Analytic semigroups of pseudodifferential operators on vector-valued Sobolev spaces

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Universität Konstanz

Analytic semigroups of pseudodifferential operators on vector-valued Sobolev spaces

Bienvenido Barraza Martinez Robert Denk

Jairo Hernández Monzón

Konstanzer Schriften in Mathematik Nr. 303, Mai 2012

ISSN 1430-3558

Fach D 197, 78457 Konstanz, Germany

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-193526

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OPERATORS ON VECTOR-VALUED SOBOLEV SPACES

B. BARRAZA MARTINEZ, R. DENK, AND J. HERN ´ANDEZ MONZ ´ON

Abstract. In this paper we study continuity and invertibility of pseudodifferential operators with non-regular Banach space valued symbols.

The corresponding pseudodifferential operators generate analytic semigroups on the Sobolev spacesWp^k(Rⁿ, E) withk∈N⁰, 1≤p≤ ∞.Here E is an arbitrary Banach space. We also apply the theory to solve non- autonomous parabolic pseudodifferential equations in Sobolev spaces.

1. Introduction

In the present work, we regard symbols in the space S_1,0^m,ρ(Rⁿ, E) with m ∈ R, ρ ∈ N0 and E being an arbitrary Banach space. We say that a ∈ S^m,ρ_1,0 (Rⁿ, E) ifa∈C^ρ(Rⁿ, E) and if for allα ∈Nⁿ₀ with|α| ≤ρthere exists a c_α >0 such that

∂_ξ^αa(ξ)

E ≤cαhξi^m−|α| for all ξ∈Rⁿ. (1) Herehξi:=p

1 +|ξ|²,ξ∈Rⁿ. Because of Mikhlin’s theorem (see [BL76] and [NNH02]) the pseudodifferential operator or the Fourier multiplier induced by a,

F⁻¹aF :Lp(Rⁿ, E1)−→Lp(Rⁿ, E2) for 1< p <∞, (2) is continuous if a ∈ S_1,0^0,n+1(Rⁿ,L(E₁, E₂)) and E₁, E₂ are Hilbert spaces.

HereF andF⁻¹denote the Fourier transform and the inverse Fourier transform, respectively. In practice one would like to have the validity of this result for arbitrary Banach spaces E₁, E₂. But, this is impossible in light of an observation of G. Pisier: If the Mikhlin theorem is valid onL_p(Rⁿ, E) for L(E)-vector-valued symbols, then E is isomorphic to a Hilbert space (see [LLM98] for a proof).

Therefore, in order to obtain versions of Mikhlin’s theorem, one has to, for example, change the space scale Lp(Rⁿ, E), make additional conditions for (1), or impose conditions on the geometry of the Banach space E. On one hand, Amann has shown in [Am97], Theorem 6.2, that for arbitrary

B. Barraza Martinez, Universidad del Norte, Departamento de Matem´aticas, Barranquilla (Colombia)

R. Denk, Universit¨at Konstanz, Fachbereich f¨ur Mathematik und Statistik, Konstanz (Germany)

J. Hernández Monzón, Universidad del Norte, Departamento de Matemáticas, Barranquilla (Colombia)

E-mail addresses:bbarraza@uninorte.edu.co, robert.denk@uni-konstanz.de, jahernan@uninorte.edu.co.

Date: May 8th, 2012.

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Banach spaces E1 and E2 and a ∈ S_1,0^m,n+1(Rⁿ,L(E1, E2)) that the pseudodifferential operator F⁻¹aF : B_p,q^s+m(Rⁿ, E₁) −→ B_p,q^s (Rⁿ, E₂) (s ∈ R, p, q∈[1,∞]) is continuous, where B_p,q^s (Rⁿ, E) denotes the Besov space.

On the other hand, Weis proved in [We01] for UMD spaces E1 and E2

and symbols ainS_1,0^0,1(R,L(E₁, E₂)) which fulfill a stronger condition than (1), that F⁻¹aF :L_p(R, E₁) −→L_p(R, E₂) (for 1< p <∞) is continuous.

Similar results were obtained in [PS06]. By a UMD space E one under- stands a Banach space for which the Hilbert transform is continuous from Lp(R, E) toLp(R, E) for 1< p <∞, or equivalently, for which the function m(t) =|t|⁻¹tis a Fourier multiplier onL_p(R, E) (1< p <∞). This implies reflexivity of E (see [Am95], Remark 4.4.2).

In order to achieve an optimal value of ρin (1), one has to regard the geometry of the Banach spacesE₁, E₂. Girardi and Weis considered in [GW03]

Banach spaces E1, E2 of Fourier type p, 1 ≤p ≤2, (i.e. Banach spaces Ei

for which the Fourier transform is continuous ofL_p(Rⁿ, E_i) intoL_p⁰(Rⁿ, E_i) with ¹_p +_p¹0 = 1), and symbols a∈S_1,0^0,ρ(Rⁿ,L(E₁, E₂)). Under these conditions, they proved that for alls∈R,r, q∈[1,∞], the pseudodifferential operator F⁻¹aF :B_r,q^s (Rⁿ, E1)−→B_r,q^s (Rⁿ, E2) is continuous, ifρ=

hn p

i +1.

For for arbitrary Banach spaces E₁, E₂ the Fourier type equalsp = 1, and one obtains Theorem 6.2 in [Am97]. If E₁,E₂ are uniformly convex Banach spaces (thus having Fourier type p >1), see [Bo87], one can chooseρ=n.

