Cone pseudodifferential operators in the edge symbolic calculus

J.B. Gil ^{∗}  B.-W. Schulze  J. Seiler

February 1998

Abstract

We consider a class of parameter-dependent pseudodifferential operators on the model cone (with smooth compact cross section) of a wedge, which is shown to be a symmetric algebra. Its construction relies on a particular quantization of corresponding edgedegenerate symbols, based on the Mellin transform along the axial variable of the cone. These operator families are essential for the analysis of pseudodifferential operators on manifolds with edges. In particular, they serve as symbols for Laplace-Beltrami operators to wedge metrics and their parametrices if the parameter is treated as the covariable.

AMS Classification: 58G20, 58G99, 35S15

Introduction

The present paper studies an algebra of parameter-dependent pseudodifferential operators on a manifold with conical singularities, where the parameters are involved as covariables in a specific degenerate way. Such operator families serve as an adequate symbol class for pseudodifferential operators on manifolds with edges. Also the study of resolvents of differential operators of Fuchs type leads to families of a similar form.

Our results fit into the frame of pseudodifferential calculi on manifolds with singularities, particularly with piecewise smooth geometry. They belong to the idea to reflect the stratification of such a space by a hierarchy of operator algebras with symbolic structure, and to organize an iterative procedure which starts from the calculus on a given space, say a cone, and constructs a next ‘higher’ calculus on a space with higher order singularities, say a wedge. It is well-known that, for instance, boundary value problems for pseudodifferential operators can be represented as operators along the boundary with operator-valued symbols, acting along ℝ_{+}, the inner normal. In this sense, not only Boutet de Monvel’s algebra [1] (cf. also Schrohe, Schulze [7]) of boundary value problems with the transmission property is included in the context but also Vishik, Eskin’s theory [3], [12], turned into a corresponding operator algebra, cf. Schulze [10]. The iteration of calculi leads to very complex analytic phenomena and it is still a serious problem to formulate manageable operator algebras for higher singularities such as of corner type or for boundary value problems with such singular boundaries.

∗Supported by Max-Planck-Gesellschaft

The main objective of our paper is to develop an efficient new approach to the algebra of cone operator-valued edge symbols as originally introduced in [9]. One of the difficulties is that the edge covariables are involved in a degenerate form, i.e. multiplied by the axial variable of the cone (cf. [2], [10]). This edge-degenerate behaviour is a consequence of the wedge geometry, and can be observed, for instance, in the symbols of corresponding Laplace-Beltrami operators. In the new representation we can, in particular, avoid a number of extremely voluminous calculations in the precise analysis of operator-valued edge symbols by a new quantization of edge-degenerate interior symbols in which a part of the above inconvenient combinations of edge covariable and axial variable is dismissed. This relies on a form of the Mellin quantization for edge-degenerate pseudodifferential symbols from Gil, Schulze, Seiler [4].

We begin this paper with a short discussion of oscillatory integrals in the spirit of Kumano-go [6], here generalized to amplitude functions with values in Fréchet spaces. It is an important tool to obtain the composition result of the parameter-dependent cone pseudodifferential operators.

Section 2 provides the basic material about the Mellin calculus. We introduce there the class of holomorphic pseudodifferential operators (based on the Mellin transform) as well as the cone Sobolev spaces.

In Section 3 we discuss a class of parameter-dependent cone operators as used in the edge symbolic calculus, formulated in the old and new fashion. The main result of this section is Theorem 3.18, where we prove the equivalence of both representations. We also give a global Mellin quantization which plays an essential role in the proof above.

Some properties of our operator-valued edge symbols and some basic elements of the pseudodifferential calculus (e.g. composition) are treated in the last section.

Finally, a part of the results are postponed to the appendix in order to keep the exposition of this paper transparent.

Acknowledgement: We thank Prof. E. Schrohe for valuable discussions.

Basic notation

Let ℝ_{+} = \{r ∈ ℝ | r > 0\}, \overline{ℝ}_{+} = ℝ_{+} ∪ \{0\}, ℕ_{0} = ℕ ∪ \{0\}. For real β set Γ_{β} = \{z ∈ ℂ| Re z = β}.

A cut-off function is a non-negative function σ ∈ C_{0}^{∞}(\overline{ℝ}_{+}) with σ ≡ 1 near t = 0. For functions φ, ψ we write φ ≺ ψ if φψ = φ and supp φ ∩ supp (1 − ψ) = ∅. A function χ ∈ C^{∞}(ℝ^{q}) is called an excision function at η = η_{0} if 0 ≤ χ ≤ 1, χ(η) = 0 on some neighborhood of η_{0} and χ(η) = 1 for large |η − η_{0}|.

For u ∈ C_{0}^{∞}(ℝ^{n}), v ∈ C_{0}^{∞}(ℝ_{+}) the Fourier and Mellin transform respectively, are given by

Fu(ξ) = ∫_{ℝ^{n}} e^{−ixξ}u(x) dx, Mv(z) = ∫_{0}^{∞} t^{z−1}v(t) dt.

These transforms can be extended to more general (distribution) spaces.

Let U ⊂ ℝ^{m} be open, and set ⟨ξ⟩ = (1 + |ξ|^{2})^{1/2} for ξ ∈ ℝ^{n}. Then S^{μ}(U × ℝ^{n}) consists of all p ∈ C^{∞}(U × ℝ^{n}) with

sup_{x∈K, ξ∈ℝ^{n}} {|D_{ξ}^{α}D_{x}^{β}p(x, ξ)| ⟨ξ⟩^{|α|−μ}} < ∞  (1)

for all α ∈ ℕ_{0}^{n}, β ∈ ℕ_{0}^{m}, and all compact sets K ⊂ U. This is a Fréchet space. Moreover, set S^{μ}(\overline{ℝ}_{+} × U × ℝ^{n}) = S^{μ}(ℝ × U × ℝ^{n})|_{\overline{ℝ}_{+}×U×ℝ^{n}}. This is a Fréchet space if we take as semi-norms the analogous expressions as in (1), where now K is a compact set in \overline{ℝ}_{+} × U.

For Λ = ℝ^{l} or Λ = Γ_{β} × ℝ^{l} ≅ ℝ^{1+l} we associate with a symbol p ∈ S^{μ}(U × U × ℝ^{n} × Λ), U ⊂ ℝ^{n}, its parameter-dependent pseudodifferential operator op_{x}(p)(λ) : C_{0}^{∞}(U) → C^{∞}(U) by

[op_{x}(p)(λ)u](x) = ∫∫ e^{i(x−x′)ξ}p(x, x′, ξ, λ)u(x′) dx′d¯ ξ,  (2)

where d¯ ξ = (2π)^{−n} dξ.

By pasting together (via partition of unity) on a smooth compact manifold X operators of the form (2) we obtain the space L^{μ}(X; Λ) of parameter-dependent pseudodifferential operators on X.

We denote by X^{∧} the half-cylinder ℝ_{+}×X. With the symbols P (t, t′, τ, λ) ∈ C^{∞}(ℝ_{+}× ℝ_{+}, L^{μ}(X; ℝ_{τ} ×Λ)), where n = dim X, we associate the corresponding operator op_{t}(P)(λ) viewing the parameter τ as a covariable. The space of all these operators is denoted by L^{μ}(X^{∧}; Λ). For F (t, t′, z, λ) ∈ C^{∞}(ℝ_{+} × ℝ_{+}, L^{μ}(X; Γ_{\frac{1}{2} −γ} × Λ)) we also consider

[op_{M}^{γ}(F)(λ)u](t) = ∫_{Γ_{\frac{1}{2} −γ}} ∫_{0}^{∞} (\frac{t}{t′} )^{−z} F (t, t′, z, λ)u(t′)\frac{dt′}{t′} d¯ z,

where d¯ z = (2πi)^{−1}dz, and for t′ fixed, u(t′) is viewed as a function in C_{0}^{∞}(X). We further define

L^{μ}(X^{∧}; Λ)_{0} := {P (λ) ∈ L^{μ}(X^{∧}; Λ) | σP (λ)σ̃ = P (λ) for some cut-off functions σ, σ̃}.

The intersection ⋂_{μ∈ℝ} L^{μ} is denoted by L^{−∞} in all the cases.

Finally, for some Fréchet space E let C_{b,F}^{∞} (ℝ_{+}×ℝ_{+}, E) be the space of smooth E-valued functions bounded with all their totally characteristic derivatives (t∂_{t})^{k} (t′∂_{t′})^{l}.

1 Oscillatory integrals in Fréchet spaces

The purpose of this section is to generalize oscillatory integral technique to the case of amplitude functions with values in Fréchet spaces. Our approach is based on that of Kumano-go [5], [6] for the scalar case. As a minor modification we use the Mellin instead of the Fourier transform.

In the following let E be a Fréchet space.

Definition 1.1 Let T (ℝ_{+} × ℝ, E) be the space of all functions u ∈ C^{∞}(ℝ_{+} × ℝ, E) such that

(Su)(t_{1}, t_{2}) := u(e^{−t_{1}}, t_{2}) ∈ S(ℝ_{t_{1}} × ℝ_{t_{2}}).

A Fréchet topology on T (ℝ_{+} × ℝ, E) is defined by requiring the map S : T (ℝ_{+} × ℝ, E) → S(ℝ^{2}, E) to be a topological isomorphism. If E = ℂ we suppress E from the notation.

Definition 1.2 (i) The space of amplitude functions, A(ℝ_{+} × Γ_{0}, E), consists of all h ∈ C^{∞}(ℝ_{+} × Γ_{0}, E) such that for each continuous semi-norm p of E there exist reals m = m_{p}, μ = μ_{p} such that

sup {p ((∂_{z}^{k}(s∂_{s})^{l}h)(s, iξ)) ⟨log s⟩^{−m} ⟨ξ⟩^{−μ} | (s, ξ) ∈ ℝ_{+} × ℝ} < ∞  (3)

for all k, l ∈ ℕ_{0}.

(ii) By A(ℝ_{+} × ℂ, E) denote the space of holomorphic amplitude functions, i.e. all h ∈ C^{∞}(ℝ_{+}, O(ℂ, E)) such that for each continuous semi-norm p of E there exist reals m = m_{p}, μ = μ_{p} such that

sup {p ((∂_{z}^{k}(s∂_{s})^{l}h)(s, δ + iξ)) e^{−m⟨log s⟩} ⟨ξ⟩^{−μ} | (s, ξ) ∈ ℝ_{+} × ℝ, |δ| ≤ j} < ∞

for all j, k, l ∈ ℕ_{0}.

Remark 1.3 To verify that a function h is an amplitude it suffices to check the corresponding estimates for an arbitrary system {p_{n} | n ∈ ℕ} of continuous semi-norms of E that determines the topology of E.

The following facts are immediate:

Lemma 1.4 (i) If h ∈ A(ℝ_{+} × Γ_{0}, E) then ∂_{z}^{k}(s∂_{s})^{l}h ∈ A(ℝ_{+} × Γ_{0}, E);

(ii) let E_{j} be Fréchet spaces and the topology in E be the projective topology with respect to the linear maps T_{j} : E → E_{j}. Then h ∈ A(ℝ_{+} × Γ_{0}, E) if and only if T_{j}(a) ∈ A(ℝ_{+} × Γ_{0}, E_{j}) for each j;

(iii) let E_{0}, E_{1} be two Fréchet spaces and (·, ·) : E_{0} × E_{1} → E be a continuous and bilinear map. If h_{j} ∈ A(ℝ_{+} × Γ_{0}, E_{j}), j = 0, 1, then (h_{0}, h_{1}) ∈ A(ℝ_{+} × Γ_{0}, E);

(iv) let V be a closed subspace of E. For e ∈ E let [e] = e + V. Then h ∈ A(ℝ_{+} × Γ_{0}, E)

implies that [h] ∈ A(ℝ_{+} × Γ_{0}, E/V). Analogous statements are true for holomorphic amplitude functions.

Definition 1.5 A function χ_{ε}(s, z) : ]0, 1] × ℝ_{+} × Γ_{0} → ℂ is called regularizing, if

(i) χ_{ε} ∈ T (ℝ_{+} × Γ_{0}) for each ε;

(ii) sup{|(∂_{z}^{k}(s∂_{s})^{l}χ_{ε})(s, ξ)| | 0 < ε ≤ 1, (s, ξ) ∈ ℝ_{+} × ℝ} < ∞ for each k, l ∈ ℕ_{0};

(iii) ∂_{z}^{k}(s∂_{s})^{l}χ_{ε}(s, ξ) ⟶ \{ \begin{matrix}1, & k + l = 0 \\ 0, & k + l ≠ 0 \end{matrix} pointwise on ℝ_{+} × ℝ as ε tends to 0.

Example 1.6 Let χ ∈ T (ℝ_{+} × ℝ) with χ(1, 0) = 1, and set χ_{ε}(s, iξ) = χ(s^{ε}, εξ). Then χ_{ε} is regularizing in the sense of Definition 1.5.

Definition 1.7 A function χ_{ε}(s, z) : ]0, 1] × ℝ_{+} × ℂ → ℂ will be called holomorphically regularizing, if

(i) (ε, s, iξ) ↦ χ_{ε}(s, δ + iξ) is regularizing in the sense of Definition 1.5 for each δ ∈ ℝ;

(ii) z ↦ χ_{ε}(s, z) is entire, and ξ ↦ χ_{ε}(s, δ + iξ) ∈ S(ℝ) uniformly for δ in compact intervals;

(iii) for each ε there is a compact set K_{ε} ⊂ ℝ_{+} such that χ_{ε}(s, z) = 0 whenever s ∉ K_{ε}.

Example 1.8 Let φ ∈ C_{0}^{∞}(ℝ_{+}) with φ(1) = Mφ(0) = 1. Here, M is the Mellin transform. The function χ_{ε}(s, z) = φ(s^{ε})Mφ(εz) is regularizing in the sense of Definition 1.7.

Theorem 1.9 Let h ∈ A(ℝ_{+} × Γ_{0}, E) and let χ_{ε} be regularizing. Then the limit

Os[h] = ∫∫_{0}^{∞} s^{iξ}h(s, iξ) \frac{ds}{s}d¯ ξ := lim_{ε→0} ∫∫_{0}^{∞} s^{iξ}χ_{ε}(s, iξ)h(s, iξ) \frac{ds}{s}d¯ ξ

exists in E and is independent of the choice of χ_{ε}. The same holds for h ∈ A(ℝ_{+} × ℂ, E) with χ_{ε} being holomorphically regularizing. In particular, both definitions of the oscillatory integral coincide on A(ℝ_{+} × Γ_{0}, E) ∩ A(ℝ_{+} × ℂ, E).

This result essentially follows from integration by parts.

Remark 1.10 Let {p_{n} | n ∈ ℕ} be a fixed system of continuous semi-norms that determines the topology of E, and μ̄ = (μ_{n}), m̄ = (m_{n}) be fixed sequences of reals. If we denote by A^{μ̄,m̄}(ℝ_{+} × Γ_{0}, E) the space of all amplitude functions h that satisfy the estimates (3) for p_{n} with m = m_{n} and μ = μ_{n}, then A^{μ̄,m̄}(ℝ_{+}×Γ_{0}, E) carries a natural Fréchet topology. The map h ↦ Os[h] : A^{μ̄,m̄}(ℝ_{+} × Γ_{0}, E) → E is linear and continuous. Moreover,

A(ℝ_{+} × Γ_{0}, E) = ∪_{μ̄,m̄} A^{μ̄,m̄}(ℝ_{+} × Γ_{0}, E).

Here, the union is taken over all sequences μ̄, m̄. Analogous statements hold for holomorphic amplitude functions.

