Pseudodifferential operators with non-regular operator-valued symbols

(1)

Universität Konstanz

Pseudodifferenzial operators with non-regular operator-valued symbols

Bienvenido Barraza Martínez Robert Denk

Jairo Hernández Monzón

Konstanzer Schriften in Mathematik Nr. 307, September 2012

ISSN 1430-3558

Fach D 197, 78457 Konstanz, Germany

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-202597

(2)

(3)

OPERATOR-VALUED SYMBOLS

BIENVENIDO BARRAZA MARTÍNEZ, ROBERT DENK, AND JAIRO HERN ÁNDEZ MONZ ÓN

Abstract. In this paper, we consider pseudodifferential operators with operator- valued symbols and their mapping properties, without assumptions on the un- derlying Banach spaceE. We show that, under suitable parabolicity assumptions, theW_p^k(Rⁿ, E)-realization of the operator generates an analytic semigroup. An application to non-autonomous pseudodifferential Cauchy problems gives the existence and uniqueness of a classical solution. Our approach is based on oscillatory integrals and kernel estimates for them.

1. Introduction

In this paper, we consider pseudodifferential operators with operator-valued symbols, their mapping properties, generation of an analytic semigroup and the application to non-autonomous vector-valued evolution equations. Operator-valued symbols and vector-valued function spaces appear in a natural way in several appli- cations, e.g. in models of coagulation-fragmentation processes where an additional parameter (the cluster size) appears in the model, see [Am00]. Another example is given by boundary value problems in cylindrical domains which can be treated by a Fourier multiplier approach which leads to operator-valued symbols, too (see [NS11]). Therefore, during the last decade, the investigation of vector-valued function spaces and differential equations with operator-valued coefficients has gained increasing interest.

Our work is motivated by two directions of research: On one hand, operator- valued Fourier multipliers inarbitrary Banach spaces and their mapping properties have been considered in [Am97]. Based on these Fourier multiplier results, the generation of an analytic semigroup for differential operators could be shown in [Am01]. On the other hand, under an additional geometric assumption on the Ba- nach space (to be a UMD space), Weis could establish in [We01] a vector-valued Mikhlin type theorem (see also [NNH02], [GW03]). This was the basis for a large number of results on maximal regularity for differential and pseudodifferential operators in UMD spaces. As references for operator-valued differential boundary value problems in UMD spaces and the connection toR-sectoriality and maximal regularity, we mention [DHP03] and [KW04]. For vector-valued pseudodifferential operators in UMD spaces, we refer to [DK07], [PS06] and the references therein.

Date: August 31, 2012.

2010Mathematics Subject Classification. 35S05, 47D06, 35R20.

Key words and phrases. Pseudodifferential operators, operator-valued symbols, generation of analytic semigroup.

1

(4)

The restriction to UMD spaces, however, excludes natural state spaces asL¹, and therefore the present paper deals withpseudodifferential operators with operator- valued symbols in arbitrary Banach spaces. We consider operator-valued symbols ain the standard H¨ormander class with limited smoothness both in the variablex and in the covariableξand with positive orderm >0. Similar operators were also considered in [Ki01],[Ki03] with additional assumptions on the symbol (e.g., order m >1, existence of a homogeneous principal part, infinite smoothness inξ).

One of the main results in the present paper states that, under suitable parabolicity assumptions, theW_p^k(Rⁿ, E)-realization of the symbolagenerates an analytic semigroup in W_p^k(Rⁿ, E), see Theorem 4.2 and Corollary 4.3 below. An application to non-autonomous pseudodifferential Cauchy problems gives the existence and uniqueness of a classical solution (Theorem 5.3). Our approach is based on oscillatory integrals and careful kernel estimates for them, the technical key result being Lemma 3.2. For the case of constant coefficients, results on vector-valued pseudodifferential operators were obtained in our paper [BDH12].

2. Vector-valued pseudo-differential operators In the following, we set |x| := Pn

j=1|xj|²)^1/2, |x, µ| := (|x|²+µ²)^1/2, hxi:=

(1 +|x|²)^1/2, and hx, µi := (1 +|x|² +µ²)^1/2 for x = (x₁, . . . , x_n) ∈ Rⁿ and µ∈R. Throughout this paper, (E,k · k) denotes an arbitrary Banach space, and for locally convex spacesX, Y we writeL(X, Y) for the space of all continuous linear operators fromX toY, and setL(X) :=L(X, X). LetC_b^∞(Rⁿ, E) be the space of allu:Rⁿ→Esuch that∂^αuis bounded and continuous for allα∈Nⁿ0 where we use standard multi-index notation. The spaceC_b^∞(Rⁿ, E) is endowed with the locally convex topology given by the seminorms kukk := max_|α|≤ksup_x∈_Rnk∂^αu(x)kE, k∈N0. The Schwartz space of rapidly decreasingE-valued functions is denoted by S(Rⁿ, E), and fork ∈N0 and 1≤p <∞, we write W_p^k(Rⁿ, E) for the E-valued Sobolev space endowed with the norm

kuk_Wk

p(Rⁿ,E):= X

|α|≤k

k∂^αuk^p_Lp(Rⁿ,E)

1/p

.

Here k · k_Lp(Rⁿ,E) stands for the norm in the Lebesgue-Bochner space L^p(Rⁿ, E).

We setRⁿ+:=Rⁿ×[0,∞),D^α:= (−i)^|α|∂^α andd¯(ξ, y) := (2π)⁻ⁿd(ξ, y).

We start with the definition of the symbol class (which is a non-smooth parameter- dependent version of the standard H¨ormander classS_1,0^m) and the related pseudodifferential operators. In the following, for n ∈ N and m ∈ R, we set ρn :=

(n+ 1, ifnis odd,

n+ 2, ifnis even. andρn,m:=

([n+m] + 1, if [n+m] is odd, [n+m] + 2, if [n+m] is even.