Many applications to problems of physics and biology, e.g. models for reaction-diffusion processes (see [Am00]), suggest the necessity to regard Banach space-valued symbols. The space considered in [Am00] is the vector- valued Sobolev-Slobodeckii space W₁^s(Rⁿ, L₁(Y, µ)), s /∈ N, and it is well- known that L1(Y, µ) is not a reflexive space. For this reason, we would like to obtain a version of the Mikhlin theorem (like in [Am97]) for Ba- nach space-valued symbols, in which the correspondent pseudodifferential operator generates an analytic semigroup on W_p^k(Rⁿ, E), 1≤p ≤ ∞. As a consequence, one obtains existence and uniqueness of solutions of parabolic pseudodifferential equations.

Therefore, we regard arbitrary Banach spaces Ei, i = 0,1,2, the vector space

V(Rⁿ, E_i) := [

(s,p,q)∈R×[1,∞]×[1,∞]

B_p,q^s (Rⁿ, E_i) and symbols a∈S_1,0^m,ρⁿ(Rⁿ, E1) with regularity

ρ_n:=

n+ 1, n∈N odd,

n+ 2, n∈N even. (3)

We will show (see Theorem 3.1) that there exists a unique linear map a](D) :V(Rⁿ, E₂)−→V(Rⁿ, E₀)

with a](D) ∈ L(B_p,q^s+m(Rⁿ, E₂), B_p,q^s (Rⁿ, E₀)) such that its restriction on every intersection C_b^∞(Rⁿ, E₂)∩B_p,q^s+m(Rⁿ, E₂) (s ∈ R, p, q ∈ [1,∞]) co- incides with the classical definition of the pseudodifferential operator a(D) in [Ku81], which has been defined by means of an oscillatory integral. In

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this way, we show that suitable restrictions of the operator −a](D) generate analytic semigroups on the Besov spaces B_p,q^s (Rⁿ, E0) andC^∞-semigroups on the Sobolev spaces W_p^k(Rⁿ, E₀). Thus we obtain results similar to those in [Am97]: Using a](D) ∈ L(B_p,q^s+m(Rⁿ, E₂), B_p,q^s (Rⁿ, E₀)), the continuous embedding

B_p,1^k (Rⁿ, E),→W_p^k(Rⁿ, E),→B^k_p,∞(Rⁿ, E) (1≤p≤ ∞),

and some properties of Banach space interpolation, we will show that the W_p^k−realization of parabolic pseudodifferential operators are the negative generators of analytic semigroups on W_p^k(Rⁿ, E0) with 1 ≤ p ≤ ∞ (see Theorem 3.14). Therefore, we obtain the existence and uniqueness of solutions for a non-autonomous Cauchy problem in Sobolev spaces W_p^k(Rⁿ, E) (see Theorem 4.3).

For vector-valued differential operators, generation of an analytic semigroup on L_p(Rⁿ, E₀),C₀(Rⁿ, E₀) andBU C(Rⁿ, E₀) was shown by Amann in [Am01]. Pseudodifferential operators with smooth symbols (with respect to the dual variable ξ) were considered in [Ki03]. In [Ki01], the generation of an analytic semigroup in Lp(Rⁿ, E0) (1 ≤ p < ∞) was shown for suitable parabolic vector-valued pseudodifferential operators where the symbols are assumed to have a homogeneous principal part and regularity greater than or equal to 2n+ 1. Apart from generalizing these results, the method used in the present paper will allow us to consider x-dependent symbols in a forthcoming publication.

The plan of the paper is as follows: After some preliminary definitions and remarks in Section 2, we prove in Section 3 the main results of the present paper on continuity (Theorem 3.1) and on generation of an analytic semigroup (Theorem 3.14). As an application, we prove in Theorem 4.3 the existence and uniqueness of the solution of a non-autonomous Cauchy problem in the Sobolev spaces W_p^k(Rⁿ, E). The constants C1, . . . , C15, M1

are rigourously calculated, because we will explore stability conditions for an adequate family of pseudodifferential operators in Remark 4.4.

2. Preliminary definitions and remarks

In the following, E and E_i always denote arbitrary Banach spaces with norm k·k_E and k·k_E

i, respectively, and L(E1, E0) the space of linear, continuous maps of E₁ into E₀. The definitions and properties of functions spaces are taken from [Am97]. In particular,S(Rⁿ, E) denotes the Schwartz space of rapidly decrasing functions, C_b^k(Rⁿ, E), k ∈ N0, is the space of all functions u : Rⁿ → E such that ∂^αu is bounded and continuous on Rⁿ for all |α| ≤ k, BU C^k(Rⁿ, E) is the space of all u ∈ C^k(Rⁿ, E) such that ∂^αu is bounded and uniformly continuous on Rⁿ for all |α| ≤ k and BU C(Rⁿ, E) :=BU C⁰(Rⁿ, E).