2 Mellin pseudodifferential operators

Pseudodifferential operators based on the Mellin transform constitute a special ingredient of the cone algebra (cf. [2] and [10]). They are adequate for the treatment of differential operators of Fuchs type which are the natural ones on manifolds with conical singularities. Such differential operators are roughly speaking polynomials in the totally characteristic derivative −t∂_{t}. For this reason microlocalization by means of the Mellin transform M_{t→z} is natural since −t∂_{t} corresponds in the Mellin image to multiplication by the complex variable z. For an introductory exposition of the cone calculus we refer the reader to the chapters 7 and 8 of [2].

2.1 Weighted cone Sobolev spaces

For a compact smooth manifold X we construct the Sobolev spaces on X^{∧} = ℝ_{+} × X in a way that they satisfy a number of natural conditions, in particular, for smoothness s ∈ ℝ they are subspaces of H_{loc}^{s}(X^{∧}) and contain H_{comp}^{s}(X^{∧}) as a subspace. The cone Sobolev spaces can be understood as a suitable modification of the usual spaces H^{s}, defined with help of the Mellin transform in the axial direction t ∈ ℝ_{+}.

First of all, for s, γ ∈ ℝ define H^{s,γ}(ℝ_{+} × ℝ^{n}) as the closure of C_{0}^{∞}(ℝ_{+} × ℝ^{n}) with respect to the norm

∥u∥_{H^{s,γ}}^{2} = ∫_{Γ_{\frac{n+1}{2} −γ}} ∫_{ℝ^{n}}(1 + |z|^{2} + |ξ|^{2})^{s}|(M_{t→z}F_{x→ξ}u)(z, ξ)|^{2} |dz|dξ.

Definition 2.1 For s, γ, ϱ ∈ ℝ we introduce the cone Sobolev spaces H^{s,γ}(X^{∧}) and K^{s,γ}(X^{∧})^{ϱ} as the closure of C_{0}^{∞}(X^{∧}) with respect to the norms

∥u∥_{H^{s,γ}(X^{∧})}^{2} = ∑_{j=1}^{N} ∥(φ_{j}u) ∘ (1 × θ_{j})^{−1}∥_{H^{s,γ}(ℝ_{+}×ℝ^{n})}^{2} and

∥u∥_{K^{s,γ}(X^{∧})^{ϱ}}^{2} = ∥σu∥_{H^{s,γ}(X^{∧})}^{2} + ∑_{j=1}^{N} ∥[(1 − σ)φ_{j}u] ∘ κ_{j}^{−1}∥_{H^{s,ϱ}(ℝ^{1+n})}^{2},

respectively. Here, {φ_{j}}_{j=1,... ,N} is a partition of unity subordinate to the covering X = ⋃_{j=1}^{N} U_{j}, θ_{j} : U_{j} → V_{j} ⊂ ℝ^{n} are charts, and

κ_{j}(t, x) = tκ̃_{j}(x) : ℝ_{+} × U_{j} → ℝ^{1+n}

for diffeomorphisms κ̃_{j} : U_{j} → Ṽ_{j} ⊂ S^{n}, with S^{n} being the unit sphere in ℝ^{1+n}. Moreover, H^{s,ϱ}(ℝ_{y}^{1+n}) = ⟨y⟩^{−ϱ} H^{s}(ℝ_{y}^{1+n}) are the usual weighted Sobolev spaces on ℝ^{1+n} and σ ∈ C_{0}^{∞}(\overline{ℝ}_{+}) is a cut-off function. In case ϱ = 0 we suppress it from the notation. Note that other localization data yield equivalent norms.

Remark 2.2 The following elementary properties are valid:

(i) H^{s,γ}(X^{∧}) and K^{s,γ}(X^{∧})^{ϱ} are Hilbert spaces. In particular, H^{0,0}(X^{∧}) = K^{0,0}(X^{∧}) = t^{− \frac{n}{2}} L^{2}(X^{∧}), where X^{∧} is equipped with the product metric;

(ii) t^{−μ}H^{s,γ}(X^{∧}) = H^{s,γ−μ}(X^{∧}) and t^{−μ}K^{s,γ}(X^{∧})^{ϱ} = K^{s,γ−μ}(X^{∧})^{ϱ+μ};

(iii) H_{comp}^{s}(X^{∧}) ↪ H^{s,γ}(X^{∧}) ↪ H_{loc}^{s}(X^{∧}), and the same is true for K^{s,γ}(X^{∧})^{ϱ}; (iv) the scalar product in t^{− \frac{n}{2}} L^{2}(X^{∧}) extends to non-degenerate sesquilinear pairings

H^{s,γ}(X^{∧}) × H^{−s,−γ}(X^{∧}) → ℂ and K^{s,γ}(X^{∧})^{ϱ} × K^{−s,−γ}(X^{∧})^{−ϱ} → ℂ, respectively;

(v) the embedding H^{s,γ}(X^{∧}) ↪ H^{s′,γ}(X^{∧}) is continuous if s ≥ s′;

(vi) the embedding K^{s,γ}(X^{∧})^{ϱ} ↪ K^{s′,γ′}(X^{∧})^{ϱ′} is continuous if s ≥ s′, γ ≥ γ′, ϱ ≥ ϱ′, and compact if s > s′, γ > γ′, ϱ > ϱ′.

Proposition 2.3 Let s, γ ∈ ℝ. Then there exist ϱ, ϱ′ ∈ ℝ such that

H^{s,γ}(X^{∧}) ↪ K^{s,γ}(X^{∧})^{ϱ}, K^{s,γ}(X^{∧})^{ϱ′} ↪ H^{s,γ}(X^{∧}).

Proof: Follows from Lemmas 3.1.20 and 4.2.2 in [7]. □

Definition 2.4 Using the pairing from Remark 2.2(iv), we associate to every operator A ∈ L(K^{s,γ}(X^{∧})^{ϱ}, K^{s′,γ′}(X^{∧})^{ϱ′}) the formal adjoint A^{∗} ∈ L(K^{−s′,−γ′}(X^{∧})^{−ϱ′}, K^{−s,−γ}(X^{∧})^{−ϱ}) that satisfies

(Au, v) = (u, A^{∗}v) for all u, v ∈ C_{0}^{∞}(X^{∧}).

Analogously, for every operator A ∈ L(H^{s,γ}(X^{∧}), H^{s′,γ′}(X^{∧})) we also define the formal adjoint A^{∗} ∈ L(H^{−s′,−γ′}(X^{∧}), H^{−s,−γ}(X^{∧})).

Next we introduce a family of isomorphisms on the cone Sobolev spaces defined above.

Definition 2.5 For each λ > 0 define the linear map κ_{λ} : C_{0}^{∞}(X^{∧}) → C_{0}^{∞}(X^{∧}) by

(κ_{λ}u)(t, x) := λ^{\frac{n+1}{2}} u(λt, x),

where n = dim X. These mappings extend by continuity to linear operators on K^{s,γ}(X^{∧})^{ϱ} and H^{s,γ}(X^{∧}) for all s, γ, ϱ ∈ ℝ, and the set {κ_{λ}}_{λ∈ℝ_{+}} is a (strongly continuous) group of isomorphisms, that means,

(i) κ_{λ}κ_{ϱ} = κ_{λϱ} for all λ, ϱ > 0;

(ii) for each u ∈ K^{s,γ}(X^{∧})^{ϱ} the function λ ↦ κ_{λ}u : ℝ_{+} → K^{s,γ}(X^{∧})^{ϱ} is continuous, and analogously for u ∈ H^{s,γ}(X^{∧}).

2.2 Mellin operators

Before we pass to the parameter-dependent version of the cone algebra we recall some definitions and properties of the Mellin pseudodifferential operators with operator-valued symbols.

Definition 2.6 For γ ∈ ℝ let T_{γ}(ℝ_{+}, C^{∞}(X)) denote the space of all functions u ∈ C^{∞}(ℝ_{+}, C^{∞}(X)) such that

(S_{γ}u)(t) := e^{(γ− \frac{1}{2} )t}u(e^{−t}) ∈ S(ℝ_{t}, C^{∞}(X)).

The topology is that inherited from S(ℝ, C^{∞}(X)) via the isomorphism S_{γ}. We view T_{γ}(ℝ_{+}, C^{∞}(X)) as a subspace of C^{∞}(X^{∧}).

To each symbol h ∈ C_{b,F}^{∞} (ℝ_{+} × ℝ_{+}, L^{μ}(X; Γ_{\frac{1}{2} −γ})) we associate a continuous Mellin pseudodifferential operator

op_{M}^{γ}(h) : T_{γ}(ℝ_{+}, C^{∞}(X)) → T_{γ}(ℝ_{+}, C^{∞}(X)),

which is defined by

[op_{M}^{γ}(h)u](t) = ∫∫_{0}^{∞} (\frac{t}{t′} )^{−( \frac{1}{2} −γ+iτ)} h(t, t′, \frac{1}{2} − γ + iτ)u(t′) \frac{dt′}{t′} d¯ τ.

Here, the integrand is viewed as an amplitude function in A(ℝ_{+} × Γ_{0}, C^{∞}(X)) for each fixed t > 0.

Remark 2.7 Let h ∈ C_{b,F}^{∞} (ℝ_{+} ×ℝ_{+}, L^{μ}(X; Γ_{\frac{1}{2} −γ})). One can check (as we shall do it later on in a similar situation, cf. Lemma 4.1) that

a(s, iξ) := ( (t, z) ↦ h(t, st, z + iξ) )

belongs to A(ℝ_{+} ×Γ_{0}, C_{b,F}^{∞} (ℝ_{+}, L^{μ}(X; Γ_{\frac{1}{2} −γ})), i.e. the function a is an amplitude function with values in C_{b,F}^{∞} (ℝ_{+}, L^{μ}(X; Γ_{\frac{1}{2} −γ})). Hence the oscillatory integral

h_{L}(t, z) = ∫∫_{0}^{∞} s^{iξ}h(t, st, z + iξ) \frac{ds}{s}d¯ ξ

converges in C_{b,F}^{∞} (ℝ_{+}, L^{μ}(X; Γ_{\frac{1}{2} −γ})). Actually, h_{L} is the left-symbol satisfying op_{M}^{γ}(h) = op_{M}^{γ}(h_{L}). Analogously, we get the existence of the corresponding right-symbol h_{R}.

Proposition 2.8 Each h ∈ C_{b,F}^{∞} (ℝ_{+} × ℝ_{+}, L^{μ}(X; Γ_{\frac{1}{2} −γ})) induces a continuous operator

op_{M}^{γ}(h) : H^{s,γ+ \frac{n}{2}} (X^{∧}) → H^{s−μ,γ+ \frac{n}{2}} (X^{∧})

for each s ∈ ℝ. Moreover, we have continuity of the mapping

h ↦ op_{M}^{γ}(h) : C_{b,F}^{∞} (ℝ_{+} × ℝ_{+}, L^{μ}(X; Γ_{\frac{1}{2} −γ})) → L(H^{s,γ+ \frac{n}{2}} (X^{∧}), H^{s−μ,γ+ \frac{n}{2}} (X^{∧})).

Proof: By order reduction and conjugation with t^{γ}, the proof can be reduced to the case op_{M}^{0}(h) : L^{2}(X^{∧}) → L^{2}(X^{∧}) and μ = 0. Since L^{2}(X^{∧}) = L^{2}(ℝ_{+}, L^{2}(X)), the result follows from an easy extension of the usual Calderón-Vaillancourt Theorem (in the formulation for Mellin pseudodifferential operators) to the operator-valued case. □

Definition 2.9 For μ ∈ ℝ ∪ {−∞} we denote by M_{O}^{μ}(X; ℝ^{q}) the subspace of all holomorphic functions h(·, η) : ℂ → L^{μ}(X; ℝ^{q}) such that

h(β + iτ, η) ∈ L^{μ}(X; ℝ_{τ} × ℝ_{η}^{q})

uniformly for β in compact intervals. This is a Fréchet space with the following system of semi-norms:

sup_{|β|≤N} p_{N}(h(β + iτ, η)), N ∈ ℕ,

where {p_{N}}_{N∈ℕ} is a system of semi-norms in L^{μ}(X; ℝ × ℝ^{q}). In particular, in the case dim X = 0 we replace L^{μ}(X; ℝ^{q}) by S^{μ}(ℝ^{q}) and write M_{O}^{μ}(ℝ^{q}). The class without the parameter η ∈ ℝ^{q} will be denoted by M_{O}^{μ}(X).

It is easy to obtain the following properties:

Lemma 2.10 For h ∈ C_{b,F}^{∞} (ℝ_{+} × ℝ_{+}, M_{O}^{μ}(X)) is valid:

(i) κ_{λ}^{−1} op_{M}^{γ}(h) κ_{λ} = op_{M}^{γ}(h_{λ}) with h_{λ}(t, t′, z) = h(λ^{−1}t, λ^{−1}t′, z);

(ii) if we set (T^{σ}h)(t, t′, z) = h(t, t′, z + σ) then op_{M}^{γ}(h) t^{−σ} = t^{−σ} op_{M}^{γ+σ}(T^{σ}h);

(iii) for arbitrary γ, δ ∈ ℝ we have op_{M}^{γ}(h) = op_{M}^{δ}(h) on C_{0}^{∞}(X^{∧}).

Lemma 2.11 Let X be a smooth compact manifold. Then

M_{O}^{−∞}(X; ℝ^{q}) = C^{∞}(X × X) ˆ⊗_{π}M_{O}^{−∞}(ℝ^{q}).

Proof: In view of L^{−∞}(X; ℝ^{q}) = S(ℝ^{q}, C^{∞}(X × X)) and by a direct comparison of semi-norms we obtain M_{O}^{−∞}(X; ℝ^{q}) = C^{∞}(X × X, M_{O}^{−∞}(ℝ^{q})). Then the assertion follows from the nuclearity of C^{∞}(X × X) and the completeness of M_{O}^{−∞}(ℝ^{q}), see Treves [11]. □

3 A class of parameter-dependent cone operators

3.1 Edge degenerate symbols

Edge-degenerate symbols in the sense of the following definition are motivated by the analysis of pseudodifferential operators on a manifold with edges, cf. [9]. They are operator-valued functions depending on an additional covariable η multiplied by t, the distance variable to the conical singularity. The same kind of degeneracy appears when η is interpreted as a spectral parameter to a Fuchs type operator on a manifold with conical singularities. As a crucial result for the structure of our calculus we also formulate in this section the Mellin quantization for such edge-degenerate symbols.

Definition 3.1 Let us introduce the classes

C_{deg}^{∞}(\overline{ℝ}_{+}, L^{μ}(X; ℝ^{1+q})) := \{p | p(t, τ, η) = p̃(t, tτ, tη) with p̃ ∈ C^{∞}(\overline{ℝ}_{+}, L^{μ}(X; ℝ^{1+q}))\} ,

C_{deg}^{∞}(\overline{ℝ}_{+}, M_{O}^{μ}(X; ℝ^{q})) := \{h | h(t, z, η) = h̃(t, z, tη) with h̃ ∈ C^{∞}(\overline{ℝ}_{+}, M_{O}^{μ}(X; ℝ^{q}))\}

of edge degenerate symbols of Fourier and holomorphic Mellin type. The operators of freezing of coefficients at 0 are defined as

p ↦ p_{∧}(t, τ, η) = p̃(0, tτ, tη), h ↦ h_{∧}(t, z, η) = h̃(0, z, tη).

Note that C_{deg}^{∞}(\overline{ℝ}_{+}, F) ⊂ C^{∞}(ℝ_{+}, F) for F = L^{μ}(X; ℝ^{1+q}) or F = M_{O}^{μ}(X; ℝ^{q}). In particular, we may consider the operator-families

op_{t}(p)(η), op_{M}^{γ}(h)(η) : C_{0}^{∞}(X^{∧}) → C^{∞}(X^{∧}),

which are elements of L^{μ}(X^{∧}; ℝ^{q}). Here op_{t} indicates the pseudodifferential operator based on the Fourier transform with respect to the variable t.

