Der-Chen Chang, N. Habal and B.-W. Schulze
Abstract. Boundary value problems and mixed problems play a role in many applications of physics, cf. [6], and parametrices and asymptotics of solutions may be expressed within suitable pseudo-differential algebras. The boundary-wedge approach of [17], [18] belongs to the calculus on singular manifolds. The present article continues this concept and develops new quantisations in the framework of operators on manifolds with singular edges.
Mathematics Subject Classification (2000). Primary 35J70; Secondary 35S35, 47G30, 58J40.
Keywords. Mellin quantisations, edge pseudo-differential operators.
1. Introduction
Elliptic mixed and transmission problems may be expressed in terms of operators on manifolds with singularities, i.e., conical points, edges, corners. The construc-tion of parametrices requires pseudo-differential operators and quantisaconstruc-tions that turn symbols to operators in a controlled way. We study here Mellin quantisa-tions for symbols taking values in pseudo-differential operators on a manifold B with edge Y . Such operators form an algebra, constituted by spaces Lµ(B, g) of
operators of order µ over B, where g denotes weight data, to be explained be-low. The variables and covariables in such symbols are denoted by t, y, τ, η for t = (t1, . . . , tk) ∈ (R+)k, y = (y1, . . . , yk) ∈ Rq, q = P k j=0qj, yj ∈ Rqj, and τ ∈ Rk, η = (η1, . . . , ηk ) ∈ Rq. We consider symbols p(t, y, τ, η) = ˜p(t, y, t1τ1, . . . , t1· · · tkτk, t1η1, . . . , t1· · · tkηk) (1.1) for ˜ p(t, y, ˜τ1, . . . , ˜τk, ˜η1, . . . , ˜ηk) ∈ C∞((R+)k× Rqy, L µ (B, g; Rk+qτ ,˜˜η )), (1.2) ˜
τ = (˜τ1, . . . , ˜τk), ˜η = (˜η1, . . . , ˜ηk). Operator-valued symbols containing parameters
stratified spaces, cf. also [6, Chapter 10] in connection with mixed and transmis-sion problems with singular interfaces.
The present article is organised as follows.
In Section 2 we develop the tools from the edge calculus with parameters, i.e., we study spaces of the kind Lµ(B, g; Rl). In Section 3 we establish a Mellin quantisa-tion result for edge operator-valued amplitude funcquantisa-tions. This employs, in partic-ular, the results of [4]. In Section 4 we pass to Mellin quantisation for k = 2, and we show how to iterate the process to reach the case of an arbitrary singularity order k. An iteration of Mellin quantisations for a smooth base has been obtained in [5]. We will see how other elements of the higher corner symbolic and operator calculus occur as necessary tools, such as [4], [21]. The edge calculus from Section 2 and properties of the kernel cut-off allow us to trace back different symbols from the principal symbolic hierarchy under the higher Mellin quantisation.
Let us finally note that many traditional ideas from the analysis of boundary value problems are involved in the edge pseudo-differential calculus, see, in particular, Agranovich and Vishik [1] or Eskin’s book [3]. In addition Kondratyev’s paper [7] played a large role in the development of the cone pseudo-differential calculus, later on also combined with edges, cf. [14], [15], [16], [17]. Holomorphic Mellin quantisa-tions in edge and corner operators of lower singular orders have been studied from different point of views also in [2], [11], [18], [19].
2. Edge operators with parameters
We first recall some material on the calculus of pseudo-differential operators on manifolds with edge.