The Besov space B_p,q^s , the homogeneous Besov space ˚B_p,q^s and the small Besov space spaces b^s_p,q are defined as follows: For s ∈R and p, q ∈ [1,∞]

one defines the E-valued Besov space of order sby B_p,q^s (Rⁿ, E) :=

u∈ S⁰(Rⁿ, E) :

2^jskψ_j(D)uk_L

p(Rⁿ,E)

l^q <∞ (4)

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with norm

kuk_Bs

p,q(Rⁿ,E):=

2^jskψ_j(D)uk_L

p(Rⁿ,E)

l^q,

where S⁰(Rⁿ, E) is the space of theE-valued tempered distributions, ψ_j(D)u:=F⁻¹(ψ_jFu) (5) and (ψ_j)_j∈

N0 is a resolution of unity which is constructed in the following way: For ψ∈ S(Rⁿ) :=S(Rⁿ,C) with

suppψ⊂ {x∈Rⁿ:|x| ≤2} and ψ(x) = 1 on |x| ≤1, (6) one definesψ(x) :=e ψ(x)−ψ(2x),ψj(x) :=ψ(2e ^−jx) forx∈Rⁿandψ0 :=ψ, ψ−1 := 0. The sequence (ψ_j)_j∈

N0 satisfies:

supp(ψ0)⊂Ω0 :={x∈Rⁿ:|x| ≤2}, supp(ψ_j)⊂Ω_j :=

x∈Rⁿ: 2^j−1 ≤ |x| ≤2^j+1 for all j∈N, (7) as well as

∞

X

j=0

ψ_j(ξ) = 1 for all ξ∈Rⁿ. (8) Moreover, for each α∈Nⁿ0 there exists a constant cα>0 such that¹

D^α_ξψ_j(ξ)

≤c_α2^−j|α|1_Ω_j(ξ) for allξ ∈Rⁿ and j∈N0. (9) One defines the Banach spaces ˚B^s_p,q(Rⁿ, E) and b^s_p,q(Rⁿ, E) as the closures of S(Rⁿ, E) and B_p,q^s+1(Rⁿ, E), respectively, where the closure is taken with respect to the normk · k_B^s

p,q.

The following three lemmas will be crucial for the main results in Sec- tion 3.

Lemma 2.1. Let s∈R andp, q∈[1,∞]. Then we have:

a) S(Rⁿ, E),→ B_p,q^s (Rⁿ, E) ,→ S⁰(Rⁿ, E) and S(Rⁿ, E) ,→^d B_p,q^s (Rⁿ, E) if p, q ∈ [1,∞). Here ,→ denotes continuous embedding and ,→^d denotes continuous and dense embedding.

b) For each 1≤q₀ ≤q₁ ≤ ∞,

B_p,q^s ₀(Rⁿ, E),→B^s_p,q₁(Rⁿ, E). (10) c) For each ε >0 and 1≤q₀, q₁ ≤ ∞,

B_p,q^s+ε₀(Rⁿ, E),→B_p,q^s ₁(Rⁿ, E). (11) d) Let s1 ≤s0 and 1≤p0≤p1 ≤ ∞with s1−_pⁿ

1 =s0−_pⁿ

0. Then

B_p^s₀⁰_,q(Rⁿ, E),→B^s_p¹₁_,q(Rⁿ, E). (12) e)

b^s_p,q(Rⁿ, E) =

B_p,q^s (Rⁿ, E), if1≤p≤ ∞, 1≤q <∞, B˚_p,∞^s (Rⁿ, E), if 1≤p <∞, q=∞.

f ) For 1≤p <∞ andk∈N,

B˚_p,1^k (Rⁿ, E),→^d W_p^k(Rⁿ, E),→^d B˚_p,∞^k (Rⁿ, E) =b^k_p,∞(Rⁿ, E), (13)

11Ωdenotes the characteristic function of a set Ω.

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B_∞,1^k (Rⁿ, E),→W_∞^k (Rⁿ, E),→B^k_∞,∞(Rⁿ, E), (14) b^k_∞,1(Rⁿ, E) =B^k_∞,1(Rⁿ, E),→^d BU C^k(Rⁿ, E),→^d b^k_∞,∞(Rⁿ, E), (15) B˚_∞,1^k (Rⁿ, E),→^d C₀^k(Rⁿ, E),→^d B˚_∞,∞^k (Rⁿ, E), (16) B_∞,1^k (Rⁿ, E),→C_b^k(Rⁿ, E),→B_∞,∞^k (Rⁿ, E). (17) g) If p, q, q₁, q₂ ∈[1,∞], −∞< s₁< s₂ <∞ andθ∈(0,1), then²

B_p,q^s¹₁(Rⁿ, E), B_p,q^s²₂(Rⁿ, E)

θ,q

∼=B_p,q^(1−θ)s¹^+θs²(Rⁿ, E). (18)

Proof. See [Am95], Chap. 5 or [Sc86].

By Definition and Lemma 2.1 a), it follows immediately that S(Rⁿ, E),→^d B˚_p,q^s (Rⁿ, E),→B_p,q^s (Rⁿ, E) (s∈R, p, q∈[1,∞]), B_p,q^s+1(Rⁿ, E),→^d b^s_p,q(Rⁿ, E),→B_p,q^s (Rⁿ, E) (s∈R, p, q∈[1,∞]), B˚^s_p,q(Rⁿ, E) =B_p,q^s (Rⁿ, E) (s∈R, p, q∈[1,∞)).

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Also one can easily see, using again the previous Lemma, that if B ∈ {b,B, B}, 1˚ ≤p, q₁, q₂≤ ∞,t > s and E₁,→E₀, then

B_p,q^t

1(Rⁿ, E1),→ B_p,q^s

2(Rⁿ, E0) . (20)

If E1

,→d E0, the continuous embedding (20) is always dense, except when B=B andq₁ orq₂ is ∞. Moreover

B_p,q^s ₁(Rⁿ, E1),→ B_p,q^s ₂(Rⁿ, E0) (1≤q1≤q2 ≤ ∞). (21) Lemma 2.2. Let s∈R, p∈[1,∞] andq ∈[1,∞). Then

C_b^∞(Rⁿ, E)∩B_p,q^s (Rⁿ, E) ;k·k_Bs p,q

_d

,→B_p,q^s (Rⁿ, E). (22) Proof. For alls∈R,p∈[1,∞] and q∈[1,∞) it is clear that

C_b^∞(Rⁿ, E)∩B_p,q^s (Rⁿ, E) ;k·k_Bs p,q

,→B_p,q^s (Rⁿ, E).