Definition 2.1. Let m∈R,ν ∈[0,∞], andr∈N⁰.

a) We defineS^m,ν,r :=S^m,ν,r(Rⁿx×Rⁿ+, L(E))as the set of all functionsa:Rⁿx× Rⁿ+→L(E)such that

(i) x7→a(x, ξ, µ)∈C^r(Rⁿ, L(E))for all(ξ, µ)∈Rⁿ+, (ii) (ξ, µ)7→a(x, ξ, µ)∈C^[ν](Rⁿ+, L(E))for all x∈Rⁿ, and (iii) |a|^(ν,r)m <∞ where

|a|^(ν,r)_m := sup

x∈Rⁿ

sup

(ξ,µ)∈Rⁿ₊

max

|α|+k≤νmax

|β|≤rhξ, µi^−m+|α|+k

∂_x^β∂_ξ^α∂_µ^ka(x, ξ, µ) _L(E).

(5)

b) Fora∈S^m,ν,rwithν ≥ρn, the pseudo-differential operatorop(a) :=a(x, D, µ) is defined by

[a(x, D, µ)u](x) := Os− Z Z

R²ⁿ

e^iξ·ya(x, ξ, µ)u(x−y)d¯(ξ, y) (x∈Rⁿ)

foru∈C_b^∞(Rⁿx, E).

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

In the following, we will also study parameter-dependent double symbols a = a(x, y, ξ, µ) witha∈S^m,ν,r(R²ⁿx,y×Rⁿ+, L(E)) withν ≥ρ_n andr > ρ_n,m. Here we set

(1) [a(x, y, D, µ)u](x) := Os− Z Z

R²ⁿ

e^iξ·ya(x, x−y, ξ, µ)u(x−y)d¯(ξ, y) (x∈Rⁿ)

foru∈C_b^∞(Rⁿ, E). In the particular case wherea(x, y, ξ, µ) does not depend onx (dual symbol), we will writea(y, D, µ).

Remark 2.2. a) Forν≥ρ_n, Lemma A.4 shows that the oscillatory integral in b) exists and due to Lemma A.5 too, that a(x, D, µ)∈L(C_b^∞(Rⁿ, E), C_b^r(Rⁿ, E))for every fixedµ∈[0,∞).

b) Forν ≥ρ_n andr > ρ_n,m, the oscillatory integral in (1) exists. Note that for n+m≤0 the conditionr > ρn,m is satisfied for allr∈N.

The following result is taken from [Ki03], Remark 2.2.6.

Lemma 2.3. Let a_i ∈ S^mⁱ^,ν,r, i = 1,2, with m₁, m₂ ∈ R, ν ≥ ρ_n and r ≥ ρ_2n,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 leta∈ S^m,ν,r. Then ais called parameter-elliptic if there are constants κ > 0 and ω ≥0 such that for all (x, ξ, µ)∈Rⁿ×Rⁿ×[0,∞)with|ξ, µ| ≥ω we have

(i) a(x, ξ, µ) :E→E is bijective, and (ii) ka(x, ξ, µ)⁻¹k_L(E)≤κhξ, µi^−m.

The set of alla∈S^m,ν,r satisfying (i)–(ii) will be denoted byE_κ,ω^m,ν,r:=E_κ,ω^m,ν,r(Rⁿx× Rⁿ+, L(E)).

b) Letm∈(0,∞)andr, ν∈N0, and let a∈S^m,ν,r(Rⁿx×Rⁿξ, L(E))be a symbol independent of µ∈(0,∞). Thenais called parabolic with constants κ >0,ω≥0 if (x, ξ, µ)7→a(x, ξ) +µ^me^iθ belongs toE_κ,ω^m,ν,r for all|θ| ≤ ^π₂.

For a symbol a ∈E_κ,ω^m,ν,r, we define a smooth version of the inverse symbol in the following way: Letψ ∈C^∞(Rⁿ⁺¹) with 0≤ψ≤1,ψ(ξ, µ) = 0 for |ξ, µ| ≤ ¹₂ andψ(ξ, µ) = 1 for|ξ, µ| ≥1 be a zero extinction function. Forω0> ω we set

a^#(x, ξ, µ) :=

(ψ(_2ω^ξ

0,_2ω^µ

0)a(x, ξ, µ)⁻¹, ifx∈Rⁿ,|ξ, µ| ≥ω0, 0, ifx∈Rⁿ,|ξ, µ| ≤ω₀. Thena^#(x, ξ, µ) = 0 if|ξ, µ| ≤ω₀anda^#(x, ξ, µ) =a(x, ξ, µ)⁻¹ if|ξ, µ| ≥2ω₀.

(6)

3. Key estimates for parameter-elliptic symbols

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

Lemma 3.1. Let m, ν∈(0,∞),r∈N0, and letA ⊂S^m,ν,r be bounded, i.e. there exists a constant K > 0 such that |a|^(ν,r)m ≤K (a ∈ A). Assume moreover that A ⊂E_κ,ω^m,ν,r. Then there exists a constantC=C(K, κ, ν, r)>0 such that

k∂_x^β∂_ξ^α∂^k_µa^#(x, ξ, µ)kL(E)≤Chξ, µi^{−m−|α|−k}

for all x∈Rⁿ,(ξ, µ)∈Rⁿ+,|α|+k≤ν,|β| ≤r, and a∈ A, i.e., {a^# :a∈ A} ⊂ S^−m,ν,r is bounded.

Proof. Note that{hξ, µi^m+|α|+kk∂^β_x∂_ξ^α∂^k_µa^#(x, ξ, µ)k_L(E):x∈Rⁿ,|ξ, µ| ≤2ω0,|α|+

k≤ν,|β| ≤r} is bounded by continuity and compactness. Therefore, it suffices to consider the case|ξ, µ| ≥2ω0.

Letx∈Rⁿ,|ξ, µ| ≥2ω0, α, β ∈Nⁿ0 \ {0}, k∈N0 with |β| ≤rand |α|+k≤ν.

