On the maximal Lp-regularity of parabolic mixed order systems

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Universität Konstanz

On the maximal Lp-regularity of parabolic mixed order systems

Robert Denk Jörg Seiler

Konstanzer Schriften in Mathematik Nr. 266, Mai 2010

ISSN 1430-3558

Fach D 197, 78457 Konstanz, Germany

Konstanzer Online-Publikations-System (KOPS) URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-119246

URL: http://kops.ub.uni-konstanz.de/volltexte/2010/11924

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ON THE MAXIMAL L_p-REGULARITY OF PARABOLIC MIXED ORDER SYSTEMS

ROBERT DENK AND J ¨ORG SEILER

Abstract. We study maximal Lp-regularity for a class of pseudodifferential mixed order systems on a space-time cylinderRⁿ×RorX×R, whereXis a closed smooth manifold. To this end we construct a calculus of Volterra pseudodifferential operators and characterize the parabolicity of a system by the invertibility of certain associated symbols. A parabolic system is shown to induce isomorphisms between suitableLp-Sobolev spaces of Bessel potential or Besov type. If the cross section of the space-time cylinder is compact, the inverse of a parabolic system belongs to the calculus again. As applications we discuss time-dependent Douglis-Nirenberg systems and a linear system arising in the study of the Stefan problem with Gibbs-Thomson correction.

1. Introduction

Motivated by many applications arising in mathematics, mathematical physics and applied sciences, parabolic initial boundary value problems have been studied systematically from a general point of view at least since the 1960’s. Classical works in this direction are Agranovich- Vishik [1], Solonnikov [21] and Eidel’man [6]. Generalizations covering wider classes of systems and more general boundary conditions have been obtained subsequently, for example by Kozhevnikov [10], Gindikin, Volevich [9], Volevich [24], and Denk, Mennicken, Volevich [2] . A standard approach in the analysis of boundary value problems are reduction to the boundary techniques. In this way the investigation of an initial boundary value problem is reduced to studying a system on the domain’s boundary corresponding to the so-called Lopatinskij matrix. This system has two characteristic properties: it has a pseudodifferential structure (even if the underlying boundary value problem is purely differential) and it is, in general, a mixed order system. In this paper we develope a calculus of Volterra pseudodifferential operators that allows to obtain solvability results for a wide class of such kind of systems.

Our approach combines pseudodifferential techniques with ideas of Volevich [24] and Denk, Volevich [5], where boundary value problems with dynamical boundary conditions are investi- gated. Though quite general in nature the analysis in [5] is limited to anL2-setting for model problems on a half-space and all involved operators have constant coefficients, both in time and space (i.e., the framework is that of Fourier multipliers rather than general pseudodifferential operators). We avoid these restrictions and treat more general problems on smoothly bounded domains and work within scales of L_p-Sobolev spaces both of Bessel potential and of Besov type. Our results also extend those of Denk, Saal, Seiler [4] where the authors use a

2010Mathematics Subject Classification. 35G40, 35S05, 35R35.

Key words and phrases. Pseudodifferential operators, mixed-order systems, maximal regularity.

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Kalton-Weis approach to investigate in anL_p-setting mixed order systems on the half-space, where the reduced system on the boundary has constant coefficient symbols depending onτ and |ξ|only (with τ and ξ being the time and space co-variable, respectively).

The use of pseudodifferential analysis for studying partial differential equations is by now a rather classic method. The principal idea is to embed problems of a certain class of interest in an algebra of operators with a ‘symbol structure’ and to obtain qualitative statements on solvability by characterizing invertibility (possibly modulo good remainders) of operators within the algebra in terms of conditions on the associated symbols. By knowing the precise symbolic structure of the inverses, at the same time one also obtains information on the shape of the solutions. It was Piriou [17], [18] who introduced the concept of Volterra pseudodifferential operators to tackle parabolic problems. Roughly speaking, the novel feature of a Volterra calculus is that the pseudodifferential symbols extend in the time co-variable holomorphically to the complex lower half-plane, and satisfy there certain symbol estimates.

This Volterra property has two striking consequences: it allows to modify parametrices (in the sense of elliptic theory) to become exact inverses and it ensures that the operators preserve ‘vanishing in the past’, i.e., if a function vanishes before a time t0, so does its image.

The latter condition is important in handling initial values, while the first implies unique solvability. The concept of the Volterra property has been employed in Krainer, Schulze [13]

and Krainer [12] in the study of long-time asymptotics for parabolic equations. Though the modification of holomorphy in the covariable looks rather innocent, it brings along certain technical difficulties; for instance the standard procedure of asymptotic summation cannot be performed in the usual way, since excisions in the covariable destroy the holomorphy. We develope a calculus of Volterra pseudodifferential operators that is adapted to the kind of mixed order systems as described above, and find explicit ‘parabolicity’ conditions on a system that imply the existence of an inverse within the calculus, leading to unique solvability in exponentially weighted spaces.

As a straight-forward application of our calculus, we establish maximal regularity (in the sense of isomorphisms between suitable Sobolev spaces) of time-dependent Douglis-Nirenberg systems. This generalizes results of [3] where we have discussed the existence of a bounded H^∞-calculus for a (stationary) system of Douglis-Nirenberg type. Moreover, we show that the linearized Stefan problem with Gibbs-Thomson correction fits into our framework. This free boundary value problem with dynamic boundary conditions leads to an inhomogeneous structure of the Lopatinskij matrix and cannot be dealt with classical parabolic theory. We discuss when our results can be applied to problems with dynamical boundary conditions of more general form.

2. Volterra pseudodifferential operators on Rⁿ×R

In this section we develop a pseudodifferential calculus for symbols having theVolterra property, i.e., the time co-variable extends holomorphically to the lower complex half-plane. We discuss all standard properties of a pseudodifferential calculus, like composition, asymptotic summation and continuity in associated Sobolev spaces.

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2.1. Weight functions and symbol spaces. In the sequel we letEdenote a Fr´echet space and set

H:={τ ∈C|Imτ ≤0}.

ory belonging toRⁿ orCⁿ with somen∈N, we write hyi:= (1 +|y|²)^1/2.