Theorem 3.2 (Mellin quantization) Let P ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, L^{μ}(X; ℝ^{1+q})) and let ϕ ∈ C_{0}^{∞}(ℝ_{+}) be a function such that ϕ ≡ 1 near to 1. Then there exists an operator-valued symbol H ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, M_{O}^{μ}(X; ℝ^{q})) such that

op_{t}(P)(η) − opM^{\frac{1}{2}}(H)(η) = op_{t}(Q)(η) ∈ L^{−∞}(X^{∧}; ℝ^{q})

with Q(t, t′, τ, η) = (1 − ϕ(t′/t))P (t, τ, η) for all t, t′ ∈ ℝ_{+}, τ ∈ ℝ, η ∈ ℝ^{q}.

A proof of this theorem is given in the appendix A.1.

3.2 Operator-valued symbols

Our parameter-dependent operator functions on X^{∧} are operator-valued symbols in which the typical rescaling properties of the cone operators together with the degeneracy in η are reflected by a corresponding type of symbol estimates. In this section we summarize some properties of a class, suitable to describe this effect.

In the sequel assume E_{j} = K^{s_{j},γ_{j}}(X^{∧})^{ϱ_{j}} or E_{j} = H^{s_{j},γ_{j}}(X^{∧}) for some reals s_{j}, γ_{j}, ϱ_{j}, and {κ_{λ}|λ > 0} as in Definition 2.5. Moreover, let us fix a smooth function η ↦ [η] : ℝ^{q} → ℝ_{+} with [η] = |η| for |η| > c_{0} for some constant c_{0} > 0.

Definition 3.3 Let μ ∈ ℝ. By S^{μ}(ℝ^{q}; E_{0}, E_{1}) denote the space of all symbols a ∈ C^{∞}(ℝ^{q}, L(E_{0}, E_{1})) that satisfy

∥κ_{[η]}^{−1}(D_{η}^{α}a(η))κ_{[η]}∥_{L(E_{0},E_{1})} ≤ c_{α} [η]^{μ−|α|}

for all α ∈ ℕ_{0}^{q}, and c_{α} some constant independent of η ∈ ℝ^{q}. Further, we set

S^{−∞}(ℝ^{q}; E_{0}, E_{1}) := ⋂_{μ∈ℝ}S^{μ}(ℝ^{q}; E_{0}, E_{1}) = S(ℝ^{q}, L(E_{0}, E_{1})).

If we replace both spaces E_{0}, E_{1} by ℂ and suppose κ_{λ} ≡ 1, we obtain the standard scalar-valued Hörmander class S_{1,0}^{μ}(ℝ^{q}) of symbols with constant coefficients.

Of course, such operator-valued symbols can be formulated on Ω × ℝ^{q} for some open set Ω ⊂ ℝ^{q} or Ω ⊂ ℝ^{q} × ℝ^{q}. It is also possible to admit arbitrary Banach spaces E_{j} with strongly continuous group actions, cf. [2], [10].

Remark 3.4 The following properties hold:

i) S^{μ_{1}}(ℝ^{q}; E_{1}, E_{2}) · S^{μ_{0}}(ℝ^{q}; E_{0}, E_{1}) ⊂ S^{μ_{0}+μ_{1}}(ℝ^{q}; E_{0}, E_{2});

ii) D_{η}^{α}S^{μ}(ℝ^{q}; E_{0}, E_{1}) ⊂ S^{μ−|α|}(ℝ^{q}; E_{0}, E_{1});

iii) S^{μ}(ℝ^{q}; E_{1}, E_{2}) ⊂ S^{μ}(ℝ^{q}; E_{0}, E_{3}) if E_{0} ↪ E_{1} and E_{2} ↪ E_{3}.

Example 3.5 A function f ∈ C^{∞}(ℝ^{q} \ {0}, L(E_{0}, E_{1})) is called (twisted) homogeneous of order μ, if

f(λη) = λ^{μ}κ_{λ}f(η)κ_{λ}^{−1} for all λ > 0 and η ≠ 0.

Then, if χ(η) is a excision function at 0 we have χf ∈ S^{μ}(ℝ^{q}; E_{0}, E_{1}).

This concept of homogeneity allows us to introduce classical symbols:

Definition 3.6 A symbol a ∈ S^{μ}(ℝ^{q}; E_{0}, E_{1}) is called classical, if there is a sequence of functions a_{(μ−j)} ∈ C^{∞}(ℝ^{q} \ {0}, L(E_{0}, E_{1})) that are homogeneous of degree μ − j, such that for any excision function χ(η) at η = 0

a − ∑_{j=0}^{N−1} χa_{(μ−j)} ∈ S^{μ−N}(ℝ^{q}; E_{0}, E_{1})

for every N ∈ ℕ. In this case we write a ∈ S_{cℓ}^{μ}(ℝ^{q}; E_{0}, E_{1}). For j ∈ ℕ_{0} we set

σ_{∧}^{μ−j}(a)(y, η) := a_{(μ−j)}(y, η).

In particular, σ_{∧}^{μ}(a) plays the role of the homogeneous principal symbol of a.

Example 3.7 Let us consider an element f̃(z, η) ∈ L^{μ}(X; Γ_{0} × ℝ_{η}^{q}) and set f(t, z, η) = f̃(z, tη). Then, if ω(t) and ω̃(t) are arbitrary cut–off functions, and ν, ν̃ ∈ ℝ,

a(η) := ω(t[η]) [η]^{ν}tν̃opM^{\frac{1}{2}}(f)(η)ω̃(t[η]) : K^{s, \frac{n+1}{2}} (X^{∧})^{ϱ} → K^{s−μ, \frac{n+1}{2} +ν̃}(X^{∧})^{ϱ′}

is a smooth family in η of continuous operators for every fixed s, ϱ, ϱ′ ∈ ℝ, and we have

a(λη) = λ^{ν−ν̃}κ_{λ}a(η)κ_{λ}^{−1}

for all λ ≥ 1, |η| ≥ c for a constant c > 0. Then

a(η) ∈ S_{cℓ}^{ν−ν̃}(ℝ^{q}; K^{s, \frac{n+1}{2}} (X^{∧})^{ϱ}, K^{s−μ, \frac{n+1}{2} +ν̃}(X^{∧})^{ϱ′}).

Furthermore,

σ_{∧}^{ν−ν̃}(a)(η) = ω(t|η|) |η|^{ν}tν̃opM^{\frac{1}{2}}(f)(η)ω̃(t|η|).

3.3 Green symbols

Algebra operations between operator families of the above kind lead to additional remainder terms, the so-called Green symbols. The notation is motivated by Green’s function in elliptic boundary value problems. For the Laplacian (for instance) this function has, up to a fundamental solution, a boundary symbolic structure of the form of a Green symbol, cf. [10].

Definition 3.8 Let γ, γ′ ∈ ℝ. The space R_{G}^{μ}(ℝ^{q}; (γ, γ′)) of Green symbols with weight data (γ, γ′) consists of all operator-valued functions that satisfy

g ∈ ⋂_{s,s′,ϱ,ϱ′∈ℝ}S_{cℓ}^{μ}(ℝ^{q}; K^{s,γ}(X^{∧})^{ϱ}, K^{s′,γ′}(X^{∧})^{ϱ′}),  (4)
g^{∗} ∈ ⋂_{s,s′,ϱ,ϱ′∈ℝ}S_{cℓ}^{μ}(ℝ^{q}; K^{s,−γ′}(X^{∧})^{ϱ}, K^{s′,−γ}(X^{∧})^{ϱ′}),  (5)

where ∗ denotes the pointwise formal adjoint in the sense of Definition 2.4. The subspace R_{G}^{μ}(ℝ^{q}; (γ, γ′))_{∞} consists of all Green symbols, where in (4) we can replace γ′ by γ′ + ε and in (5) −γ by −γ + ε, and the intersections are taken also over ε > 0.

The properties of operator-valued symbols, cf. Remark 3.4, carry over in an obvious way to the Green symbols.

Remark 3.9 Due to the mapping property we can conclude that every Green symbol is parameter-dependent smoothing on X^{∧}, that is, g(η) ∈ L^{−∞}(X^{∧}; ℝ^{q}).

Example 3.10 Let φ, ψ ∈ C_{0}^{∞}(ℝ_{+}) be arbitrary functions with supp φ ∩ supp ψ = ∅. Then, for f(t, z, η) as in Example 3.7, and μ, μ̃ ∈ ℝ,

g(η) := φ(t[η]) [η]^{μ}tμ̃opM^{\frac{1}{2}}(f)(η)ψ(t[η])

is a Green symbol of order μ − μ̃.

Example 3.11 For P (η) ∈ L^{−∞}(X^{∧}; ℝ^{q})_{0} and functions φ, ψ ∈ C_{0}^{∞}(ℝ_{+}) we have

φ P (·)ψ ∈ R_{G}^{−∞}(ℝ^{q}; (γ, γ′))_{∞}

for every γ, γ′ ∈ ℝ.

Remark 3.12 Let σ, σ̃ ∈ C_{0}^{∞}(\overline{ℝ}_{+}) be cut-off functions, and g ∈ R_{G}^{μ}(ℝ^{q}; (γ, γ′)). Then

(i) σ g, g σ ∈ R_{G}^{μ}(ℝ^{q}; (γ, γ′)),

(ii) (1 − σ)g, g(1 − σ) ∈ R_{G}^{−∞}(ℝ^{q}; (γ, γ′)),

(iii) σ g σ̃ − g ∈ R_{G}^{−∞}(ℝ^{q}; (γ, γ′)),

(iv) t^{k} g t^{l} ∈ R_{G}^{μ−k−l}(ℝ^{q}; (γ − l, γ′ + k)) for l, k ∈ ℝ.

Example 3.13 Let b ∈ S(ℝ_{(t,t′)}^{2}, C^{∞}(X_{x} × X_{x′})) be supported in ℝ_{+}^{2} × X^{2}, and set

(g(η)u)(t, x) = [η]^{μ+n+1} ∫_{0}^{∞} ∫_{X} b(t[η], x, t′[η], x′)u(t′, x′)(t′)^{n} dt′dx′

with μ ∈ ℝ. Then g(λη) = λ^{μ} κ_{λ}g(η)κ_{λ}^{−1} for all sufficiently large |η| and λ ≥ 1, i.e. g is twisted homogeneous of degree μ for large |η|, and we have g ∈ R_{G}^{μ}(ℝ^{q}, (γ, γ′))_{∞} for all γ, γ′ ∈ ℝ.

3.4 Complete edge symbols

We now turn to a class of operator-valued symbols that are parameter-dependent families of pseudodifferential operators on the infinite cone X^{∧}.

Definition 3.14 For γ, μ, ν ∈ ℝ, with μ − ν ∈ ℕ_{0}, let R^{ν}(ℝ^{q}; (γ, γ − μ)) be the space of all

a(η) = σ_{1}(t)(a_{0}(η) + a_{1}(η))σ_{0}(t) + (1 − σ_{1})(t)P (η)(1 − σ_{2})(t) + g(η)

with

a_{0}(η) = ω_{1}(t[η])t^{−ν}op_{M}^{γ− \frac{n}{2}} (h)(η)ω_{0}(t[η]),
a_{1}(η) = (1 − ω_{1})(t[η])t^{−ν}op_{t}(p)(η)(1 − ω_{2})(t[η]),

where p ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, L^{ν}(X; ℝ_{τ,η}^{1+q})) and h ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, M_{O}^{ν}(X; ℝ^{q})) is the Mellin quantization of p. Moreover, g ∈ R_{G}^{ν}(ℝ^{q}; (γ, γ − μ)) and P (η) ∈ L^{ν}(X^{∧}; ℝ^{q})_{0}. Apart from that, σ_{j}, ω_{j}, j = 0, 1, 2, are cut-off functions satisfying ω_{2} ≺ ω_{1} ≺ ω_{0} and σ_{2} ≺ σ_{1} ≺ σ_{0}. We associate to a an additional specific symbolic level, namely the (twisted) homogeneous principal edge symbol

σ_{∧}^{ν}(a)(η) := σ_{∧}^{ν}(a_{0})(η) + σ_{∧}^{ν}(a_{1})(η) + σ_{∧}^{ν}(g)(η), η ∈ ℝ^{q} \ {0},

with

σ_{∧}^{ν}(a_{0})(η) = ω_{1}(t|η|)t^{−ν}op_{M}^{γ− \frac{n}{2}} (h_{∧})ω_{0}(t|η|),
σ_{∧}^{ν}(a_{1})(η) = (1 − ω_{1})(t|η|)t^{−ν}op_{t}(p_{∧})(1 − ω_{2})(t|η|),

where h_{∧} and p_{∧} are as in Definition 3.1.

The elements of the space R^{ν}(ℝ^{q}; (γ, γ − μ)) are called complete edge symbols (without asymptotics, with constant coefficients) as they were originally introduced in [9] (cf. also [2],[10]). In the present form we can prove the following fact:

Proposition 3.15 For arbitrary s, ϱ ∈ ℝ we have

R^{ν}(ℝ^{q}; (γ, γ − μ)) ⊂ S^{ν}(ℝ^{q}; K^{s,γ}(X^{∧})^{ϱ}, K^{s−ν,γ−μ}(X^{∧})^{ϱ}).

In particular, σ_{∧}^{ν}(a) ∈ C^{∞}(ℝ^{q} \ {0}, L(K^{s,γ}(X^{∧})^{ϱ}, K^{s−ν,γ−μ}(X^{∧})^{ϱ})) is homogeneous of degree ν, cf. Example 3.5.

For a proof see for example [8].

3.5 New representation of complete edge symbols

This section shows that the concept of holomorphic representations in the above Mellin quantization leads to a very convenient new description of the operator-valued edge symbols, in which the η-dependent cut-off functions are removed from the non-smoothing part. In particular, we get the principal edge symbol in an equivalent simple form.

Definition 3.16 Let γ, μ, ν ∈ ℝ, with μ − ν ∈ ℕ_{0}. The space R^{ν}(ℝ^{q}; (γ, γ − μ)) consists of all operator-families of the form

a(η) = σ_{1}(t) t^{−ν}op_{M}^{γ− \frac{n}{2}} (h)(η) σ_{0}(t) + (1 − σ_{1})(t)P (η)(1 − σ_{2})(t) + g(η),  (R)

where h ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, M_{O}^{ν}(X; ℝ^{q})), P (η) ∈ L^{ν}(X^{∧}; ℝ^{q})_{0} and g ∈ R_{G}^{ν}(ℝ^{q}; (γ, γ − μ)). The functions σ_{j}, j = 0, 1, 2, are cut-off functions satisfying σ_{2} ≺ σ_{1} ≺ σ_{0}. As in Definition 3.14 we have the principal edge symbol

σ_{∧}^{ν}(a)(η) := t^{−ν}op_{M}^{γ− \frac{n}{2}} (h_{∧})(η) + σ_{∧}^{ν}(g)(η), η ∈ ℝ^{q} \ \{0\}.

Recall that h_{∧}(t, z, η) = h̃(0, z, tη). In Lemma 4.5 we will prove that every edge symbol is a usual parameter-dependent pseudodifferential operator on X^{∧}. In particular, we also have the interior principal symbol σ_{ψ}^{ν}(a) in the case of classical operators.

Remark 3.17 The localization with σ_{0} allows us to assume, without loss of generality, that in (R) the symbol h is compactly supported in t ∈ \overline{ℝ}_{+}.

Theorem 3.18 The class R^{ν}(ℝ^{q}; (γ, γ − μ)) from Definition 3.14 coincides with the class R^{ν}(ℝ^{q}; (γ, γ − μ)). The principal edge symbol σ_{∧}^{ν}(a) is independent of the representation of the corresponding edge symbol a.

Proof: Let a ∈ R^{ν}(ℝ^{q}; (γ, γ − μ)) with the notation as in (R). Setting a_{M}(η) := t^{−ν}op_{M}^{γ− \frac{n}{2}} (h)(η) we have

a(η) = σ_{1}(t)a_{M}(η)σ_{0}(t) + (1 − σ_{1})(t)P (η)(1 − σ_{2})(t) + g(η).