Then∂_x^β∂_ξ^α∂_µ^ka(x, ξ, µ)⁻¹ is a finite linear combination of terms of the form (2) a(x, ξ, µ)⁻¹

∂_x^β⁽¹⁾∂_ξ^α⁽¹⁾∂_µ^k⁽¹⁾a(x, ξ, µ)

a(x, ξ, µ)⁻¹. . . . . .

∂_x^β^(p)∂_ξ^α^(p)∂_µ^k^(p)a(x, ξ, µ)

a(x, ξ, µ)⁻¹

with 1≤p≤ |α|+|β|+kand withα⁽ⁱ⁾, β⁽ⁱ⁾∈Nⁿ0,k⁽ⁱ⁾∈N(i= 1, . . . , p) satisfying Pp

i=1α⁽ⁱ⁾=α,Pp

i=1β⁽ⁱ⁾=β,Pp

i=1k⁽ⁱ⁾=k. The norm of the operator in (2) can be estimated by

ka(x, ξ, µ)⁻¹k^p+1_L(E)

p

Y

i=1

∂^β_x⁽ⁱ⁾∂_ξ^α⁽ⁱ⁾∂_µ^k⁽ⁱ⁾a(x, ξ, µ) _L(E)

≤κ^p+1hξ, µi^−m(p+1)

p

Y

i=1

|a|^(ν,r)_m hξ, µi^m−|α⁽ⁱ⁾^|−k⁽ⁱ⁾

=κ^p+1 |a|^(ν,r)_m ^p

hξ, µi^{−m−|α|−k}. Summing up over all terms of the form (2), we obtain

k∂_x^β∂^α_ξ∂_µ^ka⁻¹(x, ξ, µ)k_L(E)≤Chξ, µi^{−m−|α|−k}

with a constantC=C(K, κ, ν, r).

From now on we set χε(ξ, y) := χ(εξ)χ(εy) for ε > 0 and ξ, y ∈ Rⁿ, where χ∈S(Rⁿ) withχ(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. For ε∈(0,1) define

K_ε(x, y, µ) :=

Z

Rⁿ

e^iξ·yχ_ε(ξ, y)b(x−y, ξ, µ)dξ (x, y, µ)∈R²ⁿ×[ω₀,∞) . a) There exists a constantC such that

kKε(x, y, µ)kL(E)≤C|b|^(ν,r)_−mµ^−m+n |µy|^θ⁰+|µy|^θ¹

|µy|ⁿ(1 +|µy|) (x, y, µ)∈R²ⁿ×[ω₀,∞)

(7)

whereθ0:= ¹₂min{m,1}andθ1:= ¹₂. Moreover,

(3) kKε(x,·, µ)kL¹(Rⁿ,L(E))≤C|b|^(ν,r)_−mµ^−m (x, µ)∈Rⁿ×[ω0,∞) holds with a constantC independent of ε,xandµ.

b) There exists a strongly measurable function K: R²ⁿ×[ω₀,∞)→L(E) with K_ε(x, y, µ) → K(x, y, µ) (ε & 0) pointwise, and the estimate (3) holds with K_ε being replaced byK.

Proof. a) Substitutingξ7→µξ, we have K_ε(x, y, µ) =µⁿ

Z

Rⁿ

e^iµξ·yχ_ε(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

Z

Rⁿ

D_ξ^γ

χε(µξ, y)b(x−y, µξ, µ) dξ= 0.

With this and partial integration, we see (µy)^γKε(x, y, µ) =µⁿ

Z

Rⁿ

(e^iµξ·y−1)D_ξ^γ

χε(µξ, y)b(x−y, µξ, µ) dξ.

To estimate the integrand, we apply the Leibniz rule noting that forα+β =γwe have

D^α_ξχ_ε(µξ, y) =

D^α_ξχ(εµξ)

|χ(εy)| ≤c_αµ^|α|hµξi^−|α|

by Lemma A.3. This and

(4) kD_ξ^β[b(x−y, µξ, µ)]kL(E)=µ^|β|k(D_ξ^βb)(x−y, µξ, µ)kL(E)

≤µ^|β||b|^(ν,r)_−mhµξ, µi^−m−|β|

gives

D^γ_ξ

χε(µξ, y)b(x−y, µξ, µ) _L(E)

≤ X

α+β=γ

cαβµ^|α|hµξi^−|α||µ|^|β|hµξ, µi^−m−|β||b|^(ν,r)_−m

≤cγµ^|γ|hµξi^−m−|γ||b|^(ν,r)_−m

where we usedhµξ, µi^−m−|β|≤ hµξi^−m−|β|. Now we apply the elementary estimate

|e^iµξy−1| ≤2|µy|^θ|ξ|^θ valid for allθ∈(0,1) and obtain k(µy)^γKε(x, y, µ)k_L(E)≤Cµ^n+|γ||µy|^θ|b|^(ν,r)_−m

Z

Rⁿ

|ξ|^θhµξi^−m−|γ|dξ

=Cµ^n−m|µy|^θ|b|^(ν,r)_−mI(θ,|γ|) withI(θ,|γ|) :=R

Rⁿ|ξ|^θ(µ⁻²+|ξ|²)^{−m/2−|γ|/2}dξ.