We shall use the standard multi-index notation for partial derivatives D^α = D^α₁¹· · ·D^α_nⁿ, whereD=−i∂.

Definition 2.1. A weight function is a function ω : Rⁿ×H → C having the following properties:

i) ω depends smoothly on ξ and holomorphically on τ, i.e.

ω∈C^∞ H,C^∞(Rⁿ)

∩A intH,C^∞(Rⁿ) ,

ii) for all multi-indices α∈Nⁿ0 and β∈N²0 there exist constants Cαβ such that

|D^α_ξD^β_τω(ξ, τ)| ≤C_αβ|ω(ξ, τ)|hξi^−|α|hτi^−|β|

(identifying C withR²),

iii) there exist constants C, M ≥0 such that

|ω(ξ+ξ⁰, τ +τ⁰)| ≤C|ω(ξ, τ)|hξ⁰i^Mhτ⁰i^M. The estimates in ii) and iii) are uniform inξ, ξ⁰ ∈Rⁿ andτ, τ⁰∈H. The third property implies that, for suitable positive constantsc, C ≥0, (2.1) chξi^−Mhτi^−M ≤ |ω(ξ, τ)| ≤Chξi^Mhτi^M. From the fact that D_ξ^αDτ^β1

ω is a finite linear combination of terms of the form (D^α_ξ¹D^β_τ¹ω)·. . .·(D^α_ξ^kD^β_τ^kω) 1

ω^k+1

with 1≤k≤ |α|+|β|andα₁+. . .+α_k=α,β₁+. . .+β_k =β, it is obvious that withω also 1/ω is a weight function. By product rule the product of two weight functions is a weight function, again.

Definition 2.2. For µ, ν ∈ R let S^µ,ν;ω(Rⁿ ×H;E) consist of all smooth functions a : Rⁿ×H→E which satisfy estimates

|||D^α_ξD^β_τa(ξ, τ)||| ≤C|ω(ξ, τ)|hξi^µ−|α|hτi^ν−|β| ∀(ξ, τ)∈Rⁿ×H

for all multi-indices α ∈ Nⁿ₀, β ∈ N²₀, and each continuous semi-norm ||| · ||| of E (with C depending only on α, β, and the semi-norm). Moreover, let

S_V^µ,ν;ω(Rⁿ×H;E) :=S^µ,ν^;ω(Rⁿ×H;E)∩A intH,C^∞(Rⁿξ, E)

be the space of all symbols from S^µ,ν;ω(Rⁿ×H;E) that additionally depend holomorphically on τ ∈ intH. For notational convenience we shall often use the short-hand notation S^µ,ν;ω and S_V^µ,ν;ω, respectively. Ifω= 1 we suppress it from the notation.

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The subscriptV stands for Volterra property; see below for further explanation. In the standard way, we may also define spacesS^µ,−∞;ω,S^−∞,ν;ω, S^{−∞,−∞;ω}, and those with subscript V. In fact, the last space does not depend on ω and therefore we shall write S_(V^{−∞,−∞}₎ . All these symbol spaces are Fr´echet spaces in a canonical way. Note that a weight function ω belongs to S_V^0,0;ω and that

S_(V^µ,ν;ω₎ (Rⁿ×H;E) =S_(V^0,0;ω₎ ^µ,ν(Rⁿ×H;E), ω_µ,ν(ξ, τ) =ω(ξ, τ)hξi^µ(1 +iτ)^ν. Hence often we can assume without loss of generality thatµ=ν = 0.

Due to Definition 2.1.(iii) and estimate (2.1) we obtain the embeddings S_(V^{µ−M,ν−M}₎ ,→S_(V^µ,ν;ω₎ ,→S_(V^µ+M,ν+M₎

and

S^µ−M,ν;ω_(V₎ ⁰ ,→S_(V^µ,ν₎^;ω ,→S_(V^µ+M,ν;ω₎ ⁰, ω⁰(τ) :=ω(0, τ), as well as

S_(V^µ,ν−M;ω₎ ⁰ ,→S_(V^µ,ν;ω₎ ,→S_(V^µ,ν+M;ω₎ ⁰, ω₀(ξ) :=ω(ξ,0).

Note that Cauchy’s integral formula implies that

S^µ,ν_V ^;ω(Rⁿ×H;E)⊂A intH, S^µ;ω⁰(Rⁿ_ξ;E) whereS^µ;ω⁰(Rⁿ_ξ;E) is defined in the obvious way.

Definition 2.3. By choosing the Fr`echet spaceE as Em:=Cb^∞ Rⁿx, S^m(Rt)

, m∈R,¹ we² define

(2.2) S_(V^µ,(ν,m);ω₎ (Rⁿ×R;Rⁿ×H) :=S_(V^µ,ν₎^;ω(Rⁿ×H;Em).

This is then a space of symbols a=a(x, t, ξ, τ) with variablesx, t and corresponding covariables ξ and τ, respectively. In case m= 0, we write S_(V^µ,ν;ω₎ instead of S_(V^µ,(ν,0);ω₎ .

The type oft-dependence of the symbols from S_(V^µ,(ν,m);ω₎ is known from the calculus of SG- pseudodifferential operators, cf. [16]. By product rule it is obvious that pointwise multiplication yields a continuous map

(2.3) S_(V^µ,(ν,m);ω₎ ×S_(V^µ⁰^,(ν₎ ⁰^,m⁰^);ω⁰ −→S_(V^µ+µ₎ ⁰^,(ν+ν⁰^,m+m⁰^);ωω⁰. For a symbola∈S^µ,(ν,m);ω the associated pseudodifferential operator

a(x, t, D_x, D_t) :S(Rⁿ×R)−→S(Rⁿ×R) is defined in the usual way by

[a(x, t, Dx, Dt)u](x, t) = Z

ei(x,t)(ξ,τ)a(x, t, ξ, τ)u(ξ, τb )d¯(ξ, τ),

2i.e. the space of all smooth functions p : Rⁿ×R →C satisfying estimates |D^αxD^ltp(x, t)| ≤ Cαlhti^m−l uniformly in (x, t)∈Rⁿ×R, for allαandl.