It only remains to prove the density. In the case p <∞, the density in (22) follows from Lemma 2.1 a) andS(Rⁿ, E)⊂C_b^∞(Rⁿ, E)∩B_p,q^s (Rⁿ, E). Now, let u∈B_∞,q^s (Rⁿ, E) withq ∈[1,∞), (ψ_j)_j∈

N0 a partition of unity as in (8) and u_N :=

N

P

k=0

ψ_k(D)u,N ∈N. By definition (4), ψ_j(D)u∈L∞(Rⁿ, E) for all j∈N0, and thereforeu_N ∈L∞(Rⁿ, E). On the other hand, let (ϕ_j)_j∈

N0

be another partition of the unity as in (8). Then one obtains from (5) and the fact that convolution corresponds to multiplication of the Fourier transform³ that

ϕ_j(D) (ψ_k(D)u) = (2π)⁻ⁿ² ϕˇ_j ∗(ψ_k(D)u)∈ O_M(Rⁿ, E) (23)

2Here (E0, E1)_θ,q denotes the real interpolation space with exponentθ and parameter q between the Banach spacesE1 andE0.

3OM(Rⁿ, E) denotes the space ofE-valued slowly increasing smooth functions onRⁿ. For the result on convolution see [Am97], Th. 3.6.

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for all j, k∈N0 where ˇϕ_j :=F⁻¹ϕ_j. Therefore ϕ_j(D)(ψ_k(D)u) is a regular tempered distribution. From (23), ˇϕj ∈ L1(Rⁿ), ψk(D)u ∈ L∞(Rⁿ, E0) and Theorem 1.9.9 in [Am03] it follows that for all j, k∈N0

kϕ_j(D) (ψ_k(D)u)k_L

∞ ≤ kϕˇ_jk_L

1kψ_k(D)uk_L

∞ =c_ϕkψ_k(D)uk_L

∞ (24) where the constant cϕ := max{kϕkˇ _L₁,k(ϕ)e ^∨k_L₁} is independent ofj and k.

Thus

ϕj(D) (u−uN) =ϕj(D)u−

N

X

k=0

ϕj(D) (ψk(D)u)∈L∞(Rⁿ, E).

From supp (ψj), supp (ϕj)⊂Ωj, (24) and

∞

P

k=0

ψk(D)u=uin S⁰(Rⁿ, E), we see that

ku−u_Nk^q_B_s

∞,q(Rⁿ,E)

=

∞

X

j=0

2^jsq

ϕ_j(D) u−

N

X

k=0

F⁻¹(ψ_kFu)

q

L∞(Rⁿ,E)

=

∞

X

j=0

2^jsq ϕj(D)

X^∞

k=0

F⁻¹(ψkFu)−

N

X

k=0

F⁻¹(ψkFu)

q

L∞(Rⁿ,E)

(5)=

∞

X

j=0

2^jsq F⁻¹

ϕj

∞

X

k=N+1

ψ_kFu

q

L∞(Rⁿ,E)

= 2^{N sq}

F⁻¹(ϕNψN+1Fu)

q

L∞(Rⁿ,E)

+ 2^(N+1)sq

F⁻¹(ϕ_N+1(ψ_N₊₁+ψ_N₊₂)Fu)

q

L∞(Rⁿ,E)

+

∞

X

j=N+2

2^jsq

F⁻¹(ϕjFu)]

q

L∞(Rⁿ,E)

since

1

X

r=−1

ϕjψj+r =ϕj

(5)= 2^{N sq}kψ_N+1(D) (ϕ_N(D)u)k^q_L

∞

+ 2^(N+1)sqk(ψ_N+1+ψ_N+2) (D) (ϕ_N+1(D)u)k^q_∞ +

∞

X

j=N+2

2^jsqkϕ_j(D)uk^q_∞}

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≤ c_n,q,s,ψn

2^{N sq}kϕ_N(D)uk^q_∞+ 2^(N+1)sqk(ϕ_N₊₁(D)u)k^q_∞ +

∞

X

j=N+2

2^jsqkϕ_j(D)uk^q_∞o

=c_n,q,s,ψ

∞

X

j=N

2^jsqkϕ_j(D)uk^q_∞ −→

N→∞0,

becauseuis inB^s_∞,q(Rⁿ, E). This implies thatu_N ∈B_∞,q^s (Rⁿ, E) andu_N → u inB_∞,q^s (Rⁿ, E). Furthermoreu_N is a regular distribution in C_b^∞(Rⁿ, E),

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because we have, setting χk :=P1

r=−1ψk+r, u_N =

N

X

k=0

F⁻¹(ψ_kFu) =

N

X

k=0

F⁻¹((χ_kψ_k)Fu)

=

N

X

k=0

F⁻¹(χkF(ψk(D)u)) =

N

X

k=0

cnχˇk∗(ψk(D)u) . Thus, it follows from Theorem 1.9.9 in [Am03], for each α∈Nⁿ0, that

∂^αu_N =c_n

N

X

k=0

(∂^αχˇ_k)

| {z }

∈L₁

∗(ψ_k(D)u)

| {z }

∈L∞

∈BU C(Rⁿ, E) .

This finishes the proof.

