂λx a)(x ′, ·, ·)|m,2,2′ ≤ q for all x ′ ∈ Ux , where q isa constant. Then

‖∂γx ′ J (x ′, ξ, y)‖E ≤ const〈y〉−(2−τ)〈ξ 〉−((1−δ)2′−m)

for all (x ′, ξ, y) ∈ Ux × Rnξ × R

ny . Since

[(ξ, y) → 〈y〉−(2−τ)〈ξ 〉−((1−δ)2′−m)

]∈ L1(R2n

ξ,y)

Non-regular operator-valued symbols

it follows from a classic result about differentiation of an integral with respect to aparameter that

∂γx

∫∫

R2n

eiξ ·y J (x, ξ, y)d̄ (ξ, y) =∫∫

R2n

eiξ ·y∂γx J (x, ξ, y)d̄ (ξ, y)

= Os −∫∫

R2n

eiξ ·y∂γx a(x, ξ, y)d̄ (ξ, y).

�

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