A manifold B with edge Y ⊂ B is defined by the properties that both Y =: s1(B)
and B \ Y =: s0(B) are smooth manifolds and every y ∈ Y has a neighbourhood
V ⊆ B such that there is a homeomorphism
χ : V → X∆× Ω (2.1)
for an open set Ω ⊆ Rq and a smooth manifold X; here
X∆:= (R+× X)/({0} × X)
is the cone with base X, written as a quotient space where the vertex is just represented by the bottom {0} × X of the cylinder R+× X. We assume that χ
restricts to diffeomorphisms
χ0: V \ Y → X∧× Ω for X∧:= R+× X (2.2)
and
χ1: V ∩ Y → Ω, (2.3)
(the latter is a chart on Y ). In addition if χ : V → X∆× Ω, ˜χ : V → X∆× ˜Ω are
two different “singular charts” of the kind (2.1) then for the associated maps (2.2) and ˜χ0, respectively, the diffeomorphism ˜χ0◦ χ−10 : X∧× Ω → X∧× ˜Ω is asked to
in the sense of manifolds with boundary X × Ω and X × ˜Ω, respectively. With B we can associate a manifold B with smooth boundary, the so-called stretched manifold, which is locally near ∂B modelled on R+× X × Ω, Ω ⊆ Rq. Moreover,
∂B is an X-bundle over the edge Y . We can identify B with the quotient space B/ ∼ where ∼ identifies points of ∂B when they are projected under the bundle projection to the same point y ∈ Y . To illustrate the situation consider the case B = X∆× Ω. Then we have B = R+× X × Ω and the trivial bundle ∂B = X × Ω,
and the bundle projection is simply the projection (x, y) → y to the second factor. Edge pseudo-differential operators play an important role in this article. Therefore we present here some essential details in concise form. More material may be found in [6], [14], [16], [17], [19].
First let X be a manifold of dimension n. Let Lµ(cl)(X; Rl
), µ ∈ R, denote the space of pseudo-differential operators on X depending on a parameter λ ∈ Rl. By
(cl) we indicate spaces of operators with classical or general symbols when consid-erations concern both cases. The local symbols a(x, ξ, λ) depend on (ξ, λ) ∈ Rn+l, treated as covariables. The space of parameter-dependent smoothing operators is denoted by L−∞(X; Rl) := S(Rl, L−∞(X)). Here L−∞(X) is identified with the Fr´echet space C∞(X × X) via a Riemannian metric on X that is fixed. Modulo
smoothing elements the operators in Lµ(cl)(X; Rl) have the form Op
x(a)(λ),
Opx(a)(λ)u(x) =
ZZ
ei(x−x0)ξa(x, ξ, λ)u(x0)dx0d¯ξ,
d¯ξ = (2π)−ndξ. If we say nothing else the operator families A(λ) ∈ Lµ(cl)(X; Rl)
are regarded as families of continuous operators A(λ) : C0∞(X) → C∞(X).
In a similar manner we will define a space Lµ(B, g; Rl) of parameter-dependent pseudo-differential operators on B \ Y with parameter, contained in Lµcl(B \ Y ; Rl), see Definition 2.6. Since B is a manifold with edge singularities we have to observe also so-called weight data g = (γ, γ − µ, Θ), where γ ∈ R refers to the weight in some weighted edge Sobolev spaces Hs,γ
(B), cf. (2.8) below, with s ∈ R being the smoothness, and our operators then will be continuous in the sense
A(λ) : Hs,γ(B) → Hs−µ,γ−µ(B) (2.4)
(here, for simplicity, in the case of compact B). Moreover, Θ := (ϑ, 0], Θ indicates the width of so-called weight strips in the complex z-plane
n + 1 2 − γ + ϑ < Re z < n + 1 2 − γ and n + 1 2 − (γ − µ) + ϑ < Re z < n + 1 2 − (γ − µ),
lying on the left of weight lines Γn+1
2 −γand Γ n+1
2 −(γ−µ), respectively. Here Γβ:=
{z ∈ C : Re z = β} for some real β. The variable z refers to the Mellin transform M u(z) =
Z ∞
0
rz−1u(r)dr.
Recall that for u ∈ C0∞(R+) the Mellin transform is an entire function, and we
have M u|Γβ ∈ S(Γβ) for every β ∈ R. The map Mγ : u → M u|Γ1/2−γ, γ ∈ R,
extends by continuity to an isomorphism
Mγ : rγL2(R+) → L2(Γ1/2−γ),
called the weighted Mellin transform with weight γ. The associated Mellin pseudo-differential operator will be denoted by opγM(f ),
opγM(f )u(r) = Z R Z ∞ 0 (r/r0)−(1/2−γ+iρ)f (r, r0, 1/2 − γ + iρ)u(r0)dr0/r0d¯ρ, d¯ρ = (2π)−1dρ, for some Mellin symbol f (the admitted symbol spaces will be specified later on).