For|γ|=n+iwithi∈ {0,1}andθ0:=¹₂min{m,1},θ1:= ¹₂, we haveθi∈(0,1) and

I(θi,|γ|) = Z

|ξ|≤1

|ξ|^θⁱ(µ⁻²+|ξ|²)^{−m/2−|γ|/2}dξ +

Z

|ξ|≥1

|ξ|^θⁱ(µ⁻²+|ξ|²)^{−m/2−|γ|/2}dξ

(8)

≤ω₀^−m−|γ|

Z

|ξ|≤1

|ξ|^θⁱdξ+ Z

|ξ|≥1

|ξ|^θⁱ^−m−|γ|dξ≤C <∞

due toθi>0 andθi−m− |γ|<−n. Therefore, for allγwith|γ|=n+i,i∈ {0,1}, we have

k(µy)^γKε(x, y, µ)k_L(E)≤Cµ^n−m|µy|^θⁱ|b|^(ν,r)_−m and consequently

|µy|ⁿ⁺ⁱkK_ε(x, y, µ)k_L(E)≤n^(n+i)/2 X

|γ|=n+i

k(µy)^γK_ε(x, y, µ)k_L(E)

≤Cµ^n−m|µy|^θⁱ|b|^(ν,r)_−m.

Summing up these inequalities fori= 0 andi= 1, we obtain the first assertion in a). Since the function

y7→µⁿ |µy|^θ⁰+|µy|^θ¹

|µy|ⁿ(1 +|µy|)

belongs toL¹(Rⁿ) and itsL¹-norm does not depend onµ, we obtain inequality (3).

b) Letε, ε⁰∈(0,1), ξ, y∈Rⁿ andµ∈[ω0,∞). From the proof of a) we see that (5)

(µy)^γ(Kε(x, y, µ)−Kε⁰(x, y, µ)

=µⁿ Z

Rⁿ

(e^iµξ·y−1)D_ξ^γ

χε(µξ, y)−χε⁰(µξ, y)

b(x−y, µξ, µ) dξ.

From Lemma A.3 we know that D^α_ξ χε(µξ, y)−χε⁰(µξ, y)

→0 for ε, ε⁰ & 0 for all α∈ Nⁿ0 and all µ, ξ, y. Therefore the integrand in (5) converges pointwise to zero for ε, ε⁰ & 0. By a), we haveKε(x,·, µ)∈ L¹(Rⁿ, L(E)) with a dominating function independent of ε, and by dominated convergence we see that for fixed (x, y, µ)∈ R²ⁿ×[ω₀,∞) we get kKε(x, y, µ)−K_ε⁰(x, y, µ)kL(E) → 0 (ε, ε⁰ & 0).

Therefore there exists a strongly measurable function K: R²ⁿ×[ω₀,∞) →L(E) such thatK_ε→K(ε&0) pointwise. By dominated convergence again, inequality

(3) holds forK(x,·, µ) instead ofK_ε(x,·, µ).

Theorem 3.3. Letm >0,1≤p <∞,r, k, ν∈N0withr≥ρ_n,−m+k+ 1,ν ≥ρn, andω₀>0. Let b∈S^−m,ν,r. Then b(y, D, µ)∈L(W_p^k(Rⁿ, E))and we have

kb(y, D, µ)k_L(Wk

p(Rⁿ,E))≤Cµ^−m|b|^(ν,r)_−m (µ∈[ω0,∞)) with a constantC independent of bandµ.

Proof. (i) Consider first the casek= 0. Foru∈S(Rⁿ, E) we have (b(y, D, µ)u)(x) = Os−

Z Z

R²ⁿ

e^iξ·yb(x−y, ξ, µ)u(x−y)d¯(ξ, y)

= lim

ε&0

Z

Rⁿ

Z

Rⁿ

e^iξ·yχε(ξ, y)b(x−y, ξ, µ)u(x−y)d¯ξd¯y

= lim

ε&0

Z

Rⁿ

Kε(x, y, µ)u(x−y)d¯y with K_ε(x, y, µ) := R

Rⁿe^iξ·yχ_ε(ξ, y)b(x−y, ξ, µ)d¯ξ. By Lemma 3.2, K_ε(x,·, µ) ∈ L¹(Rⁿ, L(E)) withε-independent dominating function, andK_εconverges pointwise

(9)

to a strongly measurable function K for ε & 0. By the dominated convergence theorem, we have

(b(y, D, µ)u)(x) = Z

Rⁿ

K(x, y, µ)u(x−y)d¯y= [K(x,·, µ)∗u](x)

where ∗ stands for the standard convolution. Because of theL¹-estimate of K in Lemma 3.2, we obtain forµ∈[ω0,∞)

kb(y, D, µ)ukL^p(Rⁿ,E)≤CkK(x,·, µ)kL¹(Rⁿ,L(E))kukL^p(Rⁿ,E)

≤Cµ^−m|b|^(ν,r)_−mkuk_Lp(Rⁿ,E).

(ii) Now letk∈N. By Lemma A.5, forβ ∈Nⁿ0 with|β| ≤kwe write

∂_x^β[b(y, D, µ)u]

(x) = Os− Z Z

R²ⁿ

e^iξ·y∂_x^β

b(x−y, ξ, µ)u(x−y) d¯(ξ, y)

= X

β⁰≤β

cββ⁰Os− Z Z

R²ⁿ

e^iξ·y(∂_x^β⁰b)(x−y, ξ, µ)(∂_x^β−β⁰u)(x−y)d¯(ξ, y).

From∂_x^β⁰b∈S^m,ν,r−|β⁰^|and part (i) of the proof we obtain k∂_x^β[b(y, D, µ)u]k_Lp(Rⁿ,E)≤Cµ^−m|b|^(ν,r)_−m X

|γ|≤k

k∂^γuk_Lp(Rⁿ,E)

≤Cµ^−m|b|^(ν,r)_−mkuk_Wk p(Rⁿ,E).

Remark 3.4. The last result shows that the mapping b 7→ b(y, D, µ), S^−m,ν,r → L(W_p^k(Rⁿ, E))is continuous with norm not greater than a constant timesµ^−mfor µ∈[ω0,∞).

Corollary 3.5. In the situation of Lemma 3.1, we havea^#(y, D, µ)∈L(W_p^k(Rⁿ, E)) and

sup

a∈A, µ≥0

ka^#(y, D, µ)k_L(Wk

p(Rⁿ,E))<∞ for allk∈N0 withr≥ρn,−m+k+ 1.