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whered¯(ξ, τ) =d(ξ, τ)/(2π)ⁿ⁺¹ and ubdenotes the Fourier transform of u.

The Leibniz product a#b, corresponding to the composition

a(x, t, D_x, D_t)b(x, t, D_x, D_t) = (a#b)(x, t, D_x, D_t), also induces a map as in (2.3), as can be deduced from the formula

a#b(x, t, ξ, τ) = Z Z

e−i(y,s)(η,σ)a(x, t, ξ+η, τ+σ)b(x+y, t+s, ξ, τ)d(y, s)d¯(η, σ), where integration is to be understood as an oscillatory integral over an amplitude function on Rⁿ⁺¹×Rⁿ⁺¹ (cf. [14] for details). Composition of the operator-familiesa(x, t, D_x, τ) and b(x, t, Dx, τ) yields a symbol we denote by a#xb. Analogously, by passing to the operator- families with respect tot, we get a symbola#_tb.

Proposition 2.4. Let a∈S_(V^µ,(ν,m);ω₎ and b∈S_(V^µ⁰^,(ν₎ ⁰^,m⁰^);ω⁰. Then

a#b≡ab+ (a#_xb−ab) + (a#_tb−ab) mod S_(V^µ+µ₎ ⁰^−1,(ν+ν⁰^−1,m+m⁰^−1);ωω⁰. Proof. Insert in the above formula fora#bthe expansion

a(x, t, ξ+η, τ +ρ) =a(x, t, ξ+η, τ) +ρ Z 1

0

(∂_τa)(x, t, ξ+η, τ +θ₁ρ)dθ₁

=a(x, t, ξ+η, τ) +a(x, t, ξ, τ +ρ)−a(x, t, ξ, τ)+

+ P

|α|=1

ρη^α Z 1

0

Z 1 0

(∂_ξ^α∂τa)(x, t, ξ+θ2η, τ +θ1ρ)dθ2dθ1.

The first term in this expansion yields a#_xb, the seconda#_tb, the third −ab, while the last (after integration by parts) yields the remainder term of the stated type.

The previous proposition shows that the Leibniz-producta#bis not determined modulo lower order terms by the pointwise product ab. This will have consequences on the parametrix construction, see below.

The terminology Volterra symbols stems from the fact that the (distributional) kernel of the associated pseudodifferential operators vanishes ‘above the diagonal’. In fact, ifa∈S_V^µ,(ν,m);ω then its kernel is

k(x, t, x⁰, t⁰) = Z Z

e^i(x−x⁰^,t−t⁰^)(ξ,τ⁾a(x, t, ξ, τ)d¯(ξ, τ)

=e^(t−t⁰^)y Z Z

e^i(x−x⁰^,t−t⁰^)(ξ,σ)a(x, t, ξ, σ−iy)d¯(ξ, σ)

for anyy≥0, due to the holomorphy in the covariableτ. Passing to the limity→ ∞, we see thatk(x, x⁰, t, t⁰) = 0 whenevert−t⁰ <0. A particular consequence is that pseudodifferential operators with Volterra property preserve the ‘time-forward support’ of distributions: If we set

S(Rⁿ×[t₀,∞)) ={u∈S(Rⁿ×R) |u(x, t) = 0 if t < t₀} with arbitrary t₀ ∈R, then

a(x, t, D_x, D_t) :S(Rⁿ×[t₀,∞))−→S(Rⁿ×[t₀,∞)).

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A further fundamental consequence of the Volterra property is the following fact on invertibilty of integral operators:

Theorem 2.5. LetK be an integral operator with kernelk∈S(Rⁿ⁺¹_(x,t)×Rⁿ⁺¹_(x⁰_,t⁰₎)that vanishes whenever t < t⁰. Then

1 +K :S(Rⁿ×R)−→S(Rⁿ×R)

is invertible and (1 +K)⁻¹ = 1 +K, wheree Ke has the same structure as K.

Proof. First one can show that 1+K :L₂(Rⁿ×R)→L₂(Rⁿ×R) is invertible with (1+K)⁻¹= 1+K, wheree Ke has a kernel inL2(Rⁿ⁺¹_(x,t)×Rⁿ⁺¹_(x0,t⁰)) that vanishes whenevert < t⁰ (see Theorem 4.2.6 in [11]). However, then

1 +Ke = (1 +K)⁻¹ = 1−K+K(1 +K)⁻¹K

shows that both Ke and the L₂-adjoint Ke^∗ map L₂(Rⁿ×R) into S(Rⁿ×R). This implies that the kernel ofKe is rapidly decreasing in all variables. In fact, first we see thatKe has an integral kernel ekin

S(Rⁿ)⊗b_πL₂(Rⁿ)

∩ L₂(Rⁿ)⊗b_πS(Rⁿ) ,

where ⊗b_π denotes the completed projective tensor product. However, this space coincides withS(Rⁿ×Rⁿ), since the inequalityr^as^b ≤ ¹₂(r^2a+s^2b) together with Plancherel’s formula allows to estimate theL2(R²ⁿ)-norm ofx^α⁰y^β⁰D_x^αDy^βek(x, y) by theL2-norms ofx^α⁰⁰D_x^α⁰ek(x, y)

and y^β⁰⁰Dy^β⁰ek(x, y), respectively.

2.2. Asymptotic summation of Volterra symbols. An important feature in any pseudodifferential calculus is the possibility of summing asymptotically a sequence of symbols of decreasing order. The standard technique to achieve this involves excision of symbols in the co-variables. In the present context this technique is not applicable, since excision destroys the holomorphy of Volterra symbols. Hence an alternative approach is used, based on the so-called ‘kernel cut-off’ procedure, cf. [20]. Again, let E denote a Fr`echet space.