Lemma 2.3. Let (ψ_j)_j∈

N0 as in (8), B :=

x∈Rⁿ: ¹₂ ≤ |x| ≤2 , m∈ R and a∈S_1,0^m,n+1(Rⁿ, E). Then (ψ_ja)^∨ ∈L₁(Rⁿ, E) and

(ψ_ja)^∨

L1(Rⁿ,E)≤C₁2^jmkak_Sm,n+1

1,0 (Rⁿ,E) (j∈N0), where the constantC1is given byC1 := 2^3m² ωnC₁^∗max

vol(B), P

|α|=n+1

nⁿ⁺¹² with C₁^∗ := kψke _Cn+1

b

P

|α|=n+1 β≤α

α β

2^|β| and ωn denoting the volume of the unit ball in Rⁿ.

Proof. For allj∈N0, we have

k(ψ_ja)^∨k_L₁₍_Rn,E)=k(ψa(2e ^j·))^∨k_L₁₍_Rn,E) (25) where ψe= ψ in the case j = 0. Now, let 1B be the characteristic function of B, and let α ∈Nⁿ0 with|α| ≤n+ 1 and j ∈N. Then for all ξ ∈Rⁿ we obtain⁴

D^α(ψa(2e ^j·))(ξ) E

≤ X

β≤α

α β

2^j|β|

|{z}

=2^|β|2^(j−1)|β|

|D^α−βψ(ξ)|e 1B(ξ)

(D^βa)(2^jξ) _E

≤ X

β≤α

α β

2^|β|kψke _Cn+1

b 1B(ξ) max

|β|≤n+1sup

Ωj

hξi^|β|

D^βa(ξ) E

≤C₁^∗kak_Sm,n+1

1,0 1B(ξ) sup

Ωj

hξi^m

≤C₁^∗kak_Sm,n+1

1,0 1_B(ξ)·

( 1 + 2^2(j+1)m/2

ifm≥0, 1 + 2^2(j−1)m/2

ifm <0,

≤2^3m² C₁^∗2^jmkak_Sm,n+1

1,0 1_B(ξ) .

An analogous result is obtained for j= 0 with ψe=ψ. Therefore, D^α ψa(2e ^j·)

∈L₁(Rⁿ, E),

4D^α:= (−i)^|α|∂^α.

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and for all j∈N0 and |α| ≤n+ 1 we have

D^α(ψa(2e ^j·))

L1(Rⁿ,E)≤2^3m² C₁^∗2^jmkak_Sm,n+1 1,0 . From the last inequality, the equality

x^αF⁻¹(ψa(2e ^j·)) = (−1)^|α|F⁻¹ D^α(ψa(2e ^j·))

∈C0(Rⁿ, E), (25) and the inequality |x|ⁿ⁺¹ ≤nⁿ⁺¹² P

|α|=n+1|x^α|one concludes that

(ψja)^∨

L1(Rⁿ,E)

= Z

B1(0)

F⁻¹(ψa(2e ^j·))(x) Edx+

Z

Rⁿ\B₁(0)

F⁻¹(ψa(2e ^j·))(x) Edx

≤ Z

B1(0)

Z

B

|ψ(ξ)| ka(2e ^jξ)k_Edξ dx

+ Z

Rⁿ\B₁(0)

nⁿ⁺¹² P

|α|=n+1|x^α|

|x|ⁿ⁺¹

F⁻¹

ψae 2^j· (x)

_Edx

≤2^3m² C₁^∗ωnvol(B)2^jmkak_Sm,n+1 1,0

+ X

|α|=n+1

nⁿ⁺¹² Z

Rⁿ\B1(0)

1

|x|ⁿ⁺¹ F⁻¹

D^α

ψae 2^j· (x)

Edx

+ X

|α|=n+1

nⁿ⁺¹² Z

Rⁿ\B1(0)

1

|x|ⁿ⁺¹ Z

B

D^α

ψae 2^j· (ξ)

Edξ

dx

+ 2^3m² C₁^∗



 X

|α|=n+1

nⁿ⁺¹²



2^jmkak_Sm,n+1 1,0

Z

Rⁿ\B₁(0)

1

|x|ⁿ⁺¹dx

≤C₁2^jmkak_Sm,n+1 1,0 , with C1 := 2^3m² ωnC₁^∗max

(

vol(B), P

|α|=n+1

nⁿ⁺³² )

.

Remark 2.4. Leta∈S_1,0^m,ρ(Rⁿ, E₁) andu∈C_b^∞(Rⁿ, E₂). For the definition of

[a(D)u](x) = Os− Z Z

R²ⁿ

e^iξ·ηa(ξ)•u(x−η)d(ξ, η)

(2n)ⁿ, x∈Rⁿ, (26) (cf. [Ku81]), the condition ρ≤n+ 1is not sufficient, because

b_x(ξ, η) :=a(ξ)•u(x−η)∈ A^m_δ,τ

withδ = 0and τ = 0, and the oscillatory integral in (26)exists ifn+τ <2l and n+m

1−δ < 2l⁰, where 2l ( l ∈ N0) denotes the necessary number of derivatives in the variable ξ (see [Ku81]). We obtain the condition n < 2l

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and therefore we consider in this paperρ≥ρ_n, whereρ_n is the smallest even number greater than n (see (3)).