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

4. Generation of an analytic semigroup

We will need the following slight generalization of Lemma 3.2.

Lemma 4.1. Letm >0,m1, m2∈Rwithm1+m2=−m,ν, r∈N0withν≥n+1, ω0>0. Let b1∈S^m¹^,ν,r andb2∈S^m²^,ν,r. For ε∈(0,1)define

Kε(x, y, µ) :=

Z

Rⁿ

e^iξ·yχε(ξ, y)b(x, y, ξ, µ)dξ with b(x, y, ξ, µ) := R1

0 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 .

(10)

Proof. In the proof of Lemma 3.2, we replace inequality (4) by (6) kD_ξ^β[b(x, y, µξ, µ)]k_L(E)≤µ^|β||b₁|^(ν,r)_m

1 |b₂|^(ν,r)_m

2 hµξ, µi^−m−|β|. This can be seen by

kD^β_ξ[b(x, y, µξ, µ)]kL(E)

≤µ^|β| X

β⁰≤β

c_ββ0

Z 1

0

(D_ξ^β⁰b₁)(x−τ y, µξ, µ)(D_ξ^β−β⁰b₂)(x−y, µξ, µ)dτ _L(E)

≤µ^|β| X

β⁰≤β

cββ⁰|b1|^(ν,r)_m₁ hµξ, µi^m¹^−|β⁰^||b2|^(ν,r)_m₂ hµξ, µi^m²^{−(|β|−|β}⁰^|)

≤µ^|β||b₁|^(ν,r)_m

1 |b₂|^(ν,r)_m

2 hµξ, µi^−m−|β|.

With (6) instead of (4) we can follow the proof of Lemma 3.2 almost literally.

Throughout the remainder of this section, we fix k ∈N0, m > 0, 1≤ p <∞, r, ν ∈N0 withr≥ρn,m+k+ 1 andν ≥n+ 2. We will consider theW_p^k(Rⁿ, E)- realizationA_k of a(x, D, µ), i.e. the restriction of a(x, D, µ) to D_max(A_k) :={u∈ W_p^k(Rⁿ, E) :a(x, D, µ)u∈W_p^k(Rⁿ, E)}.

Theorem 4.2. Let a ∈ S^m,ν,r be parameter-elliptic, i.e., a ∈ E_κ,ω^m,ν,r for some κ > 0 and ω ≥ 0. Then there exist constants M₁ > 0 and µ₁ >0 such that the W_p^k-realizationA_k of a(x, D, µ)is invertible for µ≥µ₁ and

kA⁻¹_k k_L(Wk

p(Rⁿ,E))≤ M1

1 +µ^m (µ≥µ1).

Proof. Letω0> ω, µ≥2ω0, andu∈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) Letx∈Rⁿ. By a Taylor expansion we obtain

p(x, y, D, µ)u (x)

= Os− Z Z

R²ⁿ

e^iξ·y a(x, ξ, µ)−a(x−y, ξ, µ)

a(x−y, ξ, µ)⁻¹u(x−y)d¯(ξ, y)

= Os− Z Z

R²ⁿ

e^iξ·y

n

X

j=1

y_j Z 1

0

(∂x_ja)(x−τ y, ξ, µ)a(x−y, ξ, µ)⁻¹dτ u(x−y)d¯(ξ, y)

=

n

X

j=1

Os− Z Z

R²ⁿ

e^iξ·y Z 1

0

Dξ_j

(∂xja)(x−τ y, ξ, µ)a(x−y, ξ, µ)⁻¹

dτ u(x−y)d¯(ξ, y)

=

n

X

j=1 ε&0lim

Z

Rⁿ

K_ε^(j,1)(x, y, µ) +K_ε^(j,2)(x, y, µ)

u(x−y)d¯y with

K_ε^(j,m)(x, y, µ) :=

Z

Rⁿ

e^iξ·yχε(ξ, y)b^(j,m)(x, y, ξ, µ)d¯ξ,

(11)

b^(j,1)(x, y, ξ, µ) :=

Z 1

0

(Dξ_j∂x_ja)(x−τ y, ξ, µ)a(x−y, ξ, µ)⁻¹dτ, b^(j,2)(x, y, ξ, µ) :=

Z 1

0

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

We haveDξ_j∂x_ja∈Sm−1,r−1,ν−1 and Dξ_ja⁻¹ ∈S^{−m−1,r,ν−1}. By Lemma 4.1 we obtain forj= 1, . . . , nandm= 1,2 the existence of a strongly measurable function K^(j,m) withKε^(j,m)→K^(j,m)(ε&0) pointwise and

kK^(j,m)(x,·, µ)k_L1(Rⁿ,L(E)) ≤Cµ⁻¹|a|^(ν,r)_m |a^#|^(ν,r)_−m forµ≥2ω0. Therefore,

kp(x, y, D, µ)ukL^p(Rⁿ,E)≤

n

X

j=1 2

X

m=1

k[K^(j,m)(x,·, µ)∗u]kL^p(Rⁿ,E)

≤Cµ⁻¹|a|^(ν,r)_m |a^#|^(ν,r)_−mkuk_Lp(Rⁿ,E).

(ii) Now letβ ∈Nⁿ0 with|β| ≤k. In the same way as in the proof of part (ii) of Theorem 3.3, we write

(∂^β_xp(x, y, D, µ)u)(x)

= X

β⁰≤β

c_ββ⁰Os− Z Z

R²ⁿ

e^iξ·y(∂_x^β⁰p)(x, y, ξ, µ)(∂_x^β−β⁰u)(x−y)d¯(ξ, y).