Definition 2.6. Forµ, ν ∈RletΛ^µ,ν(Rⁿ×H;E)consist of all smooth functionsa:Rⁿ×H→ E which satisfy estimates

|||D_ξ^αD_τ^βa(ξ, τ)||| ≤Chξi^µ−|α|hξ, τi^ν−|β|

uniformly in (ξ, τ)∈Rⁿ×H, for all α∈Nⁿ0, β ∈N²0, and each continuous semi-norm of E.

Similarly as above, we denote by Λ^µ,ν_V (Rⁿ×H) the subspace of all symbols that, in addition, depend holomorphically onτ ∈intH.

We may also define spaces Λ^µ,−∞, Λ^−∞,ν, Λ^{−∞,−∞}, and those with subscript V. Note that S^{−∞,−∞}= Λ^{−∞,−∞}= Λ^µ,−∞.

Lemma 2.7. The map

a(ξ, τ)7→ba(ξ, τ) :=a(ξ,hξi⁻¹τ)

induces(topological)isomorphismsS_(V^µ,ν₎(Rⁿ×H;E)→Λ^µ−ν,ν_(V₎ (Rⁿ×H;E)with inverse induced by

a(ξ, τ)7→ea(ξ, τ) :=a(ξ,hξiτ).

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In particular, a 7→ ba : S_(V^{−∞,−∞}₎ (Rⁿ×H;E) → Λ^{−∞,−∞}_(V₎ (Rⁿ×H;E) with inverse given by a7→ea.

Proof. Clearly, holomorphy inτ is preserved. Hence it suffices to consider the classes without subscriptV. We shall now use the (equivalent) identities

hξ,hξiτi=hξihτi, hhξi⁻¹τi=hξi⁻¹hξ, τi.

From

D_ξ_jba(ξ, τ) =(D_ξ_ja)(ξ,hξi⁻¹τ) +τ1(D_ξ_jhξi⁻¹)(Dτ1a)(ξ,hξi⁻¹τ) +τ₂(D_ξ_jhξi⁻¹)(D_τ₂a)(ξ,hξi⁻¹τ)

=(D_ξ_ja)(ξ,hξi⁻¹τ) +hξi(D_ξ_jhξi⁻¹)(τ1Dτ1a)(ξ,hξi⁻¹τ) +hξi(D_ξ_jhξi⁻¹)(τ₂D_τ₂a)(ξ,hξi⁻¹τ)

and induction it easily follows thatD^α_ξbais a linear combination of termsbbwithb∈S^µ−|α|,ν. Consequently, D_ξ^αDτ^βbais a linear combination of termshξi^−|β|Dd^βτb. Therefore,

|||D^α_ξD_τ^βba(ξ, τ)||| ≤Chξi^−|β|hξi^µ−|α|hhξi⁻¹τi^ν−|β|

=Chξi^−|β|hξi^µ−|α|hξ, τi^ν−|β|hξi^|β|−ν

=Chξi^{µ−ν−|α|}hξ, τi^ν−|β|,

provided a∈S^µ,ν. Vice versa, if a∈Λ^µ,ν, then D^α_ξeais a linear combination of termsebwith b∈Λ^µ−|α|,ν. Hence D^α_ξD^βτea is a linear combination of termshξi^|β|(D^βτb)e. It follows that

|||D_ξ^αD^β_τea(ξ, τ)||| ≤Chξi^|β|hξi^µ−|α|hξ,hξiτi^ν−|β|

=Chξi^|β|hξi^µ−|α|hξi^ν−|β|hτi^ν−|β|

=Chξi^µ+ν−|α|hτi^ν−|β|.

This shows that eabelongs to S^µ+ν,ν.

The main reason for introducing the symbol spaces Λ^µ,ν is that differentiation with respect to τ improves the behaviour of the symbols both in τ and ξ. This property shall be used frequently below.

Proposition 2.8 (Kernel cut-off). Let ϕ∈Ccomp^∞ (R) withϕ≡1in some neighborhood of 0.

The map

(2.4) a(ξ, τ)7→[h(ϕ)a](ξ, τ) :=

Z Z

e^−isσϕ(s)a(ξ, τ −σ)dsd¯σ (oscillatory integral) has the following properties:

a) h(ϕ) : Λ^µ,ν_V (Rⁿ×H;E)−→Λ^µ,ν_V (Rⁿ×H;E),

b) 1−h(ϕ) : Λ^µ,ν_V (Rⁿ×H;E)−→Λ^{−∞,−∞}_V (Rⁿ×H;E).

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Proof. The proof is completely analogous to the standard kernel cut-off construction due to [20] (see also [8]). A detailed exposition which can be followed quite closely can be found in [11]. For convenience of the reader we indicate the main steps of the proof: First of all, the definition of h(ϕ) makes sense for any function ϕ ∈ C_b^∞(R), and one shows by explicit regularization of the oscillatory integral that

|||[h(ϕ)a](ξ, τ)||| ≤Chξi^µhξ, τi^ν.

Together with ∂_ξ^α∂_τ^β[h(ϕ)a] =h(ϕ)(∂_ξ^α∂_τ^βa) this yields that h(ϕ) maps into Λ^µ,ν. Also holomorphy is preserved, i.e. h(ϕ) maps into Λ^µ,ν_V . A Taylor expansion of ϕ in s = 0 implies that

h(ϕ)a=

N−1

P

k=0

(−1)^k

k! D^k_sϕ(0)

∂_τ^ka+h(ψ_(N₎)(∂_τ^Na),

where ψ_(N) ∈ C_b^∞(R). As we have seen above, h(ψ_(N₎)(∂_τ^Na) ∈ Λ^µ,ν_V ^−N. Hence, if ϕ ∈ Ccomp^∞ (R) and ϕ≡1 neart= 0,

a−h(ϕ)a∈

∩

N∈N

Λ^µ,ν−N_V = Λ^{−∞,−∞}_V .

This proves the claim.