3. Main results

Theorem 3.1 (and definition of the operator a](D)). Let m ∈ R, a ∈ S^m,ρ_1,0 ⁿ(Rⁿ;E₁) and •:E₁×E₂ →E₀ a multiplication⁵. Then it holds:

a) There exists a map T :V(Rⁿ, E₂)−→V(Rⁿ, E₀), such that i) T is linear.

ii) T

Bp,q^s+m:B_p,q^s+m(Rⁿ, E₂)−→B^s_p,q(Rⁿ, E₀) is continuous for all s∈R and p, q∈[1,∞].

iii) T

(^B^s+m^p,q ⁽Rⁿ,E2)∩C_b^∞(Rⁿ,E2)) =a(D)for all(s, p, q)∈R×[1,∞]×

[1,∞], where [a(D)u] (x) =os−

Z Z

e^iξ·ηa(ξ)•u(x−η)d(ξ, η)

(2π)ⁿ , u∈C_b^∞(Rⁿ, E2). b) LetS :V(Rⁿ, E2)−→V(Rⁿ, E0) be another map satisfaying i)-iii).

Then T =S. We denote this operator T by a](D).

Proof. The proof of a) is done in four steps:

I) We will show that for all (s,(p, q))∈R×[1,∞]² a(D) :

B_p,q^s+m(Rⁿ, E₂)∩C_b^∞(Rⁿ, E₂) ;k·k_Bs+m p,q (Rⁿ,E2)

→B_p,q^s (Rⁿ, E₀) is continuous. For this purpose one calculatesψ_j(D) (a(D)u) with u∈B_p,q^s+m(Rⁿ, E2)∩C_b^∞(Rⁿ, E2):

[ψj(D) (a(D)u)] (x) = [(ψja) (D)]u(x)

= (2π)⁻ⁿ²





(ψja)^∨∗_•χj(D)u

| {z }

∈Lp(Rⁿ,E0)





(x) , where χj := P1

i=−1ψj+i and ∗• is the convolution relative to the multiplication • (see [Am97]). From this and from Lemma 2.3 it follows for allu∈B_p,q^s+m(Rⁿ, E2)∩C_b^∞(Rⁿ, E2) that

ka(D)uk_Bs

p,q(Rⁿ,E0)≤C2kak_Sm,n+1

1,0 (Rⁿ,E1)kuk_Bs+m

p,q (Rⁿ,E2), withC2 := 3(2π)⁻ⁿ²C1.

II) Because we have the embedding

B_p,q^s (Rⁿ, E)∩C_b^∞(Rⁿ, E) ;k·k_Bs p,q

_d

,→B_p,q^s (Rⁿ, E) (27) for all 1 ≤ p ≤ ∞ and 1≤q < ∞ (see Lemma 2.2), there exists a unique linear continuous extension

as,p,q(D) :B_p,q^s+m(Rⁿ, E2)→B_p,q^s (Rⁿ, E0) of a(D) giving I).

5that is, a continuous bilinear map with an operator norm smaller than or equal to 1.

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III) One defines T :V(Rⁿ, E₂) −→V(Rⁿ, E₀) by T

Bp,q^s+m :=a_s,p,q(D) for all (s, p, q)∈R×[1,∞]×[1,∞) and by as−1,p,q(D) in the other cases. T is well-defined on V(Rⁿ, E2) and fulfillsa) for all q <∞.

IV) The case q =∞ can be treated using real interpolation theory.⁶ In fact, from B^s+m_p,∞ (Rⁿ, E2) = (B_p,1^s+m−1(Rⁿ, E2), B_p,1^s+m+1(Rⁿ, E2))¹

2,∞, B_p,∞^s (Rⁿ, E0) = (B_p,1^s−1(Rⁿ, E0), B^s+1_p,1 (Rⁿ, E0))¹

2,∞, and from B_p,1^s+m−1(Rⁿ, E2) −−−−−−−→^a^s−1,p,1^(D)

continuous B_p,1^s−1(Rⁿ, E0)

∪ ∪

B^s+m_p,∞ (Rⁿ, E₂) −−−−−−−→^a^s−1,p,1^(D) B_p,∞^s (Rⁿ, E₀)

∪ ∪

B_p,1^s+m+1(Rⁿ, E2) −−−−−−−→^a^s−1,p,1^(D)

continuous B_p,1^s+1(Rⁿ, E0)

it follows thatas−1,p,1(D) :B_p,∞^s+m(Rⁿ, E₂)→B_p,∞^s (Rⁿ, E₀) is conti- nouos. To proveb), letu∈V(Rⁿ, E₂). Then there exists a (s, p, q)∈ R×[1,∞]×[1,∞) such thatu∈B_p,q^s+m(Rⁿ, E2). From this, (27) and a) assertion b) follows.

Corollary 3.2. Let m, s∈R,1≤p, q≤ ∞ and B ∈{B,B, b}. Then˚

a7→a(D))] ∈ L

S_1,0^m,ρⁿ(Rⁿ, E₁),L B^m+s_p,q (Rⁿ, E₂),B_p,q^s (Rⁿ, E₀) with

a](D)u _B_s

p,q(Rⁿ;E0)≤C3kak_Sm,n+1

1,0 (Rⁿ;E1)kuk_Bs+m p,q (Rⁿ;E2)

for all u∈ B_p,q^s+m(Rⁿ;E₂), where C₃ = 3(2π)⁻ⁿ²C₁.

3.1. Analytic semigroups. In this subsection we will assume thatE1 ,→ E₀. Fors∈R,p, q∈[1,∞],m∈R⁺,B ∈{B,B, b}˚ anda∈S_1,0^m,ρⁿ(Rⁿ,L(E₁, E₀)), we denote A := a(D)]

_Bs+m

p,q (Rⁿ,E1) with domain D(A) =B_p,q^s+m(Rⁿ, E₁) and E :=B^s_p,q(Rⁿ, E₀). Since E₁,→E₀ andm >0, it holds that

D(A),→ B_p,q^s+m(Rⁿ, E₀),→ B_p,q^s (Rⁿ, E₀) =E.