Replacing in (i)pby∂^β_x⁰panduby∂_x^β−β⁰u, we obtain k∂_x^β[p(x, y, D, µ)u]kL^p(Rⁿ,E)≤C X

β⁰≤β

k(∂_x^β⁰p)(x, y, D, µ)(∂_x^β−β⁰u)kL^p(Rⁿ,E)

≤C X

β⁰≤β

µ⁻¹|a|^(ν,r)_m |a^#|^(ν,r)_−m X

|γ|≤k

k∂_x^γuk_Lp(Rⁿ,E)

and therefore kp(x, y, D, µ)uk_Wk

p(Rⁿ,E) ≤Cµ⁻¹|a|^(ν,r)m |a^#|^(ν,r)_−mkuk_Wk

p(Rⁿ,E) forµ≥ 2ω₀ with a constant C independent of a and µ. Taking µ ≥ µ₁ with µ₁ sufficiently large, we obtain kp(x, y, D, µ)kL(W_p^k(Rⁿ,E)) ≤ ¹₂. By a Neumann series argument, we see thatI+p(x, y, D, µ)∈L(W_p^k(Rⁿ, E)) is invertible and therefore a(x, D, µ)a^#(y, D, µ)(I+p(x, y, D, µ))⁻¹= id_Wk

p(Rⁿ,E)forµ≥µ1. This shows that Ak is surjective.

(iii) In the same way, we see thata^#(x, D, µ)a(y, D, µ) =I+ ˜p(x, y, D, µ) with

˜

p(x, y, ξ, µ) := (a(x, ξ, µ)⁻¹−a(y, ξ, µ)⁻¹)a(y, ξ, µ), and that kp(x, y, D, µ)k˜ _L(Wk

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

˜

p(x, y, D, µ))⁻¹a^#(x, D, µ)a(y, D, µ) = id_D_max_(A_k₎. This shows thatAk is injektive, and we obtain the invertibility ofAk. Moreover, we have for sufficiently largeµ

kA⁻¹_k k_L(Wk

p(Rⁿ,E))≤ ka^#(y, D, µ)k_L(W^p

k(Rⁿ,E))k(I+p(x, y, D, µ))⁻¹k_L(Wk p(Rⁿ,E))

≤ 2M0

1 +µ^m

forµ≥µ₁ withµ₁sufficiently large, whereM₀ does not depend onµ.

(12)

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

Corollary 4.3. Let a∈S^m,ν,r(Rⁿx×Rⁿξ, L(E))be parabolic with constants κ >0 and ω ≥ 0, and let Ak denote the W_p^k(Rⁿ, E)-realization of a(x, D). Then there are constantsM₂>0,ϑ∈(^π₂, π)andR >0 such that for the resolvent setρ(−Ak) of −A_k we have Σ_ϑ,R⊂ρ(−A_k),

(7) k(λ+A_k)⁻¹kL(E)≤ M₂

1 +|λ| (λ∈Σ_ϑ,R).

Therefore, −Ak:W_p^k(Rⁿ, E)⊃ D(Ak)→ W_p^k(Rⁿ, E) generates an analytic semigroup onW_p^k(Rⁿ, E).

Proof. By Definition 2.4 b), the symbola(x, ξ)+µ^me^iθis parameter-elliptic for allθ withθ∈Σ^π

2,R. By a standard continuity argument, we see that the set of anglesθ where parameter-ellipticity holds is open. Therefore, we may assume that the above symbol is parameter-elliptic for allθ ∈Σ_ϑ,R with ϑ > ^π₂. Now the invertibility of λ+Ak,λ=µ^me^iθ and the estimate (7) follows directly from Theorem 4.2.

Remark 4.4. The proof of Theorem 4.2 shows that the constants µ1 and M1 in Theorem 4.2 can be chosen independently of a for alla∈ A where A ⊂S^m,ν,r is bounded andA ⊂E^m,ν,r_κ,ω for fixedκ >0andω≥0. In the same way, the constants ϑ, M₂, and R in Corollary 4.3 can be chosen independently of a for all a ∈ A⁰ where A⁰ ⊂S^m,ν,r(Rⁿx×R_ξⁿ, L(E)) is bounded and where all a∈ A⁰ are parabolic with the same constantsκ >0 andω≥0.

5. An application

Let T > 0, J := [0, T] a closed interval in R and t ∈ J. In the following t in a(t,·)∈S^m,ν,r R²ⁿ, L(E)

and ina(t, x, D) denotes only a parameter. Moreover we will consider in this section a family A := {a(t,·) :t∈J} ⊂ E^m,ν,r_κ,ω for fixed κ >0 andω≥0, such that

J 3t7−→a(t,·)∈S^m,ν,r R²ⁿ, L(E)

is a H¨older 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 uniqueness of solutions for the Cauchy problem

(8)

∂_tu+A_k(t)u=f(t), t∈J\ {0}, u(0) =u₀,

inW_p^k(Rⁿ, E). There,Ak(t),t∈J, is theW_p^k(Rⁿ, E)-realization ofa(t, x, D) and f : [0, T]→W_p^k(Rⁿ, E) is a given H¨older continuous function. A function

u∈C¹(]0, T], E)∩C([0, T], E)

is calledclassical solutionof (8) in [0, T], ifu(t)∈D(Ak(t)) andu⁰(t)+Ak(t)u(t) = f(t) for allt∈]0, T], andu(0) =u0. More precisely, we will use Corollary 4.3, The- orem 2.5.1 of Chapter IV in [Am95] and Lemma 5.2 to obtain results of existence and uniqueness of solutions of the Cauchy problem (8) inW_p^k(Rⁿ, E) if 1≤p <∞.

In the following ˚B_p,q^s (Rⁿ, E),s∈R,p, q∈[1,∞], denote theE-valued homogeneous Besov spaces of ordersand parameterspandq(see [Am97] for their definition and

(13)

properties).

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

a(x, D) : ˚B^s+m_p,q (Rⁿ, E)→B˚_p,q^s (Rⁿ, E)

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

Moreover it holds the following estimate:

ka(x, D)k_L(^B^˚^s+mp,q (Rⁿ,E),B˚_p,q^s (Rⁿ,E))≤c|a|^(2n+1,r)_m . Proof. See [Ki01], Theorem 3.5.