Theorem 2.9. Given symbolsa_j ∈S_V^µ−l^j^,ν−l^j^;ω(Rⁿ×H;E),j∈N0, with a strictly increasing sequence0≤lj → ∞, there exists a symbol a∈S_V^µ,ν;ω(Rⁿ×H;E) such that

a−

N−1

P

j=0

aj ∈ S^µ−l_V ^N^,ν−l^N^;ω(Rⁿ×H;E) ∀N ∈N0. We write a∼_V

∞

P

j=0

aj. The symbola is uniquely determined modulo S_V^{−∞,−∞}.

Proof. By multiplication with 1/ω we may assume without loss of generality that ω = 1.

Using Lemma 2.7 we have ba_j ∈ Λ^{µ−ν,ν−l}_V ^j for each j ∈ N0. Following the proof of Theorem 2.1.16 of [11], there exists a sequence of real numbers (c_j)_j with 1 ≤ c_j → ∞ such that b_k :=

∞

P

j=k

h(ϕ_c_j)ba_j converges in Λ^{µ−ν,ν−l}_V ^k for any k∈N0, where ϕ_c(t) :=ϕ(ct) with a fixed ϕ∈Ccomp^∞ (R) being equal to 1 in a neighborhood of t= 0. Due to Proposition 2.8.b),

b₀−

N−1

P

j=0 ba_j =b_N −

N−1

P

j=0

(1−h(ϕ_c_j))ba_j ∈ Λ^{µ−ν,ν−l}_V ^N. Then the result follows fora:=eb₀, because we have

a−

N−1

P

j=0

aj =

b0−

N−1

P

j=0

baj

˜

^∈ ^SV^µ−l^N^,ν−l^N Let us mention here an alternative way of proving the previous theorem (and, in fact, a slight generalization). To this end assume without loss of generality thatω= 1 and let us denote by H(ϕ) the kernel cut-off operator given by (2.4), but now acting on symbolsS_V^µ,ν(Rⁿ×H;E).

This yields a mapH(ϕ) :S_V^µ,ν(Rⁿ×H;E)→S_V^µ,ν(Rⁿ×H;E) with 1−H(ϕ) :S_V^µ,ν(Rⁿ×H;E)−→S_V^µ,−∞(Rⁿ×H;E).

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If ψ = ψ(ξ) is a zero excision function, multiplication with ψ yields a map ψ : S_V^µ,ν(Rⁿ× H;E)→S_V^µ,ν(Rⁿ×H;E) with

1−ψ:S_V^µ,ν(Rⁿ×H;E)−→S_V^−∞,ν(Rⁿ×H;E).

Note thatψ and H(ϕ) are commuting. Now let us define

χ(c) :=ψ_1/c+H(ϕc)−ψ_1/cH(ϕc), c >0, wheref_c:=f(c·). Then χ(c) :S_V^µ,ν(Rⁿ×H;E)→S_V^µ,ν(Rⁿ×H;E) and

1−χ(c) :S_V^µ,ν(Rⁿ×H;E)−→S_V^{−∞,−∞}(Rⁿ×H;E), since

1−χ(c) = (1−ψ_1/c)(1−H(ϕ_c)).

Moreover, one can show that

χ(c)a−−−→^c→∞ 0 inS_V^µ,ν(Rⁿ×H;E) whenever a∈S^µ−ε,ν_V (Rⁿ×H;E)∩S_V^µ,ν^−ε(Rⁿ×H;E) for some ε >0.

Given a sequence of symbolsaj ∈S_V^µ^j^,ν^j(Rⁿ×H;E),j∈N0, with strictly decreasing sequences µj, νj

−−−→ −∞, one can construct a sequencej→∞ cj

−−−→ ∞j→∞ such that

∞

P

j=k

χ(cj)aj converges in S_V^µ^k^,ν^k(Rⁿ ×H;E) for any k ∈ N0, see Proposition 1.1.17 of [20]. Thus a∼_V

∞

P

j=0

aj for a:=

∞

P

j=0

χ(c_j)a_j.

2.3. Sobolev and Besov spaces. In this section pseudodifferential operators are shown to extend from mappings between the rapidly decreasing functions to continuous maps in suitable distributional spaces.

Definition 2.10. For1< p <∞ the Sobolev spaces(in the sense of Bessel potential spaces) with respect to a weight function κ are defined as

(2.5) H_p^κ(Rⁿ×R) :=

u∈S⁰(Rⁿ×R) |κ(D_x, D_t)u∈L_p(Rⁿ×R) with the canonical norm

kuk_κ,p:=kκ(D_x, D_t)uk_L_p_(Rn×R). If t0 ∈R we set

H_p^κ(Rⁿ×[t0,∞)) :=

u∈H_p^κ(Rⁿ×R)| suppu⊂Rⁿ×[t0,∞) . This is a closed subspace of H_p^κ(Rⁿ×R). Analogously we introduce the Besov spaces

B_pp^κ(Rⁿ×R), B_pp^κ (Rⁿ×[t0,∞)) by replacing L_p(Rⁿ×R) in (2.5) by B_pp⁰ (Rⁿ×R).

We need the following simple observation concerning the invertibility of maps between interpolation spaces.

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Lemma 2.11. LetX₀, X₁, Y₀, Y₁ be Banach spaces contained inS⁰(R^l) which containS(R^l) as a dense subset. Let

T :S⁰(R^l)→S⁰(R^l)

be a (topological) isomorphism that restricts to isomorphisms S(R^l) → S(R^l), X₀ → Y₀, and X1 →Y1. Then T restricts to an isomorphism

T : (X0, X1)θ,p −→(Y0, Y1)θ,p,

where (·,·)_θ,p denotes the real interpolation method. The inverse is obtained by restricting T⁻¹:S⁰(R^l)→S⁰(R^l) to (Y0, Y1)_θ,p.

Proof. Let us write X = (X₀, X₁)_θ,p and Y = (Y₀, Y₁)_θ,p. By interpolation,T restricts to a continuous mapT :X→Y. Also,T⁻¹restricts to a continuous mapT⁻¹:Y →X. However, on S(R^l) we haveT T⁻¹ =T⁻¹T = 1. Hence the result follows from density of S(R^l) both

inX and Y.