Let ω_{2} ≺ ω_{1} ≺ ω_{0} be cut-off functions. Then

\begin{aligned}a_{M}(η) & = ω_{1}(t[η])a_{M}(η)ω_{0}(t[η]) + (1 − ω_{1})(t[η])a_{M}(η)(1 − ω_{2})(t[η]) \\ & + ω_{1}(t[η])a_{M}(η)(1 − ω_{0})(t[η]) + (1 − ω_{1})(t[η])a_{M}(η)ω_{2}(t[η]) \\ & ≡ ω_{1}(t[η])a_{M}(η)ω_{0}(t[η]) + (1 − ω_{1})(t[η])a_{M}(η)(1 − ω_{2})(t[η]) (mod R_{G}^{ν}) \end{aligned} (6)

due to Proposition A.8 and Remark 3.12. Therefore,

\begin{aligned}a(η) & ≡ σ_{1}(t)(a_{0}(η) + a_{1}(η))σ_{0}(t) + (1 − σ_{1})(t)P (η)(1 − σ_{2})(t) \\ & + σ_{1}(1 − ω_{1})(t[η])\{a_{M}(η) − t^{−ν}op_{t}(p)(η)\}(1 − ω_{2})(t[η])σ_{0} (mod R_{G}^{ν}), \end{aligned} (7)

where p and the symbol h of a_{M} are related via the Mellin quantization, and a_{0}(η), a_{1}(η) are as in Definition 3.14. Now, the latter term in (7) is in R_{G}^{ν} due to Proposition A.4. Hence

a(η) = σ_{1}(t)(a_{0}(η) + a_{1}(η))σ_{0}(t) + (1 − σ_{1})(t)P (η)(1 − σ_{2})(t) + g̃(η)  (8)

implying that R^{ν} ⊂ R^{ν}. The other inclusion follows similarly from the relation

a_{0}(η) + a_{1}(η) = ω_{1}(t[η])a_{M}(η)ω_{0}(t[η]) + (1 − ω_{1})(t[η])a_{M}(η)(1 − ω_{2})(t[η])
+ (1 − ω_{1})(t[η]){t^{−ν}op_{t}(p)(η) − a_{M}(η)}(1 − ω_{2})(t[η]),

using (6) and applying again Proposition A.4. Finally, by the same calculations for the principal edge symbol (with t|η| instead of t[η]) we obtain as in (8)

σ_{∧,new}^{ν}(a) = σ_{∧,old}^{ν}(a_{0}) + σ_{∧,old}^{ν}(a_{1}) + σ_{∧}^{ν}(g̃) = σ_{∧,old}^{ν}(a)

with trivial meaning of the notation. □

4 Elements of the calculus for complete edge symbols

4.1 Calculus for degenerate holomorphic Mellin symbols

Lemma 4.1 Let h̃ ∈ C^{∞}(\overline{ℝ}_{+}, M_{O}^{μ}(X; ℝ^{q})) be independent of t ∈ \overline{ℝ}_{+} for large t. Then

a(s, w) := ( (t, z, η) ↦ h̃(st, w + z, sη) ) ∈ A(ℝ_{+} × ℂ, C^{∞}(\overline{ℝ}_{+}, M_{O}^{μ}(X; ℝ^{q}))),

i.e., a is a holomorphic amplitude function with values in C^{∞}(\overline{ℝ}_{+}, M_{O}^{μ}(X; ℝ^{q})).

Proof: Step 1: Consider the local situation, i.e. h̃ ∈ S^{μ}(\overline{ℝ}_{+} × ℝ^{n} × ℂ × ℝ^{n+q}). Since the following calculations are not affected by the variables of ℝ^{n} × ℝ^{n}, we omit them in the notation. It is clear that a ∈ C^{∞}(ℝ_{+}, S^{μ}(\overline{ℝ}_{+} × ℂ × ℝ^{q})) ˆ⊗_{π}O(ℂ), where O(ℂ) is the Fréchet space of entire functions. On S^{μ}(\overline{ℝ}_{+} × ℂ × ℝ^{q}) we consider the semi-norms

p_{n}(h̃) = sup {|∂_{t}^{l′}∂_{z}^{k′}∂_{η}^{α′} h̃(t, σ_{0} + iτ, η)| ⟨τ, η⟩^{|α′|+k′−μ}} ,

where the supremum is taken over all t ≤ n, |σ_{0}| ≤ n, (τ, η) ∈ ℝ^{1+q}, and l′ + k′ + |α′| ≤ n. We have to show the existence of μ_{n}, m_{n}, such that for all k, l, N ∈ ℕ

sup {p_{n}((s∂_{s})^{l}∂_{w}^{k}a(s, σ_{1} + iξ))g(s)^{−m_{n}} ⟨ξ⟩^{−μ_{n}} | |σ_{1}| ≤ N, (s, ξ) ∈ ℝ_{+} × ℝ} < ∞. (9)

Here, g(s) = e^{⟨log s⟩}. If we write (η∂_{η})^{α} = (η_{1}∂_{η_{1}})^{α_{1}} … (η_{q}∂_{η_{q}})^{α_{q}} we have

(s∂_{s})^{l}a(s, w) = ∑_{m+|α|=l} c_{mα}[(t∂_{t})^{m}(η∂_{η})^{α}h̃](s·, w + ·, s·).

Since the map h̃ ↦ (t∂_{t})^{m}(η∂_{η})^{α}h̃ : S^{μ}(\overline{ℝ}_{+} × ℂ × ℝ^{q}) → S^{μ}(\overline{ℝ}_{+} × ℂ × ℝ^{q}) is continuous and (t∂_{t})^{l}(η∂_{η})^{α} h̃ also satisfies the assumptions on h̃, we can assume in (9) that l = 0. The estimate (9) is valid if we can show that

∂_{t}^{l′}∂_{z}^{k+k′}∂_{η}^{α}[h̃(st, σ_{0} + σ_{1} + iτ + iξ, sη)] ≤ |||h̃||| g(s)^{m_{n}} ⟨ξ⟩^{μ_{n}} ⟨τ, η⟩^{μ−|α′|−k′}  (10)

uniformly in t ≤ n, |σ_{0}| ≤ n, |σ_{1}| ≤ N, l′+k′+|α′| ≤ n, (τ, η) ∈ ℝ^{1+q}, (s, ξ) ∈ ℝ_{+}×ℝ, with a semi-norm |||·||| of S^{μ}(\overline{ℝ}_{+}×ℂ×ℝ^{q}), depending only on k, n, and N. Using the elementary inequality ⟨τ, sη⟩^{μ} ≤ max{s^{−1}, s}^{|μ|} ⟨τ, η⟩^{μ}, the left-hand side of (10) is dominated by

s^{l′+|α′|}|(∂_{t}^{l′}∂_{z}^{k+k′}∂_{η}^{α}h̃)(st, σ_{0} + σ_{1} + iτ + iξ, sη)|
≤ |||h̃||| s^{l′+|α′|} ⟨τ + ξ, sη⟩^{μ−k−k′−|α′|} ≤ c |||h̃||| s^{l′+|α′|} ⟨τ, sη⟩^{μ−k′−|α′|} ⟨ξ⟩^{|μ−|α′|−k′|}
≤ c |||h̃||| g(s)^{l′+|α′|+|μ−k′−|α′||} ⟨ξ⟩^{|μ−k′−|α′||} ⟨τ, η⟩^{μ−k′−|α′|}
≤ c |||h̃||| g(s)^{m_{n}} ⟨ξ⟩^{μ_{n}} ⟨τ, η⟩^{μ−k′−|α′|} .

Here we have set μ_{n} = max_{j=1}^{n} |μ − j|, m_{n} = n + max_{j=1}^{n} |μ − j|. Moreover,

|||h̃||| = sup {|(∂_{t}^{l′}∂_{z}^{k+k′}∂_{η}^{α′} h̃)(t, σ + iτ, η)| ⟨τ, η⟩^{k+k′+|α′|−μ}} ,

with the supremum over all t ∈ ℝ, |σ| ≤ n + N, l′ + k′ + |α′| ≤ n, and (τ, η) ∈ ℝ^{1+q}. Step 2: Assume that h̃ ∈ C^{∞}(\overline{ℝ}_{+}, M_{O}^{−∞}(X; ℝ^{q})). We introduce the following notation: for a Hilbert space E let S^{μ}(\overline{ℝ}_{+} ×Γ_{δ} ×ℝ^{q}; E) be the space of all f ∈ C^{∞}(\overline{ℝ}_{+} ×Γ_{δ} ×ℝ^{q}, E) such that

sup {∥∂_{t}^{l}∂_{τ}^{k}∂_{η}^{α}f(t, δ + iτ, η)∥_{E} ⟨τ, η⟩^{α+k−μ} | (τ, ξ) ∈ ℝ^{1+q}, t ∈ K, l + k + |α| ≤ N} < ∞

for all N ∈ ℕ and all compact sets K ⊂ \overline{ℝ}_{+}. Analogously, we generalize S^{μ}(\overline{ℝ}_{+} × ℂ × ℝ^{q}) to S^{μ}(\overline{ℝ}_{+} × ℂ × ℝ^{q}; E). These are Fréchet spaces, and

C^{∞}(\overline{ℝ}_{+}, M_{O}^{−∞}(X; ℝ^{q})) = pr. lim_{j∈ℕ}S^{−j}(\overline{ℝ}_{+} × ℂ × ℝ^{q}; H^{j}(X × X)).

Precisely as in Step 1 we have a ∈ A(ℝ_{+} × ℂ, C^{∞}(S^{−j}(\overline{ℝ}_{+} × ℂ × ℝ^{q}, H^{j}(X × X)))) for each j ∈ ℕ. Hence a ∈ A(ℝ_{+} × ℂ, C^{∞}(\overline{ℝ}_{+}, M_{O}^{−∞}(X; ℝ^{q}))) by Lemma 1.4(ii). Step 3: In the general case, i.e. h̃ ∈ C^{∞}(\overline{ℝ}_{+}, M_{O}^{−∞}(X; ℝ^{q})), we write

h̃(t, z, η) = ∑_{j=1}^{N} Φ_{j}(θ_{j}^{−1})_{∗} op_{x}(h̃_{j})(t, z, η) Ψ_{j} + h̃_{N+1}(t, z, η)

with h̃_{j} ∈ S^{μ}(\overline{ℝ}_{+} × ℝ^{n} × ℂ × ℝ^{n+q}) and h̃_{N+1} ∈ C^{∞}(\overline{ℝ}_{+}, M_{O}^{−∞}(X; ℝ^{q})). Here, θ_{j}, Φ_{j}, and Ψ_{j} are as in the proof of Theorem 3.2. Thus, with obvious meaning of notation,

a(s, w) = ∑_{j=1}^{N} Φ_{j}(θ_{j}^{−1})_{∗} op_{x}(a_{j}(s, w)) Ψ_{j} + a_{N+1}(s, w).

In view of Steps 1, 2 and Lemma 1.4(iv), a is an amplitude function as desired. □

Proposition 4.2 Let h̃(t, z, η) ∈ C^{∞}(\overline{ℝ}_{+}, M_{O}^{μ}(X; ℝ^{q})) be independent of t for large t, and h(t, z, η) = h̃(t, z, tη). Then

h̃_{R}(t′, z, η) := ∫∫ s^{−iξ}h̃(st′, z + iξ, sη) \frac{ds}{s}d¯ ξ

(convergent in C^{∞}(\overline{ℝ}_{+}, M_{O}^{μ}(X; ℝ^{q})) ). Setting h_{R}(t′, z, η) := h̃_{R}(t′, z, t′η) we obtain that, for each real γ,

op_{M}^{γ}(h_{R})(η) = op_{M}^{γ}(h)(η).

Proof: By Lemma 4.1, the oscillatory integral exists in C^{∞}(\overline{ℝ}_{+}, M_{O}^{μ}(X; ℝ^{q})). Step 1: Let h̃ be compactly supported in t ∈ \overline{ℝ}_{+}. Then h(·, ·, η) ∈ C_{b,F}^{∞} (ℝ_{+}, L^{μ}(X; Γ_{\frac{1}{2} −γ})). By Remark 2.7

h_{R}(t′, z, η) = ∫∫ s^{−iξ}h(st′, z + iξ, η) \frac{ds}{s}d¯ ξ

exists in C_{b,F}^{∞} (ℝ_{+}, L^{μ}(X; Γ_{\frac{1}{2} −γ})) and op_{M}^{γ}(h_{R})(η) = op_{M}^{γ}(h)(η) for each η. Interpreting

both oscillatory integrals as such in L^{μ}(X) yields h_{R}(t′, z, η) = h̃_{R}(t′, z, t′η) in L^{μ}(X) for all t′ > 0, z ∈ ℂ, and η ∈ ℝ^{q}. Due to the continuity this is then also true for t′ = 0. Step 2: Assume h̃(t, z, η) = (1 − ω)(t)h̃_{∞}(z, η) with h̃_{∞} ∈ M_{O}^{μ}(X; ℝ^{q}) and some cut-off function ω. The holomorphy allows us to write

op_{M}^{γ}(h)(η) = op_{M}^{γ}(t^{−N} T^{N}h)(η) t^{N}.

For N ≥ max\{0, μ\} we have h^{N}(t, z, η) := t^{−N} T^{N}h(t, z, η) ∈ C_{b,F}^{∞} (ℝ_{+}, L^{μ}(X; Γ_{\frac{1}{2} −γ})) for

each η (recall that h̃ is supported away from zero). Then, proceeding as in the first step, we obtain that

h_{R}^{N}(t′, z, η) = t′^{N} ∫∫ s^{−iξ}h^{N}(st′, z + iξ, η) \frac{ds}{s}d¯ ξ

(convergent in C_{b,F}^{∞} (ℝ_{+}, L^{μ}(X; Γ_{\frac{1}{2} −γ})) ) satisfies

op_{M}^{γ}(t^{−N} T^{N}h)(η) t^{N} = op_{M}^{γ}(h_{R}^{N})(η).

For fixed t′ > 0, z, and η,

h_{R}^{N}(t′, z, t′^{−1}η) = ∫∫ s^{−iξ}s^{−N} h̃(st′, z + N + iξ, sη) \frac{ds}{s}d¯ ξ

with convergence in L^{μ}(X). But this integral even converges in C^{∞}(\overline{ℝ}_{+}, M_{O}^{μ}(X; ℝ^{q})) and equals h̃_{R}(t′, z, η). Hence op_{M}^{γ}(h_{R})(η) = op_{M}^{γ}(h)(η). Finally, a general h̃ can be decomposed into the two parts treated above. □

With similar calculations one can prove the following results:

Proposition 4.3 Let h_{j} ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, M_{O}^{μ_{j}}(X; ℝ^{q})), j = 0, 1, be independent of t for large t. If we define h ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, M_{O}^{μ_{0}+μ_{1}}(X; ℝ^{q})) by

h̃(t, z, η) := ∫∫ s^{−iξ}h̃_{0}(t, z + iξ, η)h̃_{1}(st, z, sη) \frac{ds}{s}d¯ ξ

(convergent in C^{∞}(\overline{ℝ}_{+}, M_{O}^{μ_{0}+μ_{1}}(X; ℝ^{q})) ) then for each real γ

op_{M}^{γ}(h_{0})(η)op_{M}^{γ}(h_{1})(η) = op_{M}^{γ}(h)(η).

Proposition 4.4 Let h ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, M_{O}^{μ}(X; ℝ^{q})) be independent of t for large t. Further, define h^{(∗)} ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, M_{O}^{μ}(X; ℝ^{q})) by

h̃^{(∗)}(t, z, η) := ∫∫ s^{−iξ}h̃(st, n + 1 − z̄ + iξ, sη)^{⋆} \frac{ds}{s}d¯ ξ

(convergent in C^{∞}(\overline{ℝ}_{+}, M_{O}^{μ}(X; ℝ^{q})) ), where ⋆ is the formal adjoint in L^{μ}(X). Then for each real γ

op_{M}^{γ}(h)(η)^{∗} = op_{M}^{−γ−n}(h^{(∗)})(η).