Lemma 5.2. Let m ∈ R, ν ≥ 2n+ 1, r ≥ ρn,m+k+ 1, p ∈ [1,∞) and A :=

{a(t,·) ; t∈J} ⊂S^m,ν,r with Ak(t)the W_p^k(Rⁿ, E)-realization ofa(t, x, D). Then (9) B˚^k+m_p,1 (Rⁿ, E),→D(Ak(t)),→^d W_p^k(Rⁿ, E).

Proof. It’s clear thatD(A_k(t)),→W_p^k(Rⁿ, E) (D(A_k(t)) endowed with the graph norm) and it’s also known that

(10) S(Rⁿ, E),→B˚_p,1^k+m(Rⁿ, E),→B˚_p,1^k (Rⁿ, E),→W_p^k(Rⁿ, E), 1≤p <∞.

From this and Lemma 5.1 it follows that u, a(t, x, D)u ∈ W_p^k(Rⁿ, E) whenever u∈S(Rⁿ, E). Therefore S(Rⁿ, E)⊂D(Ak(t)). But S(Rⁿ, E),→^d W_p^k(Rⁿ, E) for 1≤p <∞and then we haveD(Ak(t)),→^d W_p^k(Rⁿ, E).

On the other hand, (10) and Lemma 5.1 imply that ˚B_p,1^k+m(Rⁿ, E)⊂D(Ak(t)) and kukD(A_k(t))=kuk_Wk

p(Rⁿ,E)+ka(t, x, D)uk_Wk p(Rⁿ,E)

≤c1kukB˚_p,1^k+m(Rⁿ,E)+c2ka(t, x, D)ukB˚^k_p,1(Rⁿ,E)

≤c1kukB˚_p,1^k+m(Rⁿ,E)+c3kukB˚_p,1^k+m(Rⁿ,E)

=ckukB˚^k+m_p,1 (Rⁿ,E). So we have ˚B^k+m_p,1 (Rⁿ, E),→D(Ak(t)).

Theorem 5.3. Letk∈N0,α, m∈R⁺ with0< α <1,p∈[1,∞),ν ≥2n+ 1and r ≥ρn,m+k+ 1. Furthermore for fixed κ >0 andω ≥0, letA :={a(t,·) ;t ∈ J} ⊂E_κ,ω^m,ν,r be a family of parabolic symbols, such that

(11) t7−→a(t,·)∈C^α J, S^m,ν,r(R²ⁿ, L(E)) and such that there is a constantM with

(12) k(λ+Ak(t))⁻¹k_{L( ˚}_Bk

p,∞(Rⁿ,E),B˚p,∞^k+m(Rⁿ,E))≤M,

for all t ∈ J and λ in the sector Σϑ,R of Corollary 4.3. There Ak(t) is the W_p^k(Rⁿ, E)-realization ofa(t, x, D).

If u0 ∈ W_p^k(Rⁿ, E) and f ∈ C^σ J, W_p^k(Rⁿ, E)

for some σ ∈ ]0,1[, then the Cauchy problem (8) has a unique solution

u∈C J, W_p^k(Rⁿ, E)

∩C¹ J\ {0}, W_p^k(Rⁿ, E) .

(14)

Proof. First we claim that D(Ak(t)) ,→ B˚_p,∞^k+m(Rⁿ, E). In fact, if u ∈ D(Ak(t)), then u, a(t, x, D)u ∈ W_p^k(Rⁿ, E) ⊂ B˚_p,∞^k (Rⁿ, E). For λ0 ∈ Σϑ,R fixed and v :=

(λ₀+A_k(t))u∈B˚_p,∞^k (Rⁿ, E), (12) implies thatu∈B˚_p,∞^k+m(Rⁿ, E) and kukB˚p,∞^k+m(Rⁿ,E)≤MkvkB˚_p,∞^k (Rⁿ,E)

≤ckvk˜ _Wk p(Rⁿ,E)

≤c˜

|λ0|kuk_Wk

p(Rⁿ,E)+kAk(t)uk_Wk p(Rⁿ,E)

≤ckuk_D(A_k_(t)),

i.e. D(Ak(t)),→B˚_p,∞^k+m(Rⁿ, E). From this, (9), Corollary 4.3 and Theorem 2.5.1 of Chapter IV in [Am95], we obtain the desired result in similar way to the proof of Theorem 4.3 in [BDH12].

Appendix 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 and results which can be found in [Ba09] and [Ku81].

Definition A.1. Let a:R²ⁿ→E be a strongly measurable function. We say that ais integrable in the oscillatory sense if for each χ∈S(R²ⁿ)with χ(0,0) = 1 the limit

Os− Z Z

R²ⁿ

e^−iξ·ya(ξ, y)d¯(ξ, y) := lim

ε&0

Z Z

R²ⁿ

e^−iξ·yχ(εξ, εy)a(ξ, y)d¯(ξ, y) exists and does not depend on the choice of χ.

Definition A.2. Let m ∈ R, τ ∈ [0,∞), ν, ρ ∈ N0 ∪ {∞}. Then the space A^(m,ν,ρ)τ (Rⁿξ ×Rⁿy, E) of amplitude functions consists of all a: Rⁿξ ×Rⁿy → E for which all derivatives∂_ξ^α∂^β_ya with|α| ≤ν,|β| ≤ρare continuous and for which all norms|a|_m,`,`⁰ with `, `⁰∈N0,`≤ν,`⁰≤ρare finite. Here

|a|_m,`,`⁰ := max

|α|≤`max

|β|≤`⁰ sup

ξ∈Rⁿ

sup

y∈Rⁿ

hξi^−m−|β|hyi^−τk∂_ξ^α∂_y^βa(ξ, y)k_E.

Lemma A.3. Forχ∈S(Rⁿ)withχ(0) = 1the following assertions hold.