Corollary 2.12. Let κ be a weight function and

κ_s(ξ, τ) = (hξi+iτ)^sκ(ξ, τ).

Then, for any real numbers s₀ 6=s₁,

B_pp^κ^s(Rⁿ×R) = H_p^κ^s⁰(Rⁿ×R), H_p^κ^s¹(Rⁿ×R)

θ,p, s=θs₀+ (1−θ)s₁.

Proof. For notational convenience, we suppress writingRⁿ×R. The previous lemma applied toT :=κ(D_x, D_t)⁻¹ shows that

κ(Dx, Dt)⁻¹ : (H_p^s⁰, H_p^s¹)_θ,p−→(Hp^κ^s⁰, Hp^κ^s¹)_θ,p isomorphically. On the other hand, by definition,

κ(Dx, Dt) :B_pp^κ^s −^∼⁼→B_pp^s = (H^s⁰, H^s¹)_θ,p.

This already implies the claim.

Theorem 2.13. Let κ and ω be weight functions, a∈S^0,0;ω(Rⁿ×R;Rⁿ×H), and t₀ ∈R. Then a(x, t, D_x, D_t) induces continuous operators

H_p^κ(Rⁿ×R)−→H_p^κ/ω(Rⁿ×R), B_pp^κ (Rⁿ×R)−→B^κ/ω_pp (Rⁿ×R).

In casea has the Volterra property these maps also restrict to H_p^κ(Rⁿ×[t₀,∞))−→H_p^κ/ω(Rⁿ×[t₀,∞)), B^κ_pp(Rⁿ×[t₀,∞))−→B^κ/ω_pp (Rⁿ×[t₀,∞)).

Proof. By definition of the Sobolev spaces, the first mapping property is equivalent to the continuity of

κ ω#a#1

κ

(x, t, D_x, D_t) :L_p(Rⁿ×R)−→L_p(Rⁿ×R).

This Leibniz-product belongs to the symbol space S^0,0;^ω^κ^ω¹^κ = S^0,0;1 = S^0,0. Hence the result follows from Theorem 1 in [25] on the continuity of pseudodifferential operators. The continuity in Sobolev spaces together with interpolation gives the continuity in Besov spaces

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(e.g., choose above s₀ =−1,s₁= 1, andθ= 1/2 to expressB_pp^κ (Rⁿ×R) as an interpolation space between two Sobolev spaces). The preservation of the ‘time-forward support’ is due to the Volterra property, cf. the above discussion at the end of Section 2.1.

Like Theorem 2.13 many of our results are valid both for Sobolev spaces and for Besov spaces.

Whenever this is the case we will indicate this by simply using the short-hand notation (2.6) H_p^κ(Rⁿ×R), H_p^κ(Rⁿ×[t0,∞)).

3. Parabolicity for symbols with the Volterra property

The main idea of any calculus of Volterra pseudodifferential operators is that under suitable invertibilty conditions on the symbol one can construct an inverse within the calculus. To this end one first constructs a parametrix (similar to standard elliptic theory, but preserving the Volterra property) and then , in a second step, modifies this parametrix to an exact inverse using the Volterra property.

3.1. Construction of a parametrix. We shall derive conditions characterizing the existence of a parametrix. These conditions are related to those obtained in [22] in connection with the analysis of ‘bisingular’ pseudodifferential operators on products of manifolds.

Definition 3.1. A symbol p ∈ S_V^0,0;1/ω(Rⁿ ×R;Rⁿ × H) is called a parametrix of a ∈ S_V^0,0;ω(Rⁿ×R;Rⁿ×H) if botha#p−1 and p#a−1 belong to S_V^{−∞,−∞}(Rⁿ×R;Rⁿ×H).

Lemma 3.2. For every real σ ≥0 the translation operator T_iσ defined by (Tiσa)(ξ, τ) :=a(ξ, τ −iσ)

induces a mapsΛ^0,0_V (Rⁿ×H;E)→Λ^0,0_V (Rⁿ×H;E)andS_V^0,0;ω(Rⁿ×H;E)→S_V^0,0;ω(Rⁿ×H;E) with the property that

1−T_iσ : Λ^0,0_V (Rⁿ×H;E)−→Λ^0,−1_V (Rⁿ×H;E), 1−T_iσ :S_V^0,0;ω(Rⁿ×H;E)−→S_V^0,−1;ω(Rⁿ×H;E).

Proof. Follows directly from

a(ξ, τ −iσ)−a(ξ, τ) =−iσ Z 1

0

(∂_τa)(ξ, τ −iθσ)dθ.

Proposition 3.3. Let a∈S_V^0,0(Rⁿ×R;Rⁿ×H) and setA(t, τ) :=a(x, t, Dx, τ), considered as an element of S⁰ R×H;L(L_p(Rⁿ))

. If A is pointwise invertible and sup

(t,τ)∈R×H

kA(t, τ)⁻¹k_L_(L_p₍_Rn))<∞, then there exists a symbol b∈S_V^0,0(Rⁿ×R;Rⁿ×H) such that

A(t, τ)⁻¹ =b(x, t, D_x, τ) ∀(t, τ)∈R×H.

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Proof. The proof of this proposition is a parameter-dependent version of the standard proof of the spectral invariance of pseudodifferential operators, see [19] for example. From the spectral invariance of pseudodifferential operators onRⁿit follows that there exists ab(x, t, ξ, τ)∈ C^∞(R×H, S⁰(Rⁿ×Rⁿ)) which also depends holomorphically onτ ∈intHsuch thatA(t, τ)⁻¹= b(x, t, Dx, τ) for all t and τ. To show that b belongs to S^0,0_V = S_V⁰ R×H;S⁰(Rⁿ×Rⁿ)

it suffices to show that b is bounded as a function R×H→S⁰(Rⁿ×Rⁿ); in fact, the bounds for derivatives then follow by chain rule. To this end let us use the following notation: If T :S(Rⁿ)→S⁰(Rⁿ) is a linear operator we define the commutators

ad(xj)T := [xj, T], ad(Dxj)T := [Dxj, T],

wherex_j refers to the operator of multiplication with the function x7→x_j. For multi-indices α, β∈Nⁿ₀ we define the iterated commutators

ad^β(x)T = ad(x1)^β¹· · ·ad(xn)^βⁿT, ad^α(D)T = ad(D1)^α¹· · ·ad(Dn)^αⁿT.