The operator on the left-hand side is the formal adjoint in the sense of Definition 2.4.

4.2 Some properties of the edge symbols

Lemma 4.5 Let a ∈ R^{ν}(ℝ^{q}; (γ, γ − μ)). Then a is an element of L^{ν}(X^{∧}; ℝ^{q}).

Proof: Let a be written as in (R). We see immediately

(1 − σ_{1})(t)P (η)(1 − σ_{2})(t) + g(η) ∈ L^{ν}(X^{∧}; ℝ^{q}).

Writing now op_{M}^{γ− \frac{n}{2}} (h)(η) as a pseudodifferential operator with respect to the Fourier transform we get

op_{M}^{γ− \frac{n}{2}} (h)(η) = op_{t}(q)(η) with q ∈ C^{∞}(ℝ_{+} × ℝ_{+}, L^{ν}(X; ℝ^{1+q}))

so that op_{t}(q)(η) ∈ L^{ν}(X^{∧}; ℝ^{q}). □

Lemma 4.6 Let φ, ψ ∈ C_{0}^{∞}(ℝ_{+}), and Q(η) ∈ L^{ν}(X^{∧}; ℝ^{q}). Then

φ Q(·)ψ ∈ R^{ν}(ℝ^{q}; (γ, γ − μ))

with vanishing principal edge symbols. Moreover, we may represent φ Q(η) ψ as in (R) with h and P being compatible, i.e. for some fixed c′ > c > 0 we have

ϕ(t^{−ν}op_{M}^{γ− \frac{n}{2}} (h)(η) − P (η))ϕ̃ ∈ L^{−∞}(X^{∧}; ℝ^{q})  (11)

for all ϕ, ϕ̃ ∈ C_{0}^{∞}(ℝ_{+}) supported in [c, c′].

Proof: Of course, P (η) := φ Q(η)ψ ∈ L^{ν}(X^{∧}; ℝ^{q})_{0}. For cut-off functions σ_{2} ≺ σ_{1} ≺ σ_{0}, the operators σ_{1}Q(η)(1 − σ_{0}) and (1 − σ_{1})Q(η)σ_{2} are smoothing so that σ_{1}P (η)(1 − σ_{0}) and (1 − σ_{1})P (η)σ_{2} are both in R_{G}^{−∞}(ℝ^{q}; (γ, γ − μ))_{∞} for all μ ∈ ℝ (cf. Example 3.11). Hence

P (η) ≡ σ_{1}P (η)σ_{0} + (1 − σ_{1})P (η)(1 − σ_{2}) (mod R_{G}^{−∞}).

We now use the Mellin quantization to write σ_{1}P (η)σ_{0} as a sum σ_{1}t^{−ν}op_{M}^{γ− \frac{n}{2}} (h)(η)σ_{0}+g(η) with suitable h and g. Due to the presence of the functions φ and ψ there is a symbol q ∈ C^{∞}(ℝ_{+}, L^{ν}(X; ℝ^{1+q})), compactly supported in ℝ_{+}, such that P (η) = op_{t}(q)(η) and

t^{ν}q(t, t^{−1}τ, t^{−1}η) ∈ C^{∞}(\overline{ℝ}_{+}, L^{ν}(X; ℝ_{τ,η}^{1+q}))

so that p(t, τ, η) := t^{ν}q(t, τ, η) ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, L^{ν}(X; ℝ_{τ,η}^{1+q})). This symbol p is also compactly supported in ℝ_{+}. From Theorem 3.2 there exists an h ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, M_{O}^{ν}(X; ℝ^{q})) such that

d(η) := op_{t}(p)(η) − op_{M}^{γ− \frac{n}{2}} (h)(η) ∈ L^{−∞}(X^{∧}; ℝ^{q}).

Note that h is compactly supported in ℝ_{+} as p, and the difference d(η) is supported away from t = 0. Together with Example 3.11 we obtain

σ_{1}P (η)σ_{0} = σ_{1} t^{−ν}op_{M}^{γ− \frac{n}{2}} (h)(η)σ_{0} + σ_{1} t^{−ν}d(η)σ_{0}
≡ σ_{1} t^{−ν}op_{M}^{γ− \frac{n}{2}} (h)(η)σ_{0} (mod R_{G}^{−∞})

what yields the desired representation

P (η) = σ_{1} t^{−ν}op_{M}^{γ− \frac{n}{2}} (h)(η)σ_{0} + (1 − σ_{1})P (η)(1 − σ_{2}) + g(η)

with g ∈ R_{G}^{−∞}(ℝ^{q}; (γ, γ − μ))_{∞}. The compatibility relation is clearly satisfied for any constants c′ > c > 0. □

Lemma 4.7 Every complete edge symbol a can be written as in (R) with compatibility of h and P as in Lemma 4.6.

Proof: Every a ∈ R^{ν} is of the form

a(η) = σ_{1} t^{−ν}op_{M}^{γ− \frac{n}{2}} (h)(η)σ_{0} + (1 − σ_{1})P (η)(1 − σ_{2}) + g(η).

By means of the (inverse) Mellin quantization we find an element P_{1}(η) ∈ L^{ν}(X^{∧}; ℝ^{q})_{0} being compatible with h. Set

a_{1}(η) = σ_{1} t^{−ν}op_{M}^{γ− \frac{n}{2}} (h)(η)σ_{0} + (1 − σ_{1})P_{1}(η)(1 − σ_{2}) + g(η).

On the other hand, a−a_{1} = (1−σ_{1})(P (η)−P_{1}(η))(1−σ_{2}) can be written with the desired compatibility condition due to Lemma 4.6. Thus we are done since a = a_{1} + (a − a_{1}). □

Corollary 4.8 Let h and P be compatible in [c, c′], and let σ_{2} ≺ σ_{1} ≺ σ_{0} and σ̃_{2} ≺ σ̃_{1} ≺ σ̃_{0} be two sets of cut-off functions supported in [0, c′] such that σ_{2} = σ̃_{2} = 1 on some open neighbourhood of [0, c]. As in (R) let us form a with {σ_{j}} as well as ã with {σ̃_{j}}. Then a − ã belongs to R_{G}^{−∞}(ℝ^{q}; (γ, γ − μ))_{∞}; in other words, the class R^{ν}(ℝ^{q}; (γ, γ − μ)) is independent of the choice of the cut-off functions whenever they respect the compatibility between h and P. The same is true if we simultaneously interchange in (R) σ_{1} ↔ σ_{0} and (1 − σ_{1}) ↔ (1 − σ_{2}).

Proof: Choose cut-off functions ω_{5} ≺ ω_{4} ≺ ω_{3} with ω_{3} ≺ σ_{2}, ω_{3} ≺ σ̃_{2} and such that ω_{5} = 1 on [0, c]. If we write

a = ω_{4} a ω_{3} + (1 − ω_{4})a(1 − ω_{5}) + ω_{4} a(1 − ω_{3}) + (1 − ω_{4})a ω_{5}

then Proposition A.13 yields

a ≡ ω_{4}a_{M}ω_{3} + (1 − ω_{4})a(1 − ω_{5}) (mod R_{G}^{−∞})  (12)

where a_{M}(η) = σ_{1} t^{−ν}op_{M}^{γ− \frac{n}{2}} (h)(η)σ_{0}. In the same manner

a ≡ ω_{1} a ω_{0} + (1 − ω_{1})P (1 − ω_{2}) (mod R_{G}^{−∞})  (13)

for functions ω_{2} ≺ ω_{1} ≺ ω_{0} with σ_{0} ≺ ω_{2} and σ̃_{0} ≺ ω_{2}. Inserting (13) into (12) we get

a ≡ (1 − ω_{4})ω_{1} a ω_{0}(1 − ω_{5}) + ω_{4}a_{M}ω_{3} + (1 − ω_{1})P (1 − ω_{2}) (mod R_{G}^{−∞}).

Doing the same for ã we obtain

a − ã ≡ \underbrace{(1 − ω_{4})ω_{1}}_{=:φ_{1}}(a − ã) \underbrace{ω_{0}(1 − ω_{5})}_{=:φ_{0}} (mod R_{G}^{−∞}).

As in the proof of Lemma 4.6 we have

φ_{1}(1 − σ_{1})P (1 − σ_{2})φ_{0} ≡ φ_{1}(P − σ_{1}P σ_{0})φ_{0} (mod R_{G}^{−∞}),
φ_{1}(1 − σ̃_{1})P (1 − σ̃_{2})φ_{0} ≡ φ_{1}(P − σ̃_{1}P σ̃_{0})φ_{0} (mod R_{G}^{−∞}),

and therefore

φ_{1}(a − ã)φ_{0} ≡ φ_{1}σ_{1}(a_{M} − P)σ_{0}φ_{0} − φ_{1}σ̃_{1}(a_{M} − P)σ̃_{0}φ_{0} ≡ 0 (mod R_{G}^{−∞})

due to the compatibility relation (11) and Example 3.11. □

As an immediate consequence of the new representation we get that the class of complete edge symbols is closed with respect to differentiation and pointwise formal adjoint:

Proposition 4.9 Let a ∈ R^{ν}(ℝ^{q}; (γ, γ − μ)). Then

D_{η}^{α}a ∈ R^{ν−|α|}(ℝ^{q}; (γ, γ − μ)) and a^{∗} ∈ R^{ν}(ℝ^{q}; (−γ + μ, −γ)),

where a^{∗} is the formal adjoint in the sense of Definition 2.4. Moreover, the principal edge symbols satisfy

σ_{∧}^{ν−|α|}(D_{η}^{α}a)(η) = D_{η}^{α}σ_{∧}^{ν}(a)(η) and σ_{∧}^{ν}(a^{∗})(η) = σ_{∧}^{ν}(a)^{∗}(η).

4.3 The composition theorem

Theorem 4.10 Let a_{j} ∈ R^{ν_{j}}(ℝ^{q}; (γ_{j}, γ_{j} − μ_{j})), j = 0, 1, with γ_{1} = γ_{0} − μ_{0}. Then

a_{1}a_{0} ∈ R^{ν_{0}+ν_{1}}(ℝ^{q}; (γ_{0}, γ_{0} − μ_{0} − μ_{1})).

Moreover, the symbols satisfy

σ_{∧}^{ν_{0}+ν_{1}}(a_{1}a_{0})(η) = σ_{∧}^{ν_{1}}(a_{1})(η)σ_{∧}^{ν_{0}}(a_{0})(η) for all η ∈ ℝ^{q} \ {0},

and in the case of classical operators σ_{ψ}^{ν_{0}+ν_{1}}(a_{1}a_{0}) = σ_{ψ}^{ν_{1}}(a_{1})σ_{ψ}^{ν_{0}}(a_{0}).

Proof: Let us write

a_{0} = a_{0M} + a_{0P} + g_{0} and a_{1} = a_{1M} + a_{1P} + g_{1}

with a_{jM}(η) = σ_{1} t^{−ν_{j}}op_{M}^{γ_{j}− \frac{n}{2}} (h_{j})(η)σ_{0} and a_{jP} (η) = (1 − σ_{1})P_{j}(η)(1 − σ_{2}). First of all we consider the term

a_{1}g_{0} = a_{1M}g_{0} + a_{1P} g_{0} + g_{1}g_{0}.

Clearly, g_{1}g_{0} ∈ R_{G}^{ν_{0}+ν_{1}}(ℝ^{q}; (γ_{0}, γ_{0} − μ_{0} − μ_{1})) with σ_{∧}^{ν_{0}+ν_{1}}(g_{1}g_{0}) = σ_{∧}^{ν_{1}}(g_{1})σ_{∧}^{ν_{0}}(g_{0}). Moreover, a_{1P} g_{0} ∈ R_{G}^{−∞}(ℝ^{q}; (γ_{0}, γ_{0} −μ_{0} −μ_{1})) due to the presence of (1−σ_{1}) and Remark 3.12. If ω_{2} ≺ ω_{1} ≺ ω_{0} are cut-off functions, Proposition A.8 yields that

\begin{aligned}a_{1M}(η)g_{0}(η) & ≡ ω_{1}(t[η])a_{1M}(η)ω_{0}(t[η])g_{0}(η) \\ & + (1 − ω_{1})(t[η])a_{1M}(η)(1 − ω_{2})(t[η])g_{0}(η) (mod R_{G}^{ν_{0}+ν_{1}}). \end{aligned} (14)

Due to elementary mapping properties of Mellin operators it is easy to see that the first term on the right-hand side belongs to R_{G}^{ν_{0}+ν_{1}}(ℝ^{q}; (γ_{0}, γ_{0} − μ_{0} − μ_{1})). By choosing ω_{1} appropriately, we may rewrite the second term as

σ_{1}\{t^{−N}(1 − ω_{1})(t[η])\}\{χ(η)t^{−ν_{1}}op_{M}^{γ− \frac{n}{2}} (T^{N}h_{1})(η)\}\{t^{N}(1 − ω_{2})(t[η])σ_{0}g_{0}(η)\}

for some excision function χ at η = 0 and each N ∈ ℕ. Choosing N sufficiently large, the mapping properties of the respective three factors yield that the second term on the righthand side of (14) also belongs to R_{G}^{ν_{0}+ν_{1}}(ℝ^{q}; (γ_{0}, γ_{0} −μ_{0} −μ_{1})). By freezing the coefficients of h̃_{1} at t = 0 it is straightforward to verify that σ_{∧}^{ν_{0}+ν_{1}}(a_{1M}g_{0}) = σ_{∧}^{ν_{1}}(a_{M1})σ_{∧}^{ν_{0}}(g_{0}). Similarly we obtain g_{1}a_{0M} + g_{1}a_{0P} ∈ R_{G}^{ν_{0}+ν_{1}}(ℝ^{q}; (γ_{0}, γ_{0} − μ_{0} − μ_{1})) so that we have

a_{1} · a_{0} ≡ a_{1M} · a_{0M} + a_{1P} · a_{0P} + a_{1M} · a_{0P} + a_{1P} · a_{0M} (mod R_{G}^{ν_{0}+ν_{1}})

Choosing a cut-off function σ̃ ≺ σ_{1} we can write

a_{1M}(η)a_{0P} (η) = {σ̃a_{1M}(η)(1 − σ_{1})}P_{0}(η)(1 − σ_{2}) + (1 − σ̃)a_{1M}(η)a_{0P} (η).

In view of Proposition A.13 the first term on the right-hand side is in R_{G}^{−∞}(ℝ^{q}; (γ_{0}, γ_{0}−μ_{0}− μ_{1}))_{∞}. The second term belongs to R^{ν_{0}+ν_{1}}(ℝ^{q}; (γ_{0}, γ_{0} − μ_{0} − μ_{1})) in view of Lemma 4.6. The product a_{1P} (η)a_{0M}(η) is treated analogously. In other words, there is a P (η) ∈ L^{ν_{0}+ν_{1}}(X^{∧}; ℝ^{q})_{0} such that

a_{1P} · a_{0P} + a_{1M} · a_{0P} + a_{1P} · a_{0M} ≡ (1 − σ_{1})P (η)(1 − σ_{2}) (mod R_{G}^{−∞}).

As a direct consequence we have that this term does not contribute to the principal edge symbol of the composition. Now, with h′_{0}(t, z, η) := σ_{1}(t)h_{0}(t, z, η) and applying Lemma 2.10 we obtain

a_{1M}(η)a_{0M}(η) = σ_{1} t^{−ν_{0}−ν_{1}}op_{M}^{γ_{1}+ν_{0}− \frac{n}{2}} (T^{ν_{0}}h_{1})(η) op_{M}^{γ_{1}+ν_{0}− \frac{n}{2}} (h′_{0})(η)σ_{0}
= σ_{1} t^{−ν_{0}−ν_{1}}op_{M}^{γ_{0}− \frac{n}{2}} (h)(η)σ_{0} on C_{0}^{∞}(X^{∧}),

where h ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, M_{O}^{ν_{0}+ν_{1}}(X; ℝ^{q})) due to Proposition 4.3. More precisely, h(t, z, η) = h̃(t, z, tη) with

h̃(t, z, η) = ∫∫ s^{−iξ} h̃_{1}(t, z + ν_{0} + iξ, η)σ_{1}(st)h̃_{0}(st, z, sη)\frac{ds}{s}d¯ ξ.