From this and Corollary 3.2 it follows that

A:D(A)⊂E →E is a continuous linear operator. (28) These notations will be used in Proposition 3.5 and Corollary 3.6 to prove that the operator −A in (28) is sectorial and hence it generates an analytic semigroup on B_p,q^s (Rⁿ, E₀), when the symbol a is parabolic in the sense of Definition 3.4. In the following we introduce the notation ρ(A) for the resolvent set of A,R(λ, A) := (λI−A)⁻¹, with λ∈ρ(A), for the resolvent operator of A and

L_is(E1, E0) :={T ∈ L(E1, E0) :T is bijective}.

6A good summary of vector-valued Besov spaces and interpolation properties of these spaces can be found in [Am97], Chapter 5 and [Lu95], Chapter 1.

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Proposition 3.3. Let M >0, r≥0, θ∈[0,2π]be constants, Stθ,r:=

n

λ=µe^iθ:µ≥r o

a ray in C, and set

A_θ,M := {A∈ L(E₁, E₀) : (λI−A)∈ L_is(E₁, E₀) and P1

j=0(1 +|λ|)^1−jkR(λ, A)k_L(E

0,Ej)≤M for allλ∈Stθ,r}.

Then there exist constants ε_θ := ε(θ, M)>0, r_θ >0 and C4 := C4(M) :=

M

M+1max{2M + 3,4M² + 7M + 1} such that Σ^ε_θ,r^θ

θ ⊂ ρ(A) holds for all A∈ A_θ,M where

Σ^ε_θ,r^θ

θ :={λ∈C:|λ| ≥r_θ and arg(λ)∈[θ−ε_θ, θ+ε_θ]}. Moreover,

1

X

j=0

(1 +|λ|)^1−jkR(λ, A)k_L(E

0,Ej)≤C4 for all λ∈Σ^ε_θ,r^θ

θ and A∈ A_θ,M. Proof. First we will show two assertions.

Assertion I: Let θ = 0 in the hypotheses. Then there exist constants ε0 > 0, r0 > r and Mf:= ^M^(2M_M₊₁⁺³⁾ so that Σ^ε_0,r⁰₀ ⊂ ρ(A) for all A ∈ A_0,M and

kR(λ, A)k_L(E

0)≤ Mf

1 +|λ| ∀λ∈Σ^ε_0,r⁰₀ and A∈ A_0,M.

Proof of Assertion I: We select a r₀ > r,ε₀ := arcsin(_2(M¹₊₁₎) and define Σ^∗_ε₀_,r₀ :={λ∈C: Reλ≥r₀ and |arg(λ)| ≤ε₀}

and

s:= |λ|²

Reλ forλ∈Σ^∗_ε₀_,r₀. (29) Then it follows from |λ| = |iReλ−Imλ| and λ−s = ^Im^λ[i^Re_Re^λ−Im_λ ^λ] that for all λ∈Σ^∗_ε₀_,r₀:

1

s|λ−s|= Reλ

|λ|²

|Imλ| |iReλ−Imλ|

Reλ =

Imλ

|λ|

=|sin(arg (λ))| ≤ 1 2 (M+ 1) (Note that ε0 ∈ 0,^π₆

and f(t) = sin (t) is increasing in −^π₆,^π₆

). Thus

|λ−s| ≤ 1 +s

2 (M+ 1) for all λ∈Σ^∗_ε₀_,r₀. (30) It is clear due to the hypotheses that {s∈R:s≥r0} ⊂St0,r ⊂ρ(A) for all A∈ A_0,M. For allλ∈Σ^∗_ε₀_,r₀ and sas in (29) we get that

(λ−s) (sI−A)⁻¹ _L(E

0)

≤ |λ−s| M 1 +s ≤ 1

2 for all A∈ A_0,M. From this and a Neumann series argument it follows that B := I + (λ− s)(sI −A)⁻¹−1

∈ L(E₀) and kBk_L(E

0) ≤ 2. On the other hand we have that

λI−A= (sI−A)h

I+ (λ−s) (sI−A)⁻¹i

= (sI−A)B⁻¹.

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Therefore (λI−A)⁻¹∈ L(E0) and kR(λ, A)k_L(E

0)≤ kBk_L(E

0)kR(s, A)k_L(E

0)

≤ 2M 1 +s

= 2M

1 +|λ|

|λ−s+s|+ 1 1 +s

≤ 2M 1 +|λ|

|λ−s|

1 +s + 1

≤ Mf

1 +|λ| for all λ∈Σ^∗_ε₀_,r₀ and A∈ A_0,M

with Mf := ^M(2M_M+1⁺³⁾. Choosing er₀ ≥ r₀ appropriately, we obtain a sector Σ^ε_0,⁰

er0 ⊂Σ^∗_ε₀_,r₀ which finishes the proof of Assertion I.

Assertion II: An analogous result to Assertion I also is valid if the hypotheses are fulfilled for aθ∈(0,2π).