It is well-known that T is a pseudodifferential operator with symbol in S⁰(Rⁿ×Rⁿ) if, and only if,

ad^α(D)ad^β(x)T ∈ L(H_p^s(Rⁿ), H_p^s+|β|(Rⁿ))

for any multi-indices α, β, and some s ∈ R (more precisely, the operator on the left-hand side, which is defined on S(Rⁿ), has a continuous extension to an operator belonging to the right-hand side). Moreover, the topology induced by the system of semi-norms

p_α,β(r) =

ad^α(D)ad^β(x)r(x, D)

L(H_p^s(Rⁿ),Hp^s+|β|(Rⁿ)), α, β∈N0,

coincides with the standard topology on the symbol space S⁰(Rⁿ×Rⁿ). Hence we have to show that p_α,β A(t, τ)⁻¹

is uniformly bounded in (t, τ) for any multi-indices α, β. To this end observe that in the assumption on A we can replace L_p(Rⁿ) by any spaceH_p^s(Rⁿ) with s∈R. In fact, if 0≤s≤1, then

Λ^−sA(t, τ)⁻¹Λ^s+ Λ^−sA(t, τ)⁻¹[A(t, τ),Λ^s]A(t, τ)⁻¹

is an inverse of A(t, τ) :H_p^s(Rⁿ) → H_p^s(Rⁿ), where Λ^s =hD_xi^s is the standard reduction of orders. Then one iterates the procedure for j≤s≤j+ 1 andj = 1,2,3, . . .. Negative sare treated similarly by using

Λ^−sA(t, τ)⁻¹Λ^s+A(t, τ)⁻¹[A(t, τ),Λ^−s]A(t, τ)⁻¹Λ^s;

for details see [19]. Now it remains to observe that ad^α(D_x)ad^β(x)A(t, τ)⁻¹ is a finite linear combination of terms of the form

A(t, τ)⁻¹ ad^α¹(D)ad^β¹(x)A(t, τ)

A(t, τ)⁻¹×. . . . . .× ad^α^k(D)ad^β^k(x)A(t, τ)

A(t, τ)⁻¹

withα1+. . . α_k=α andβ1+. . .+β_k=β.

Theorem 3.4. Fora∈S_V^0,0;ω the following statements are equivalent:

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a) There exist symbols b₁, b₂ ∈S_V^0,0;1/ω such that, for some ε >0, a#_xb₁−1, b₁#_xa−1 ∈ S^0,−ε_V ,

a#tb2−1, b2#ta−1 ∈ S^−ε,0_V . (3.1)

b) There exist symbolseb₁,eb₂ ∈S_V^0,0;1/ω such that

(Tiσa)#xeb1−1 =eb1#x(Tiσa)−1 = 0, χ2(a#teb2−1) =χ2(eb2#ta−1) = 0 (3.2)

for some σ ≥0 and some zero excision function χ₂(ξ).

c) a has a parametrix.

Proof. a)⇒b): Since r1 := 1−a#xb1 belongs to S^0,−ε, we find aσ >0 such that k(T_iσr1)(x, t, Dx, τ)k_L_(L₂₍_Rn))≤ 1

2 ∀(t, τ)∈R×H. Proposition 3.3 implies that there exists a symbolb∈S_V^0,0 such that

b(x, t, Dx, τ) = 1−(Tiσr1)(x, t, Dx, τ)−1

∀(t, τ)∈R×H.

It follows that (Tiσa)#x(Tiσb1)#xb= 1, i.e. we chooseeb1 = (Tiσb1)#xb. The construction of eb₂ is done similarly, using b(x, t, x, D_t) = 1−χ(ξ)r₂(x, t, x, D_t)−1

withr₂ := 1−a#_tb₂ and a suitable zero excision function χ. The claim then follows foreb2 := b2#tb and a χ2 with χ2χ=χ2.

b)⇒ c): By the first assumption in (3.2)

r1 := 1−(Tiσa)eb1= (Tiσa)#xeb1−(Tiσa)eb1 ∈ S_V^−1,0.

Choosing zero excision functions χ(ξ),e χe2(ξ) with χe2χe = χe2 and |χre 1| ≤ ¹₂, we obtain χe2(Tiσa)⁻¹ =χe2eb1(1−χr1)⁻¹ ∈S^0,0;ω. Without loss of generality we may assume χe2=χ2.³ We thus can define

b:=eb₁+χ₂eb₂−χ₂(T_iσa)⁻¹ ∈ S^0,0;ω. By direct computation,

eb1−b=χ2(Tiσa)⁻¹ Tiσ(1−aeb2)

−χ2(1−Tiσ)eb2.

Sinceχ₂(1−aeb₂) =χ₂(a#_teb₂−aeb₂)∈S_V^0,−1 and 1−T_iσ:S_V^0,0;1/ω →S_V^0,−1;1/ω due to Lemma 3.2, we obtaineb₁−b∈S_V^0,−1;1/ω. Similarlyeb₂−b∈S_V^−1,0;1/ω, since

eb₂−b= (1−χ₂)(eb₂−eb₁) +χ₂(T_iσa)⁻¹(1−(T_iσa)eb₁).

This yields

a#_xb≡a#_xeb₁ = (T_iσa)#_xeb₁+ (1−T_iσ)a

#_xeb₁ ≡1 modS_V^0,−1, a#tb≡a#teb2=χ2(a#teb2) + (1−χ2)(a#teb2)≡χ2 ≡1 modS_V^−1,0.

3Chooseχe2 in such a way thatχe2χ2=χe2. Then the assumption remains true forχe2instead ofχ2.