Thus the relation for the principal edge symbols is clearly satisfied as well as the relation for the interior symbols. □

A Further results from the cone theory

A.1 Proof of the global Mellin quantization

Theorem A.1 Let V ⊂ ℝ^{n} be open. Let p̃ ∈ S^{μ}(\overline{ℝ}_{+} × V × ℝ^{1+n+q}), μ ∈ ℝ, and set p(t, x, τ, ξ, η) := p̃(t, x, tτ, ξ, tη). Let further ϕ ∈ C_{0}^{∞}(ℝ_{+}) be a function such that ϕ ≡ 1 near to 1. Then there exists an h̃ ∈ S^{μ}(\overline{ℝ}_{+} × V × ℂ × ℝ^{n+q}) such that

op_{t,x}(ϕ(t′/t)p)(η) = opM^{\frac{1}{2}}(op_{x}(h))(η)

for any η ∈ ℝ^{q}, where h(t, x, z, ξ, η) := h̃(t, x, z, ξ, tη).

A proof of this theorem was given in [4, Theorem 2.3]. In fact, we have explicitly

h̃(t, x, z, ξ, η) = v_{z}(t)opM^{\frac{1}{2}}(ϕ(t′/t)g̃)(x, ξ, η)v_{−z}(t)  (15)

with g̃(t, t′, x, iτ, ξ, η) := M(t, t′)t′p̃(t, x, −M(t, t′)tτ, ξ, η) ∈ S^{μ}(ℝ_{+}^{2} × V × Γ_{0} × ℝ^{n+q}). Here M(t, t′) := \frac{log t−log t′}{t−t′} > 0 for t, t′ ∈ ℝ_{+}, and v_{z}(t) := t^{z} ∈ C^{∞}(ℝ_{+}, C^{∞}(X)).

Proof of Theorem 3.2

We begin with the trivial identity op_{t}(P) = op_{t}(ϕ(t′/t)P) + op_{t}(Q). Since

op_{t}(Q)(η) = op_{t}(Q_{N})(η) for all N ∈ ℕ

with Q_{N}(t, t′, τ, η) = (1 − ϕ(t′/t))(t′/t − 1)^{−N}(D_{τ}^{N} P̃)(t, tτ, tη) ∈ C^{∞}(ℝ_{+}^{2}, L_{cℓ}^{μ−N}(X; ℝ^{1+q})), then op_{t}(Q)(η) ∈ L^{−∞}(X^{∧}; ℝ^{q}). Next, let {U_{1}, …, U_{N}} be an open covering of X by coordinate neighborhoods with corresponding charts θ_{j} : U_{j} → V_{j} ⊂ ℝ^{n}. Further let {φ_{1}, …, φ_{N}} be a subordinate partition of unity, and {ψ_{1}, …, ψ_{N}} be a system of functions satisfying ψ_{j} ∈ C_{0}^{∞}(U_{j}) and φ_{j}ψ_{j} = φ_{j} for all j = 1, …, N. Then the non-degenerate part corresponding to P can be written

P̃(t, τ, η) = ∑_{j=1}^{N} Φ_{j} P̃_{j}(t, τ, η)Ψ_{j} + P̃_{∞}(t, τ, η), and so

op_{t}(ϕ(t′/t)P)(η) = ∑_{j=1}^{N} Φ_{j} op_{t}[ϕ(t′/t)(θ_{j}^{−1})_{∗} op_{x}(p_{j})](η)Ψ_{j} + op_{t}(ϕ(t′/t)P_{∞})(η), (16)

where p_{j}(t, x, τ, ξ, η) := p̃_{j}(t, x, tτ, ξ, tη) with local symbols p̃_{j} ∈ S^{μ}(\overline{ℝ}_{+}×V_{j} ×ℝ_{τ,ξ,η}^{1+n+q}), j = 1, …, N; P̃_{∞} ∈ C^{∞}(\overline{ℝ}_{+}, L^{−∞}(X; ℝ_{τ,η}^{1+q})), and Φ_{j}, Ψ_{j} are the operators of multiplication by the corresponding functions φ_{j}, ψ_{j}. From Theorem A.1 there are Mellin symbols h̃_{j} ∈ S^{μ}(\overline{ℝ}_{+} × V × ℂ × ℝ^{n+q}), j = 1, …, N, such that

op_{t}[ϕ(t′/t)(θ_{j}^{−1})_{∗} op_{x}(p_{j})](η) = opM^{\frac{1}{2}}(H_{j})(η) with H_{j}(t, z, η) = (θ_{j}^{−1})_{∗} op_{x}(h_{j})(t, z, η)

for any η ∈ ℝ^{q}. To handle the remainder term in (16) let us set

G̃_{∞}(t, t′, iτ, η) := ϕ(t′/t)M(t, t′)t′P̃_{∞}(t, −M(t, t′)tτ, η) and

H̃_{∞}(t, z, η) := v_{z}(t)opM^{\frac{1}{2}}(G̃_{∞})(η)v_{−z}(t).

We have op_{t}(ϕ(t′/t)P_{∞})(η) = opM^{\frac{1}{2}}(H_{∞})(η) with H_{∞}(t, z, η) := H̃_{∞}(t, z, tη), cf. (15). Moreover, we claim that H̃_{∞} belongs to C^{∞}(\overline{ℝ}_{+}, M_{O}^{−∞}(X; ℝ^{q})). To prove this we first observe that G̃_{∞} ∈ C^{∞}(ℝ_{+}^{2}, L^{−∞}(X; Γ_{0} × ℝ^{q})). Further,

L^{−∞}(X; Γ_{0} × ℝ^{q}) = C^{∞}(X × X) ˆ⊗_{π}S(Γ_{0} × ℝ^{q}) and
M_{O}^{−∞}(X; ℝ^{q}) = C^{∞}(X × X) ˆ⊗_{π}M_{O}^{−∞}(ℝ^{q}).

For that reason it is sufficient to show that H̃_{∞} ∈ C^{∞}(\overline{ℝ}_{+}, M_{O}^{−∞}(ℝ^{q})) whenever G̃_{∞} ∈ C^{∞}(ℝ_{+}^{2}, S(Γ_{0} × ℝ^{q})), that is, we have to show holomorphy in z, and the boundedness of the semi-norms

|H̃_{∞}|_{m,N} := sup {|∂_{t}^{k}∂_{ϱ,η}^{α}H̃_{∞}(t, β + iϱ, η)| ⟨ϱ, η⟩^{N} },

where the supremum is taken over all (ϱ, η) ∈ ℝ^{1+q}, |α| ≤ N, |β| ≤ N, k ≤ m, t ≤ m. Now for z = β + iϱ ∈ ℂ, using the change t′ → tr, τ → τ + ϱ, we have

H̃_{∞}(t, z, η) = ∫∫ (\frac{t}{t′} )^{−iτ+z}G̃_{∞}(t, t′, iτ, η)\frac{dt′}{t′} d¯ τ
= ∫∫ r^{iτ−β}χ(r)P̃_{∞}(t, −M(r, 1)(τ + ϱ), η)drd¯ τ

with χ(r) = ϕ(r)M(r, 1) ∈ C_{0}^{∞}(ℝ_{+}). Hence for α ∈ ℕ_{0}^{1+q} and k ∈ ℕ_{0}

∂_{t}^{k}∂_{ϱ,η}^{α}H̃_{∞}(t, β + iϱ, η) = ∫∫ r^{iτ−β}χ_{1}(r)(∂_{t}^{k}∂_{ϱ,η}^{α}P̃_{∞})(t, −M(r, 1)(τ + ϱ), η)drd¯ τ,

and so, for any l > 0 we obtain

|∂_{t}^{k}∂_{ϱ,η}^{α}H̃_{∞}(t, β + iϱ, η)| ≤ c̃_{k,α} ∫∫ r^{−β}χ_{2}(r) ⟨τ + ϱ, η⟩^{−l} drdτ,

where χ_{1,2} ∈ C_{0}^{∞}(ℝ_{+}). In particular, for l ≥ N + 2 we have ⟨τ + ϱ, η⟩^{−l} ≤ ⟨τ⟩^{−2} ⟨ϱ, η⟩^{−N}. Then |∂_{t}^{k}∂_{ϱ,η}^{α}H̃_{∞}(t, β + iϱ, η)| ≤ c_{k,α}(β) ⟨ϱ, η⟩^{−N} with c_{k,α}(β) depending continuously on β. Thus |H̃_{∞}|_{m,N} < ∞ for every m, N ∈ ℕ_{0}. Finally, writing H̃_{∞} as above but without the change τ → τ + ϱ, and looking now at the semi-norms of H̃_{∞}(t, z, ·) in S(ℝ^{q}), we can analogously verify that for every t ∈ \overline{ℝ}_{+} the function z ↦ H̃_{∞}(t, z, ·) is holomorphic. Summing up, the function

H(t, z, η) = ∑_{j=1}^{N} Φ_{j} H_{j}(t, z, η)Ψ_{j} + H_{∞}(t, z, η),

belongs to C_{deg}^{∞}(\overline{ℝ}_{+}, M_{O}^{μ}(X; ℝ^{q})) and the relation op_{t}(ϕ(t′/t)P)(η) = opM^{\frac{1}{2}}(H)(η) is valid for every η ∈ ℝ^{q}. □

A.2 Green remainders induced by the Mellin quantization

For μ, m ∈ ℝ we denote by S^{μ,m}(\overline{ℝ}_{+}^{2} × ℝ^{2n} × ℝ^{1+n} × ℝ^{q}) the Fréchet space of all functions p ∈ C^{∞}(\overline{ℝ}_{+}^{2} × ℝ^{2n} × ℝ^{1+n} × ℝ^{q}) with

sup {|∂_{t}^{β} ∂_{t′,x,x′}^{γ} ∂_{τ,ξ,η}^{α} p(t, t′, x, x′, τ, ξ, η)| ⟨t⟩^{|β|−m} ⟨(τ, ξ, η)⟩^{|α|−μ} } < ∞

for all multi-indices α, β, γ; the supremum is taken over all involved variables.

Lemma A.2 Let p̃(t, t′, x, x′, τ, ξ, η) ∈ S^{μ,m}(\overline{ℝ}_{+}^{2} × ℝ^{2n} × ℝ^{1+n} × ℝ^{q}) be compactly supported in (x, x′) ∈ ℝ^{2n}. Let χ ∈ C^{∞}(ℝ^{q}) be an excision function at 0, and ϕ ∈ C_{0}^{∞}(ℝ_{+}) with ϕ ≡ 1 near to 1. Further, set

q(t, t′, x, x′, τ, ξ, η) := (1 − ϕ(t′/t))p̃(t, t′, x, x′, tτ, ξ, tη) and
Q(η) := χ(η)(1 − ω)(t[η])op_{t,x}(q)(η)(1 − ω_{1})(t[η]).

Then for each s, δ, μ̃, m̃ ∈ ℝ

Q ∈ S^{0}(ℝ^{q}; H^{s,δ}(ℝ^{1+n}), H^{s−μ̃,δ−m̃}(ℝ^{1+n})).

If p̃ is additionally compactly supported in t ∈ \overline{ℝ}_{+}, we can omit χ in the definition of Q. The spaces H^{s,δ} are the usual weighted Sobolev spaces.

Proof: The variables (x, x′, ξ) are irrelevant in the following calculations, so without loss of generality we assume n = 0. First observe that integration by parts yields op_{t}(q)(η) = op_{t}(q_{N})(η), where

q_{N}(t, t′, τ, η) = (1 − ϕ(t′/t))(t′/t − 1)^{−N}(D_{τ}^{N} p̃)(t, t′, tτ, tη)

for every N ∈ ℕ. We define

q_{N,1}(t, t′, τ, η) := χ(η)(1 − ω)(t[η])(1 − ω_{1})(t′[η])q_{N}(t, t′, τ, η).

Clearly Q(η) = op_{t}(q_{N,1})(η). We shall prove that q_{N,1} ∈ C^{∞}(ℝ^{q}; S^{μ_{N} ,m_{N}} (ℝ_{t,t′}^{2} × ℝ_{τ})) with μ_{N} = μ − N, m_{N} = m + μ − N. To verify the corresponding symbol estimates let us set f_{N}(t, t′, η) := (1 − ω)(t[η])(1 − ω_{1})(t′[η])(1 − ϕ(t′/t))(t′/t − 1)^{−N} and study for every k, k′, l ∈ ℝ the derivatives

I_{k,k′,l}(q_{N,1}) := |∂_{t}^{k}∂_{t′}^{k′}∂_{τ}^{l} q_{N,1}(t, t′, τ)|
= |χ(η)∂_{t}^{k}∂_{t′}^{k′} t^{l}f_{N}(t, t′, η)(∂_{τ}^{N+l}p̃)(t, t′, tτ, tη)| .

Due to the Leibniz formula I_{k,k′,l} leads to a finite sum of terms like

|(∂_{t}^{k_{1}} t^{l})(∂_{t}^{k_{2}}∂_{t′}^{k′_{2}}f_{N}(t, t′, η))(∂_{t}^{k_{3}} p̃_{N,l}(t, t′, tτ, tη))| =: |I_{1}I_{2}I_{3}|

with p̃_{N,l} = ∂_{t′}^{k′_{3}}∂_{τ}^{N+l}p̃, ∑ k_{j} = k, ∑ k′_{j} = k′, and where t, t′ ∈ [c_{0}, ∞) and |η| ≥ c_{1} for some constants c_{0}, c_{1} > 0 depending on ω, ω_{1} and χ.

Obviously, |I_{1}| ≤ C_{1} ⟨t⟩^{l−k_{1}}. To estimate I_{2} we apply again the Leibniz formula and the chain rule. If some ∂_{t}(1 − ω) appears then I_{2} has compact support in t. For the remaining terms we have to consider expressions like

|t^{−k_{2}−l′}(∂_{r}^{L}ϕ)(t′/t)(t′/t − 1)^{L−N−l′} | ≤ c ⟨t⟩^{−k_{2}} |(∂_{r}^{L}ϕ)(t′/t)(t′/t − 1)^{L−N−l′} |

with c > 0, l′ ≤ k′_{2} and L ≤ k_{2} + l′. Take ε, ε̃ > 0 such that ϕ(t′/t) = 1 for |t′/t − 1| ≤ ε and ϕ(t′/t) = 0 for |t′/t − 1| ≥ ε̃, then (∂_{r}^{L}ϕ)(t′/t)|t′/t − 1|^{L−N−l′} is uniformly bounded for each L, N, l′. We conclude that |I_{2}| ≤ C_{2} ⟨t⟩^{−k_{2}} for all t ∈ ℝ_{+}. Further, we have

I_{3} = ∑_{|γ|=k_{3}} a_{γ}(η)τ^{γ_{2}}(∂_{t,τ,η}^{γ}p̃_{N,l})(t, t′, tτ, tη),

where a_{γ} ∈ S^{k_{3}}(ℝ^{q}) and γ = (γ_{1}, γ_{2}, γ_{3}) ∈ ℕ_{0}^{1+1+q}. Moreover, the absolute value of every term in the sum is bounded by

c ⟨τ⟩^{γ_{2}} ⟨t⟩^{m−γ_{1}} ⟨tτ, tη⟩^{μ−N−l−γ_{2}−|γ_{3}|} ≤ c′ ⟨t⟩^{m+μ−N−l−k_{3}} ⟨τ⟩^{μ−N−l};  (17)

here we use the inequalities

⟨tτ, tη⟩^{−L} ≤ c ⟨tτ, t⟩^{−L} ≤ c′ ⟨t⟩^{−L} ⟨τ⟩^{−L} and ⟨tτ, tη⟩^{L} ≤ c′′ ⟨t⟩^{L} ⟨τ⟩^{L} ⟨η⟩^{L}

for every L ≥ 0, which are true for t ≥ c_{0} and |η| ≥ c_{1}. Thus |I_{3}| ≤ C_{3} ⟨t⟩^{m+μ−N−l−k_{3}} ⟨τ⟩^{μ−N−l}. Combining all these estimates we obtain

I_{k,k′,l}(q_{N,1}) ≤ C_{k,k′,l}(η) ⟨t⟩^{m+μ−N−k} ⟨τ⟩^{μ−N−l}  (18)

for every k, k′, l ∈ ℝ, η ∈ ℝ^{q}. In the same way we estimate I_{k,k′,l}(∂_{η}^{α}q_{N,1}) and get expressions like (18) with another C_{k,k′,l}^{α}(η) of the same kind. Hence the assertion about q_{N,1} holds. Let us finally define

q_{N,2}(t, t′, τ, η) := χ(η)f_{N}(t[η]^{−1}, t′[η]^{−1}, η)(D_{τ}^{N} p̃)(t[η]^{−1}, t′[η]^{−1}, tτ, t[η]^{−1}η).