Proof of Assertion II: Letθ∈(0,2π) andO:C→Cdefined byO(z) :=

e^−iθz,z∈C, a rotation function. It is clear thatOis bijective andO⁻¹(z) = e^iθz, z ∈C. By hypotheses one knows that St_θ,r ⊂ρ(A) for all A ∈ A_θ,M and furthermore

kR(λ, A)k_L(E

0) ≤ M

1 +|λ| for allλ∈St_θ and A∈ A_θ,M. Thereforee^−iθA∈ A_0,M for allA∈ A_θ,M, becauseR µ, e^−iθA

=e^iθR(λ, A) for all λ=µe^iθ ∈St_θ ⊂ρ(A), and

1

X

j=0

(1 +|µ|)^1−j R

µ, e^−iθA _L(E

0,Ej) =

1

X

j=0

(1 +|λ|)^1−jkR(λ, A)k_L(E

0,Ej)≤M for all λ∈St_θ,r and A∈ A_θ,M. Thus, we get from the Assertion I that

R

w, e^−iθA _L(E

0)≤ Mf

1 +|w| for allw∈Σ^ε_0,r⁰

0 and A∈ A_θ,M. Now, let Σ^ε_θ,r^θ

θ :=O⁻¹ Σ^ε_0,r⁰₀

. Then for eachλ∈Σ^ε_θ,r^θ

θ (and hence e^−iθλ∈ Σ^ε_0,r⁰₀) and each A ∈ A_θ,M it holds that R(λ, A) = e^−iθR e^−iθλ, e^−iθA

∈ L(E₀). Then Σ^ε_θ,r^θ

θ ⊂ρ(A) and kR(λ, A)k_L(E

0) ≤ Mf

1 +|λ| for all λ∈Σ^ε_θ,r^θ

θ, A∈ A_θ,M.

Consequently Assertion II is proven. In order to finish the proof of this proposition it is sufficient to show that

kR(λ, A)k_L(E

0,E1)≤C₄ for all λ∈Σ^ε_θ,r^θ

θ and A∈ A_θ,M. (31) There C₄ := maxn

M , Mf

1 + 2Mfo

. For it, let A ∈ A_θ,M,λ ∈Σ^ε_θ,r^θ

θ and λ_θ :=r_θe^iθ ∈St_θ,r. From|λ_θ|=r_θ ≤ |λ|and

x= (λθ−A)⁻¹[(λ−A)x+ (λθ−λ)x] for all x∈E1

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it follows that kxk_E

1 ≤ kR(λθ, A)k_L(E

0,E1)

k(λ−A)xk_E

0+|λ_θ−λ| kxk_E

0

(32)

≤M

k(λ−A)xk_E

0 + 2|λ| kxk_E

0

(due to definition ofA_θ,M).

Moreover, it follows from Assertion II that kR(λ, A)yk_E

0 ≤ Mf

1 +|λ|kyk_E

0 for all y∈E₀, A∈ A_θ,M andλ∈Σ^ε_θ,r^θ

θ, and therefore for all x∈E₁ (and hence (λ−A)x∈E₀) we obtain

kxk_E

0 =kR(λ, A) [(λ−A)x]k_E

0 ≤ Mf

1 +|λ|k(λ−A)xk_E

0

for all A∈ A_θ,M, λ∈Σ^ε_θ,r^θ

θ. From this and (32) we get kxk_E

1 ≤M

k(λ−A)xk_E

0 + 2Mf |λ|

1 +|λ|k(λ−A)xk_E

0

≤M

1 + 2Mf

k(λ−A)xk_E

0

for all x ∈ E₁, A ∈ A_θ,M and λ ∈ Σ^ε_θ,r^θ

θ. As (λ−A) ∈ L_is(E₁, E₀), this

yields the assertion (31).

Definition 3.4. Let m∈ R⁺, ρ ∈N0 and a∈S_1,0^m,ρ(Rⁿ,L(E1, E0)). Then the symbol a is called parabolic in S_1,0^m,ρ(Rⁿ,L(E₁, E₀)) with constants ω andκ, if there are constantsω≥0andκ >0, so that for all(ξ, µ)∈Rⁿ×R⁺₀ with |ξ, µ| ≥ω andθ∈

−^π₂,^π₂

, a(ξ) +µ^me^iθI :E₁ −→E₀ is bijective, h

a(ξ) +µ^me^iθIi−1

∈ L(E₀, E₁)

and

h

a(ξ) +µ^me^iθIi−1

L(E0,E1)

≤κhξ, µi^−m. (33) Here, hξ, µi:=

q

1 +|ξ|²+µ² and |ξ, µ|:=

q

|ξ|²+µ².

In the following proposition, we denote for R≥0 and 0< θ≤π Σθ,R :={λ∈C:|λ| ≥R and |arg(λ)| ≤θ}, and in his proof we will use the inequality

1 +t^m ≤2 (1 +t)^m for all t≥0, (34) where m ≥ 0 (this inequality is obtained from 1 ≤ (1 +t)^m and t^m ≤ (1 +t)^m for allt≥0).

Proposition 3.5. Let s ∈ R, p, q ∈ [1,∞], m ∈ R⁺, ρ_n as in (3) and A ⊂ S_1,0^m,ρⁿ(Rⁿ,L(E₁, E₀)) be bounded. Moreover, assume that all a ∈ A are parabolic in S_1,0^m,ρⁿ(Rⁿ,L(E₁, E₀)) with the same constants ω ≥ 0 and κ >0, andA:= a(D)]

Bp,q^s+m(Rⁿ,E1),a∈ A. ThenΣ^π

2,R⊂ρ(−A)forR=ω^m and

(1 +|λ|)^1−j(λ+a)⁻¹ ∈S_1,0^−jm,ρⁿ(Rⁿ,L(E0, Ej)), j= 0,1, (35)