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Therefore,

a#b≡ab+ (a#xb−ab) + (a#tb−ab)

=







1 + (a#xb−1) + (a#tb−ab)≡1 modS_V^0,−1 1 + (a#xb−ab) + (a#tb−1)≡1 modS_V^−1,0 .

It follows thatr := 1−a#b∈S_V^−1,0∩S_V^0,−1⊂S⁻

1 2,−¹

2

V . By the standard von Neumann series argument, using Theorem 2.9, we can construct a right-parametrix p_R. Analogously there exists a left-parametrix pL. Then we can choose p both aspLor pR.

c)⇒ a): If p is a parametrix of athen

1≡a#p≡ap+ (a#xp−ap) + (a#tp−ap) modS^−1,−1.

Hence a#xp−1 ≡ −(a#_tp−ap) ≡ 0 modulo S_V^0,−1, i.e., a#xp−1 ∈ S_V^0,−1. Analogously, a#_tp−1∈S_V^−1,0. Thus (3.1) holds with b₁=b₂ =p and ε= 1.

3.2. Symbols with coupling property. It is desirable to derive the existence of a parametrix by more simple conditions than those given in Theorem 3.4. To this end let us introduce the following notion:

Definition 3.5. Let J ⊂Rbe a closed interval.⁴We call a∈S_V^0,0;ω(Rⁿ×R,Rⁿ×H)weakly parabolic on J if there exists an open intervalI containing J such that

(3.3) |a(x, t, ξ, τ)| ≥C ω(ξ, τ) ∀(x, t)∈Rⁿ×I ∀ |(ξ, τ)| ≥R, for some constants C >0 andR≥0.

In general, weak parabolicity onRof the symbol a∈S_V^0,0;ω(Rⁿ×R;Rⁿ×H) does not imply the existence of a parametrix. However, as we shall show below, this is true when additionally imposing thatahas the coupling property

(C) D_ξ_ja, D_τa ∈ S_V^{−ε,−ε;ω}(Rⁿ×R;Rⁿ×H), j= 1, . . . , n,

for someε >0, i.e., the decay improves simultaneously in both covariables even if derivatives are only taken with respect to one of the covariables. Intuitively this means that there is some coupling between the covariables ξ and τ. For example, symbols from the anisotropic symbol class considered in [11] satisfy (C) (for a suitable choice ofω).

Theorem 3.6. Leta∈S_V^0,0;ω(Rⁿ×R,Rⁿ×H)satisfy at least one of the following assumptions:

(i) a has constant coefficients, (ii) a has the coupling property (C).

Then ahas a parametrix if and only if it is weakly parabolic on R.

Proof. First assume thatahas constant coefficients. Clearly estimate (3.3) follows from the existence of a parametrix. For the reverse implication we may assume without loss of generality thatω= 1. We shall construct a parametrix. By the von Neumann series argument it suffices

4This includesJ=RandJ= [t0,∞).

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to find a parametrix modulo S_V^−1,−1. We make use of the mappings ˆ·and ˜·from Lemma 2.7.

Observe that|(ξ,hξi⁻¹τ)| ≥R whenever |ξ| ≥R. If|ξ| ≤Rthen |(ξ,hξi⁻¹τ)| ≥ hξi⁻¹|τ| ≥R whenever |τ| ≥RhRi. It follows that

|ba(ξ, τ)⁻¹| ≤C ∀ |(ξ, τ)| ≥S :=Rp

1 +hRi². Hence, forσ ≥S,

|(T_iσba)(ξ, τ)⁻¹| ≤C ∀(ξ, τ)∈Rⁿ×H, i.e. (T_iσba)⁻¹ ∈Λ^0,0_V . Now let b:= [(T_iσba)⁻¹]

˜

^{. Then} ^b^∈^S_V^0,0 ^and

abb =ba(T_iσba)⁻¹ = (ba−T_iσba)(T_iσba)⁻¹+ 1≡1 mod Λ^0,−1_V

according to Lemma 3.2. Applying the map ˜· we obtain ab−1∈ S_V^−1,−1. The argument for ba−1 is the same.

Now assume thatahas the coupling property and thata⁻¹ satisfies estimate (3.3). Then, for sufficiently largeσ,

b:= (Tiσa)⁻¹ ∈ S_V^0,0;1/ω.

By chain-rule also bsatisfies (C). By (C),a−Tiσa ∈ S_V^{−ε,−ε;ω} (see the formula in the proof of Lemma 3.2). Hence

ab= (a−Tiσa)b+ (Tiσa)b≡1 mod S_V^−ε,−ε. Moreover,

a#xb−ab≡(Tiσa)#xb−(Tiσa)b mod S_V^−ε,−ε. However, the symbol on the right-hand side equals

P

|α|=1

Z ₁

0

nZ Z

e^−iyη(∂_ξ^αa)(x, t, ξ+θη, τ −iσ)(D^α_xb)(x+y, t, ξ, τ)dyd¯ηo dθ,

(oscillatory integral) which is easily seen to belong to S_V^−ε,−ε due to (C). Thusa#_xb−ab∈ S_V^−ε,−ε. Analogously one shows thata#tb−ab∈S_V^−ε,−ε. Altogether we have obtained that

a#b−1≡(ab−1) + (a#_xb−ab) + (a#_tb−ab)≡0 mod S_V^−ε,−ε.

Arguing in the same way, also b#a−1 ∈ S_V^−ε,−ε. With the standard von Neumann series argument we then can construct a parametrix toa.

Ifahas the coupling property and possesses a parametrix p, we write

ap−1≡(a#p−1)−(a#xp−ap)−(a#tp−ap) mod S_V^−1,−1.

Due to (C), the second and third summand on the right-hand side belong toS_V^−ε,−ε (replace aboveTiσaby aandb by p). Henceap−1∈S_V^−ε,−ε and the desired estimate follows.

4. Equations on a space-time cylinder with closed cross-section

While in the previous section we considered operators on Rⁿ×R we shall now focus on operators onX×Rfor a smooth closed manifold X.