Clearly q_{N,2} ∈ C^{∞}(ℝ^{q}; S^{μ_{N} ,m_{N}} (ℝ_{t,t′}^{2} × ℝ_{τ})) too, and it holds κ_{[η]}^{−1}Q(η)κ_{[η]} = op_{t}(q_{N,2})(η). In particular, q_{N,2}(η) ∈ S^{μ̃,m̃}(ℝ_{t,t′}^{2} × ℝ_{τ}) for all N such that μ_{N} ≤ μ̃ and m_{N} ≤ m̃. Moreover, due to the Calderón-Vaillancourt Theorem, the operator norm of op_{t}(q_{N,2})(η) in L(H^{s,δ}, H^{s−μ̃,δ−m̃}) can be majorized by a finite number of expressions like

c sup_{t,t′,τ} {I_{k,k′,l}(q_{N,2}) ⟨t⟩^{k−m̃} ⟨τ⟩^{l−μ̃} },

where t, t′ run over [c_{0}, ∞), |η| ≥ c_{1}. The constant c > 0 as well as the supremum above depend continuously on p̃. Similar to (18) we have here

I_{k,k′,l}(q_{N,2}) ≤ C̃_{k,k′,l}(η) ⟨t[η]^{−1}⟩^{m} ⟨t⟩^{μ−N−k} ⟨τ⟩^{μ_{N} −l} ≤ C̃_{k,k′,l}(η) ⟨t⟩^{|m|+μ−N−k} ⟨τ⟩^{μ_{N} −l} with C̃_{k,k′,l}(η) bounded in η ∈ ℝ^{q} since η appears in q_{N,2} together with [η]^{−1}. Now, for N large enough the supremum above exists and is uniformly bounded in η ∈ ℝ^{q}. Therefore

∥κ_{[η]}^{−1}Q(η)κ_{[η]}∥_{L(H^{s,δ},H^{s−μ̃,δ−m̃})} ≤ C for every s, δ, μ̃, m̃ ∈ ℝ.

The calculations for ∂_{η}^{α}Q(η) are similar. Note that ∂_{η}^{α} generates a factor t^{|α|} (p̃ depends on tη), and κ_{[η]}^{−1}t^{|α|}κ_{[η]} = t^{|α|}[η]^{−|α|}. □

Remark A.3 If we interchange the variables t ↔ t′ and x ↔ x′ in the symbol of Q, then Lemma A.2 still holds. This will be used for adjoint operators.

Proposition A.4 Let h ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, M_{O}^{ν}(X; ℝ^{q})) and p ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, L^{ν}(X; ℝ^{1+q})) be related via the Mellin quantization (Theorem 3.2). Then

g(η) = σ(t)(1 − ω)(t[η])\{op_{t}(p)(η) − opM^{\frac{1}{2}}(h)(η)\}(1 − ω_{1})(t[η])σ_{0}(t)

is an element of R_{G}^{0}(ℝ^{q}; (γ, γ))_{∞}.

Proof: ¿From Theorem 3.2 we know that for some suitable ϕ ∈ C_{0}^{∞}(ℝ_{+})

op_{t}(p)(η) − opM^{\frac{1}{2}}(h)(η) = op_{t}(Q)(η) for all η ∈ ℝ^{q};

with Q(t, t′, τ, η) = (1 − ϕ(t′/t))p(t, τ, η). Therefore, to obtain the assertion we only need to analyze locally operators of the form

G_{k}(η) = (1 − ω)(t[η])op_{t,x}(r_{k})(η)(1 − ω_{1})(t[η]), k = 0, 1,

where r_{0}(t, t′, x, x′, τ, ξ, η) = (1 − ϕ(t′/t))q̃(t, t′, x, x′, tτ, ξ, tη) and r_{1}(t, t′, x, x′, τ, ξ, η) = (1−ϕ(t/t′))q̃(t′, t, x′, x, t′τ, ξ, t′η) with some q̃ ∈ S^{ν}(\overline{ℝ}_{+}^{2}×ℝ^{2n}×ℝ^{1+n}×ℝ^{q}) being compactly supported in \overline{ℝ}_{+}^{2} × ℝ^{2n}. Lemma A.2 then give us G_{k} ∈ S^{0}(ℝ^{q}; H^{s,δ}, H^{s−ν̃,δ−m̃}) for every s, δ, ν̃, m̃ ∈ ℝ. Since G_{k} is supported away from t = 0, it induces a continuous operator between cone Sobolev spaces on X^{∧}. Moreover, if χ ∈ C^{∞}(ℝ^{q}) is an excision function at 0 then (1 − χ)G_{k} ∈ S^{−∞}(ℝ^{q}; H^{s,δ}, H^{s−ν̃,δ−m̃}). For that reason, the function g above is a Green symbol if we can prove that χG_{k} is classical in the sense of Definition 3.6. To this end we assume for simplicity dim X = 0, put t = (t, t′), and apply the Taylor expansion

q̃(t, τ, η) = ∑_{|α|<N} t^{α}q̃_{α}(τ, η) + ∑_{|α|=N} t^{α}q̃_{α}(t, τ, η)

with q̃_{α} ∈ S^{ν}(ℝ_{τ} × ℝ_{η}^{q}) for |α| < N and q̃_{α} ∈ S^{ν,0}(\overline{ℝ}_{+}^{2} × ℝ_{τ} × ℝ_{η}^{q}) for |α| = N. Further, set G := G_{0} (for G_{1} the procedure is the same). Then

χ(η)G(η) = ∑_{j=0}^{N−1} G_{(j)}(η) + G_{(N)}(η),

where G_{(j)}(η) = χ(η)(1 − ω)(t[η]) ∑_{|α|=j} op_{t,x}(r_{α})(η)(1 − ω_{1})(t[η]) for every 0 ≤ j ≤ N, and r_{α} = (1 − ϕ(t′/t))t^{α}q̃_{α}(t, tτ, tη). In view of Lemma A.2 each G_{(j)} belongs to S^{−j}(ℝ^{q}; H^{s,δ}, H^{s−ν̃,δ−m̃}). Moreover, for each 0 ≤ j < N the function G_{(j)} is homogeneous of order −j for large |η|. Therefore χG is classical and the proof is done. □

A.3 Green symbols generated by holomorphic Mellin symbols

Lemma A.5 Let s, γ ∈ ℝ be given and choose a cut-off function ω. Then

(i) for each L ∈ ℝ and all γ′ ∈ ℝ

(1 − ω)(t[η])t^{−L} ∈ S_{cℓ}^{L}(ℝ^{q}; H^{s,γ}(X^{∧}), K^{s,γ′}(X^{∧})^{ϱ})

with an appropriate ϱ = ϱ(s, γ, L); (ii) for every γ′, ϱ ∈ ℝ there exists an L = L(s, γ, γ′, ϱ) ≥ 0 such that

(1 − ω)(t[η])t^{−L} ∈ S_{cℓ}^{L}(ℝ^{q}; K^{s,γ}(X^{∧})^{ϱ}, H^{s,γ′}(X^{∧})).

Lemma A.6 Let s, γ, ϱ, γ′, ϱ′ ∈ ℝ. Then (i) ω̃(t[η])t^{L} ∈ S_{cℓ}^{−L}(ℝ^{q}; H^{s,γ′}(X^{∧}), K^{s,γ′+L}(X^{∧})^{ϱ′}) for any L ∈ ℝ; (ii) ω̃(t[η])t^{L} ∈ S_{cℓ}^{−L}(ℝ^{q}; K^{s,γ}(X^{∧})^{ϱ}, H^{s,γ}(X^{∧})) for L ≥ 0. In fact, all symbols of the previous two lemmas are twisted homogeneous for large |η| of degree L and −L, respectively. Lemma A.7 For ω_{2} ≺ ω_{1} and N ∈ ℕ let

f(t, t′, η) = ω_{2}(t[η])(log t/t′)^{−N}(1 − ω_{1})(t′[η]).

for t, t′ ∈ ℝ_{+} and η ∈ ℝ^{q}. Then the following is true:

(i) f(λ^{−1}t, λ^{−1}t′, λη) = f(t, t′, η) for all λ ≥ 1, t, t′ > 0, and all |η| ≥ const; (ii) For each k, k′ ∈ ℕ_{0}, α ∈ ℕ_{0}^{q},

sup_{t,t′,η} {|(t∂_{t})^{k}(t′∂_{t′})^{k′}∂_{η}^{α}f(t, t′, η)|[η]^{|α|}} < ∞.

In particular, (t, t′) ↦ [η]^{|α|}∂_{η}^{α}f(t, t′, η) is bounded in η ∈ ℝ^{q} with respect to the

topology in C_{b,F}^{∞} (ℝ_{+} × ℝ_{+}).

Proof: (i) is obvious. By induction, (t∂_{t})^{k}(t′∂_{t′})^{k′}∂_{η}^{α}f(t, t′, η) is a linear combination of terms

a(η)[(t∂_{t})^{l}ω_{2}](t[η])(log t/t′)^{−(N+M)}[(t′∂_{t′})^{l′}(1 − ω_{1})](t′[η])

with l ≤ |α| + k, l′ ≤ |α| + k′, M ≤ k + k′, and a ∈ S^{−|α|}(ℝ^{q}). Since ω_{2} ≺ ω_{1}, there is an ε > 0 such that ω_{2}(t)(1 − ω)(t′) = 0 whenever |1 − t/t′| < ε. Then also

[(t∂_{t})^{l}ω_{2}](t[η])[(t′∂_{t′})^{l′}(1 − ω)](t′[η]) = 0 for all |1 − t/t′| < ε and η ∈ ℝ^{q}. Thus (ii) is valid since sup{| log t/t′|^{−1} | |1 − t/t′| ≥ ε} < ∞. □

Proposition A.8 Let h̃(t, z, η) ∈ C^{∞}(\overline{ℝ}_{+}, M_{O}^{μ}(X; ℝ^{q})) be independent of t for large t, and h(t, z, η) = h̃(t, z, tη). Moreover, let ω_{2} ≺ ω_{1}. Then both

g_{0}(η) = ω_{2}(t[η]) op_{M}^{γ− \frac{n}{2}} (h)(η) (1 − ω_{1})(t[η]) and
g_{1}(η) = (1 − ω_{1})(t[η]) op_{M}^{γ− \frac{n}{2}} (h)(η) ω_{2}(t[η])

are elements of R_{G}^{0}(ℝ^{q}; (γ, γ))_{∞}.

Proof: First we will show that

g_{0} ∈ S_{cℓ}^{0}(ℝ^{q}; K^{s,γ}(X^{∧})^{ϱ}, K^{s′,γ′}(X^{∧})^{ϱ′})  (19)

for arbitrary s, s′, γ′, ϱ, ϱ′. Integration by parts shows that

g_{0}(η) = op_{M}^{γ− \frac{n}{2}} (f(t, t′, η)∂_{z}^{N}h(t, z, η))

for each N ∈ ℕ. Here, f is the function from Lemma A.7. Now choose ω, ω̃ such that ω_{2} ≺ ω̃ and ω ≺ ω_{1}. Moreover choose L ≥ 0 so large that Lemma A.5(ii) is satisfied. Due to the holomorphy of h we get

g_{0}(η) = ω̃(t[η])t^{L}op_{M}^{γ′− \frac{n}{2}} (f(t, t′, η)T^{−L}∂_{z}^{N}h(t, z, η))(1 − ω)(t[η])t^{−L}

(as operators on C_{0}^{∞}(X^{∧})). In view of Lemma A.5(ii) and Lemma A.6(i) the relation (19) follows if we can show that

a(η) := op_{M}^{γ′− \frac{n}{2}} (f(t, t′, η)T^{−L}∂_{z}^{N}h(t, z, η)) ∈ S_{cℓ}^{0}(ℝ^{q}; H^{s,γ′}(X^{∧}), H^{s−μ+N,γ′}(X^{∧}))

for then we can choose N so large that s − μ + N ≥ s′. In order to simplify the notation we replace T^{L}∂_{z}^{N}h by h, and therefore assume that h̃ ∈ C^{∞}(\overline{ℝ}_{+}, M_{O}^{μ−N}(X; ℝ^{q})). A Taylor expansion yields

h̃(t, z, η) = ∑_{j=0}^{J−1} t^{j}h̃_{j}(z, η) + t^{J} h̃_{J}(t, z, η)

with h̃_{j} ∈ M_{O}^{μ−N}(X; ℝ^{q}) and h̃_{J} ∈ C^{∞}(\overline{ℝ}_{+}, M_{O}^{μ−N}(X; ℝ^{q})). Let h_{j} and h_{J} be the corresponding degenerate symbols. Hence we get

a(η) = ∑_{j=0}^{J} t^{j}op_{M}^{γ′− \frac{n}{2}} (f(t, t′, η)h_{j}(t, z, η)) =: ∑_{j=0}^{J} t^{j}a_{j}(η)

with obvious meaning of notation. By Lemma A.7(ii), in particular, t^{j}f(t, t′, η)h_{j}(t, z, η) is an element of C^{∞}(ℝ_{η}^{q}, C_{b,F}^{∞} (ℝ_{+} × ℝ_{+}, M_{O}^{μ−N}(X))) for each j. Therefore

t^{j}a_{j}(η) ∈ C^{∞}(ℝ_{η}^{q}, L(H^{s,γ′}(X^{∧}), H^{s−μ+N,γ′}(X^{∧}))).

Lemma A.7(i) provides that, for 0 ≤ j < J, each t^{j}a_{j}(η) is homogeneous of degree −j for large |η|. Thus it remains to verify that

t^{J}a_{J}(η) = a(η) − ∑_{j=0}^{J−1} t^{j}a_{j}(η) ∈ S^{−J}(ℝ_{η}^{q}; H^{s,γ′}(X^{∧}), H^{s−μ+N,γ′}(X^{∧})).

For α ∈ ℕ_{0}^{q} we obtain

∂_{η}^{α}[f(t, t′, η)h_{J}(t, z, η)] = ∑_{β≤α} \binom{α}{β}∂_{η}^{β}f(t, t′, η)t^{|α−β|}(∂_{η}^{α−β}h̃_{J})(t, z, tη),

where the differentiation is with respect to the topology in C_{b,F}^{∞} (ℝ_{+} × ℝ_{+}, M_{O}^{μ−N}(X)). This yields

κ_{[η]}^{−1} ∂_{η}^{α}(t^{J}a_{J}(η)) κ_{[η]}[η]^{|α|+J} = op_{M}^{γ′− \frac{n}{2}} (v_{α}(t, t′, η))

with

v_{α}(t, t′, η) = ∑_{β≤α} \binom{α}{β}(∂_{η}^{β}f)(\frac{t}{[η]}, \frac{t′}{[η]}, η)[η]^{|β|}t^{J+|α−β|}(∂_{η}^{α−β}h̃_{J})(\frac{t}{[η]}, z, t\frac{η}{[η]}).