(i) χ(εx)→1 (ε&0) uniformly on all compact subsets ofRⁿ, (ii) ∂_x^αχ(εx)→0 (ε&0) uniformly on Rⁿ for allα∈Nⁿ0\ {0},

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

|∂_x^αχ(εx)| ≤cαε^σhxi^{−(|α|−σ)} (σ∈[0,|α|], x∈Rⁿ).

Proof. See [Ku81], Lemma 6.3.

Lemma A.4. Let m∈R,τ ∈[0,∞),`, `⁰ ∈Nwith n+τ <2`, m+n <2`⁰. Let furthermoreν, ρ∈N0∪{∞}with2`≤ν,2`⁰≤ρ. Then fora∈ A^(m,ν,ρ)τ (Rⁿξ×Rⁿy, E) the oscillatory integral Os−

Z Z

R²ⁿ

e^−iξ·ya(ξ, y)d¯(ξ, y)exists, and

Os−

Z Z

R²ⁿ

e^−iξ·ya(ξ, y)d¯(ξ, y)

_E≤c|a|m,2`,2`⁰

(15)

holds with a constantc not depending ona.

Proof. See [Ba09], Satz 1.3.1.

Lemma A.5. With m, τ, ν, ρ, `, and `⁰ as in Lemma A.4, let N ∈ N0 and let a:Rⁿ_x×Rⁿξ ×Rⁿy →E be a function with ∂_x^γa(x,·,·)∈A^(m,ν,ρ)τ (Rⁿξ,Rⁿy, E) for all x∈Rⁿand|γ| ≤N. Suppose further that for everyx∈Rⁿthere is a neighbourhood Uxofxsuch that the set{(∂_x^γa)(x⁰,·,·) :x⁰ ∈Ux}is bounded inA^(m,ν,ρ)τ (Rⁿξ,Rⁿy, E).

Then for all|γ| ≤N we have

∂_x^γh Os−

Z Z

R²ⁿ

e^−iξ·ya(x, ξ, y)d¯(ξ, y)i

= Os− Z Z

R²ⁿ

e^−iξ·y∂_x^γa(x, ξ, y)d¯(ξ, y).

Proof. See [Ba09], Satz 1.3.3.

References

[Am95] H. Amann: Linear and quasilinear parabolic problems, Vol. I. Birkh¨auser Verlag, Basel (1995).

[Am97] H. Amann: Operator-valued fourier multipliers, vector-valued besov spaces, and appli- cations.Math. Nachr.186(1997), 5-56.

[Am00] H. Amann Coagulation-Fragmentation Processes, Arch. Rot. Mech. Anal. 151, 339-366 (2000).

[Am01] H. Amann, Elliptic Operators with Infinite Dimensional State Spaces, J. Evo. Equ. 1, 143-188 (2001).

[Ba09] B. Barraza Mart´ınez, Pseudodifferentialoperatoren mit nichtregul¨aren banachraumwer- tigen Symbolen, Dissertation, Johannes Gutenberg-Universit¨at Mainz, (2009).

[BDH12] B. Barraza Mart´ınez, R. Denk, J. Hern´andez Monz´on, Analytic semigroups of pseudodifferential operators on vector-valued Sobolev spaces, Preprint Konstanzer Schriften in Mathematik, Nr. 303, Mai 2012.

[DHP03] R. Denk, M. Hieber, J. Pr¨uss, R-Boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc. 166 (2003).

[DK07] R. Denk, T. Krainer,R-boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators, Manuscripta Math. 124, 319-342 (2007).

[GW03] M. Girardi and L. Weis, Operator-Valued Fourier Multiplier Theorems on Besov Spaces, Math. Nachr. 251, 34-51 (2003).

[Ki01] Chr. Kiehn, Analytic Semigroup of Pseudodifferential Operators onLp(Rⁿ, E), Preprint Reihe des Fachbereichs Mathematik Nr 10, Johannes Gutenberg-Universit¨at, (2001).

[Ki03] Chr. Kiehn, Analytic Semigroups of Pseudodifferential Operators on Vector-Valued Function Spaces, Shaker Verlag, Aachen (2003).

[Ku81] H. Kumano-go, Pseudo-Differential Operators, MIT Press, Cambridge, MA (1981).

[KW04] P. C. Kunstmann, L. Weis, Maximal Lp-regularity for parabolic equations, Fourier multiplier theorems andH^∞-functional calculus, in Da Prato et al., editor, “Functional Analytic Methods for Evolution equations”, Lecture Notes in Math. 1855, Springer, 2004.

[NS11] T. Nau, J. Saal,R-sectoriality of cylindrical boundary value problems, in J. Escher et al., editor, “Parabolic Problems: The Herbert Amann Festschrift”, Progr. Nonlinear Differential Equations Appl. 80, Birkhuser/Springer, 479-506 (2011).

[NNH02] A. Noll, R. Haller, and H. Heck, Mikhlin’s Theorem for Operator-Valued Multipliers in nVariables, Math. Nach. 244, 110-130 (2002).

[PS06] P. Portal and ˇZ. ˇStrkalj, Pseudodifferential operators on Bochner Spaces and an application, Math. Z., 253, 805-819 (2006).

[We01] L. Weis, Operator-Valued Fourier Multiplier Theorems and Maximal Lp-Regularity, Math. Ann. 319, 735-758 (2001).

(16)

Universidad del Norte, Departamento de Matem´aticas, Km 5 Via a Puerto Colombia, Barranquilla, Colombia

E-mail address:bbarraza@uninorte.edu.co

Universit¨at Konstanz, Fachbereich Mathematik, 78457 Konstanz, Germany E-mail address:robert.denk@uni-konstanz.de

Universidad del Norte, Departamento de Matem´aticas, Km 5 Via a Puerto Colombia, Barranquilla, Colombia

E-mail address:jahernan@uninorte.edu.co