Now, v_{α} is a bounded function of η with values in C_{b,F}^{∞} (ℝ_{+} × ℝ_{+}, M_{O}^{μ−N}(X)) in view of Lemma A.7(ii) and the compact t-support of v_{α}. Thus

sup_{η∈ℝ^{q}} ∥op_{M}^{γ′− \frac{n}{2}} (v_{α})(η)∥_{L(H^{s,γ′},H^{s−μ+N,γ′})} < ∞,

and (19) holds. To treat g_{1} we use Proposition 4.2 and rewrite op_{M}^{γ− \frac{n}{2}} (h) = op_{M}^{γ− \frac{n}{2}} (h_{R}) with a right-symbol h_{R} ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, M_{O}^{μ}(X; ℝ^{q})). Then we proceed analogously as above, i.e., we choose L in such a way that \frac{n+1}{2} − γ + L ≥ ϱ′, and write

g_{1}(η) = (1 − ω)(t[η])t^{−L}op_{M}^{γ− \frac{n}{2}} ((−1)^{N}f(t′, t, η)T^{L}∂_{z}^{N}h_{R}(t′, z, η))ω̃(t[η])t^{L}.

With Lemmas A.5(i), A.6(ii), A.7, and a Taylor expansion of h̃_{R} in t′ at t′ = 0 one can verify that g_{1} also satisfies (19). To consider the formal adjoints g_{j}^{∗} it suffices to observe that, after rewriting g_{1} with a right-symbol, g_{j}^{∗} looks like g_{1−j} (j = 0, 1). □

Definition A.9 Let E be a Fréchet space. We define S(ℝ_{+} × ℝ_{+}, E) as the subspace of all functions k ∈ C^{∞}(ℝ_{+} × ℝ_{+}, E) that satisfy

sup {p(∂_{t}^{l}∂_{t′}^{l′}k(t, t′)) ⟨t⟩^{N} ⟨t′⟩^{N} | t, t′ ≥ ε} < ∞

for each ε > 0, l, l′, N ∈ ℕ_{0} and each semi-norm p of E. This expressions provide a semi-norm system that induces a Fréchet topology on S(ℝ_{+} × ℝ_{+}, E).

Lemma A.10 Let h̃ ∈ C_{b,F}^{∞} (ℝ_{+} × ℝ_{+}, M_{O}^{−∞}(X; ℝ^{q})), and

h_{0}(t, t′, z, η) = h̃(t, t′, z, tη), h_{1}(t, t′, z, η) = h̃(t, t′, z, t′η).

Since M_{O}^{−∞}(X; ℝ^{q}) = M_{O}^{−∞}(ℝ^{q}) ˆ⊗_{π}C^{∞}(X × X), we may associate to each h_{j} a kernel

k_{j}(t, t′, x, x′, η) = ∫ (\frac{t}{t′} )^{−iτ} h_{j}(t, t′, x, x′, iτ, η) d¯ τ.

Then k_{j} ∈ C^{∞}(ℝ^{q}, C^{∞}(ℝ_{+}×ℝ_{+}×X×X)) and k_{j} ∈ C^{∞}(ℝ^{q}\{0}, S(ℝ_{+}×ℝ_{+}, C^{∞}(X×X))). The corresponding mappings h̃ ↦ k_{j} are continuous.

Proof: By a standard tensor-product argument we can assume that the base X is a point. The part concerning the smoothness of k_{j} is clear. It remains to verify that, for each given ε > 0, N ∈ ℕ_{0}, and each compact set K ⊂ ℝ^{q} \ {0},

sup {|∂_{t}^{l}∂_{t′}^{l′}∂_{η}^{α}k_{j}(t, t′, η)| ⟨t⟩^{N} ⟨t′⟩^{N} | t, t′ ≥ ε, η ∈ K, l, l′, |α| ≤ N} < ∞.

Consider the case j = 0. Since h_{0}(t, t′, z, η) is holomorphic in z and decreases as a Schwartz function on each Γ_{β} uniformly for β in compact intervals, the Cauchy formula implies that

k_{0}(t, t′, η) = t^{N}t′^{−N} ∫ (\frac{t}{t′} )^{−iτ} h_{0}(t, t′, iτ − N, η) d¯ τ.

Noting that

(t∂_{t})^{l}[f(t, tη)] = ∑_{|β|+k=l} c_{βk}[(t∂_{t})^{k}(η∂_{η})^{β}f](t, tη),

we see that (t∂_{t})^{l}(t′∂_{t′})^{l′}∂_{η}^{α}k_{0}(t, t′, η) is a linear combination of terms

t^{N+|α|}t′^{−N} ∫ (\frac{t}{t′} )^{−iτ} [(t∂_{t})^{n}(t′∂_{t′})^{n′}(η∂_{η})^{β}∂_{η}^{α}h̃](t, t′, iτ − N, tη)τ^{m} d¯ τ.

with n+|β| ≤ l, n′ ≤ l′, and m ≤ l+l′. For each M > 0 there is an appropriate semi-norm ||| · ||| of C_{b,F}^{∞} (ℝ_{+} × ℝ_{+}, M_{O}^{−∞}(ℝ^{q})) (independent of h̃) such that the integrand is bounded from above by c_{K} |||h̃||| ⟨τ⟩^{−2} ⟨t⟩^{−M} uniformly in η ∈ K (recall that h̃ is a Schwartz function in (τ, η) and ⟨tη⟩^{−M} ≤ c_{K} ⟨t⟩^{−M} for η ∈ K). Choosing M = 3N we then obtain that

|(t∂_{t})^{l}(t′∂_{t′})^{l′}∂_{η}^{α}k_{0}(t, t′, η)| ≤ c_{ε,K}|||h̃||| ⟨t⟩^{−N} ⟨t′⟩^{−N}

uniformly for t, t′ ≥ ε and η ∈ K. This is then also true if we replace (t∂_{t})^{l}(t′∂_{t′})^{l′} by ∂_{t}^{l}∂_{t′}^{l′} since

∂_{s}^{k} = s^{−k} ∑_{j=1}^{k} c_{j}(s∂_{s})^{j}

for certain constants c_{j}, and t^{−l}t′^{−l′} is uniformly bounded for t, t′ ≥ ε. The case j = 1 can be treated in the same manner. □

Proposition A.11 Let h̃ ∈ C^{∞}(\overline{ℝ}_{+}, M_{O}^{−∞}(X; ℝ^{q})), and set

h_{0}(t, z, η) = h̃(t, z, tη), h_{1}(t′, z, η) = h̃(t′, z, t′η).

If χ(η) is an excision function at η = 0 then

g_{j}(η) = χ(η)(1 − ω)(t[η]) op_{M}^{γ− \frac{n}{2}} (h_{j})(η) (1 − ω_{1})(t[η]) ∈ R_{G}^{0}(ℝ^{q}; (γ, γ))_{∞}.

Moreover, if σ, σ_{0} ∈ C_{0}^{∞}(\overline{ℝ}_{+}) are cut-off functions then

g̃_{j}(η) = σ(t) (1 − ω)(t[η]) op_{M}^{γ− \frac{n}{2}} (h_{j})(η) (1 − ω_{1})(t[η]) σ_{0}(t) ∈ R_{G}^{0}(ℝ^{q}; (γ, γ))_{∞}.

Proof: At first we will show that, for arbitrary s, s′, γ′, ϱ, ϱ′ ∈ ℝ,

g_{0} ∈ S_{cℓ}^{0}(ℝ^{q}; K^{s,γ}(X^{∧})^{ϱ}, K^{s′,γ′}(X^{∧})^{ϱ′}).  (20)

A Taylor expansion yields

h̃(t, z, η) = ∑_{j=0}^{N−1} t^{j}f̃_{j}(z, η) + t^{N} f̃_{N}(t, z, η)

with f̃_{j} ∈ M_{O}^{−∞}(X; ℝ^{q}) for 0 ≤ j < N, and f̃_{N} ∈ C_{b,F}^{∞} (ℝ_{+}, M_{O}^{−∞}(X; ℝ^{q})). Let f_{j} be the degenerate symbols corresponding to f̃_{j}. Due to the holomorphy of the involved symbols,

g_{0}(η) = ∑_{j=0}^{N} χ(η)(1 − ω)(t[η]) t^{j} K_{j}(η) (1 − ω_{1})(t[η]) =: ∑_{j=0}^{N} g_{(j)}(η)

with integral operators K_{j}(η) that have kernels

k_{j}(t, t′, η) = t′^{−1} ∫ (\frac{t}{t′} )^{−iτ} f_{j}(t, iτ, η) d¯ τ  (21)

with respect to the metric dt′dx (for convenience we suppress the x, x′-variables from the notation). Because of Lemma A.10 and the presence of the excision function χ, each g_{(j)} is a family of integral operators with kernel in C^{∞}(ℝ_{η}^{q}, S(ℝ_{+} × ℝ_{+}, C^{∞}(X × X))), whose kernel is supported away from ({t = 0} × X) × ({t′ = 0} × X) locally uniformly in η. This implies that

g_{0} ∈ C^{∞}(ℝ^{q}; L(K^{s,γ}(X^{∧})^{ϱ}, K^{s′,γ′}(X^{∧})^{ϱ′})).

For 0 ≤ j < N each g_{(j)} is homogeneous of degree −j for large |η|, since f̃_{j} is independent of t. Hence (20) is true if we can show that g_{(N)} ∈ S^{−N}(ℝ^{q}; K^{s,γ}(X^{∧})^{ϱ}, K^{s′,γ′}(X^{∧})^{ϱ′}), i.e. we have to verify that

∥κ_{[η]}^{−1}{∂_{η}^{α}g_{(N)}(η)}κ_{[η]}∥_{L(K^{s,γ}(X^{∧})^{ϱ},K^{s′,γ′}(X^{∧})^{ϱ′})} ≤ c[η]^{−N−|α|}  (22)

with some constant c > 0 independent of η ∈ ℝ^{q}. When calculating ∂_{η}^{α}g_{(N)} we can omit all terms where χ is differentiated, since the resulting terms are operator-families that are compactly supported in η and thus satisfy (22). The remaining terms of interest are of the form

χ(η)t^{N+n_{1}+|β|}t′^{n_{2}}a_{n_{3}}(η)ψ_{n_{1}}(t[η])ψ̃_{n_{2}}(t′[η])K_{β}(η),

where n_{1} + n_{2} + n_{3} + |β| = |α|, ψ_{n_{1}} = ∂_{t}^{n_{1}}(1 − ω), ψ̃_{n_{2}} = ∂_{t′}^{n_{2}}(1 − ω_{1}), and a_{n_{3}} ∈ S^{−n_{3}}(ℝ^{q}). Furthermore, K_{β}(η) is the integral operator with kernel as in (21), where f_{N}(t, iτ, η) is replaced by (∂_{η}^{β}f̃_{N})(t, iτ, tη). Conjugation with κ_{[η]} yields operator-families [η]^{−(N+n_{1}+|β|+n_{2})}a_{n_{3}}(η)K̃(η) with integral operators K̃(η) that have kernels

k̃(t, t′, η) = t^{N+n_{1}+|β|}t′^{n_{2}−1}ψ_{n_{1}}(t)ψ̃_{n_{2}}(t′)a_{n_{3}}(η)ϕ(η) ∫ (\frac{t}{t′} )^{−iτ} (∂_{η}^{β}f̃_{N})(\frac{t}{[η]}, iτ, t\frac{η}{[η]}) d¯ τ.

As in Lemma A.10 one can show that this is a bounded function of η ∈ ℝ^{q} with values in S(ℝ_{+} × ℝ_{+}, C^{∞}(X × X)), obviously being supported away from ({t = 0} × X) × ({t′ = 0} × X) uniformly in η ∈ ℝ^{q}. Hence K̃(η) is a bounded family of operators in L(K^{s,γ}(X^{∧})^{ϱ}, K^{s′,γ′}(X^{∧})^{ϱ′}). This, together with the fact that [η]^{−(N+n_{1}+|β|+n_{2})}|a_{n_{3}}(η)| ≤ c[η]^{−(N+|α|)}, implies (22). Analogously, g_{1} fulfills (20). Finally, g_{j}^{∗} is of the form g_{1−j} for j = 0, 1. To handle g̃_{j}, write

g̃_{j}(η) = (1 − χ)(η) g̃_{j}(η) + σ(t) g_{j}(η) σ_{0}(t).

The first term on the right-hand side has a kernel in C_{0}^{∞}(ℝ^{q}, C_{0}^{∞}(ℝ_{+} × ℝ_{+} × X × X)). In particular, (1 − χ)(η)g̃_{j}(η) ∈ R_{G}^{−∞}(ℝ^{q}; (γ, γ))_{∞}. In view of Remark 3.12, the second term is an element of R_{G}^{0}(ℝ^{q}; (γ, γ))_{∞}. □

Remark A.12 Let h ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, M_{O}^{μ}(X; ℝ^{q})), and let φ, ψ ∈ C_{0}^{∞}(\overline{ℝ}_{+}) be functions with supp φ ∩ supp ψ = ∅. Then there is an element h_{∞} ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, M_{O}^{−∞}(X; ℝ^{q})) such that

φ op_{M}^{γ− \frac{n}{2}} (h)(η)ψ = φ op_{M}^{γ− \frac{n}{2}} (h_{∞})(η)ψ.

Proposition A.13 Let h ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, M_{O}^{μ}(X; ℝ^{q})), and let σ, σ̃, σ_{1}, σ_{2} ∈ C_{0}^{∞}(\overline{ℝ}_{+}) be cut-

off functions with σ_{2} ≺ σ_{1}. Further set a_{M}(η) = σ op_{M}^{γ− \frac{n}{2}} (h)(η) σ̃. Then

σ_{2} a_{M}(η)(1 − σ_{1}) and (1 − σ_{1})a_{M}(η)σ_{2}

are elements of R_{G}^{−∞}(ℝ^{q}; (γ, γ))_{∞}.

Proof: Choose cut-off functions ω_{2} ≺ ω_{1} such that ω_{2} ≺ σ_{2} and ω_{1} ≺ σ_{1}. Then

σ_{2}(t)a_{M}(η)(1 − σ_{1})(t) = σ_{2}(t)(g_{1}(η) + g_{2}(η))(1 − σ_{1})(t) =: g(η)

with g_{1}(η) = ω_{2}(t[η])a_{M}(η)(1 − ω_{1})(t[η]) and g_{2}(η) = (1 − ω_{2})(t[η])a_{M}(η)(1 − ω_{1})(t[η]). Proposition A.8 yields g_{1} ∈ R_{G}^{0}(ℝ^{q}; (γ, γ))_{∞}. Using Lemma A.12 we may assume that h ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, M_{O}^{−∞}(X; ℝ^{q})) so that g_{2} ∈ R_{G}^{0}(ℝ^{q}; (γ, γ))_{∞} because of Proposition A.11. Finally, g ∈ R_{G}^{−∞}(ℝ^{q}; (γ, γ))_{∞} in view of Remark 3.12. The family (1 − σ_{1})a_{M}(η)σ_{2} can be treated in the same way. □

Remark A.14 For h ∈ C_{deg}^{∞}(\overline{ℝ}_{+}, M_{O}^{−∞}(X; ℝ^{q})), and cut-off functions σ and σ̃ we have

σ op_{M}^{γ− \frac{n}{2}} (h)(η) σ̃ ∈ R_{G}^{0}(ℝ^{q}; (γ, γ)) for every γ ∈ ℝ.

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Address: Arbeitsgruppe “Partielle Differentialgleichungen und Komplexe Analysis” Institut für Mathematik, Universität Potsdam Postfach 60 15 33 14415 Potsdam, Germany.