Let Hs,γ
(R+× Rn) for s, γ ∈ R defined to be the completion of C0∞(R+× Rn)
with respect to the norm kukHs,γ(R+×Rn):= nZ Rn Z Γn+1 2 −γ hz, ξi2s|(M F u)(z, ξ)|2d¯zdξo1/2
where M = Mr→z is the Mellin transform, F = Fx→ξ the Fourier transform, and
d¯z := (2πi)−1dz. Here hz, ξi := (1 + |z|2+ |ξ|2)1/2. From the identity M (rβu)(z) =
(M u)(z + β) it follows that Hs,γ(R+× Rn) = rγHs,0(R+× Rn). Moreover, for
a smooth compact manifold X of dimension n we define the space Hs,γ(X∧) as follows. Choose a system of charts χj : Uj → Rn for a covering {U1, . . . , UN} by
coordinate neighbourhoods. Moreover, let {ϕ1, . . . , ϕN} be a subordinate partition
of unity. Then we define Hs,γ(X∧) 3 u with the norm
kukHs,γ(X∧)=
nXN
j=1
k(1 × χ−1j )∗ϕjuk2Hs,γ(R+×Rn)
o1/2
which is generated by the Hilbert scalar product (u, v)Hs,γ(X∧)= N X j=1 (1 − χ−1j )∗ϕju, (1 − χ−1j ) ∗ϕ jv Hs,γ(R+×Rn) where (f, g)Hs,γ(R+×Rn)= Z Rn Z Γn+1 2 −γ hz, ξi2sM F f (z, ξ)(M F g)(z, ξ)d¯zdξ.
In the rest of the paper a cut-off function will be an element of C0∞(R+) such that
ω ≡ 1 close to r = 0.
We also need spaces Hcones (X∧) that are defined as follows. Choose
diffeo-morphisms υj : Uj → Vj ⊂ Sn, for open sets Vj ⊂ Sn, set Vj∧ := {˜x ∈ R n+1 :
˜ x
|˜x| ∈ Vj} and form the diffeomorphism υ ∧
j : R+× Uj→ Vj∧, (r, x) → rυj(x). Then
Hs
cone(X∧) is characterised as the set of all u ∈ Hlocs (R×X)|X∧ such that for every
ϕj∈ C0∞(Uj)
((υ∧j)−1)∗(1 − ω)ϕju ∈ Hs(Rn+1) for j = 1, . . . , N.
Then we set
Ks,γ(X∧) := {u = ωu0+ (1 − ω)u∞: u0∈ Hs,γ(X∧), u∞∈ Hcones (X∧)}
for any fixed cut-off function ω. Also the spaces Ks,γ(X∧) can be equipped with Hilbert scalar products. For s = γ = 0 the scalar products will be defined (up to equivalence) in such a way that
H0,0(X∧) = K0,0(X∧) = r−n/2L2
(R+× X)drdx,
with dr being the restriction of the Lebesgue measure on R to R+and dx associated
with the Riemannian metric on X.
Later on we employ the action of a one-parameter group action κ = {κδ}δ∈R+
of isomorphisms
κδ : Ks,γ(X∧) → Ks,γ(X∧), (κδu)(r, x) := δ
n+1
2 u(δr, x), δ ∈ R+. (2.5)
A Hilbert space H is said to be endowed with a group action κ = {κδ}δ∈R+ if
κδ : H → H are isomorphisms, such that κδκν = κδν, and δ → κδh belongs to
C(R+, H) for every h ∈ H. We define the “abstract” edge space of smoothness
s ∈ R as the completion of S(Rq, H) with respect to the norm
kukWs(Rq,H)= nZ hηi2skκ−1 hηiu(η)kˆ 2 Hdη o1/2 . (2.6)
This space depends on the choice of κ; if necessary we write Ws
(Rq, H)
κ rather
than Ws(Rq, H). More generally, instead of H we also admit a Fr´echet space E written as a projective limit lim
←−j∈NE
j of Hilbert spaces Ej with continuous
embeddings Ej ,→ E0 such that E0 is endowed with a group action κ as before such that κ|Ej is also a group action on Ej for every j ∈ N = {0, 1, . . . }. Using
the spaces Ws (Rq, Ej) we set Ws (Rq, E) := lim ←− j∈N Ws (Rq, Ej).
Let us now recall the notion of pseudo-differential symbols with twisted symbolic estimates.
Given Hilbert spaces H and ˜H equipped with group actions κ and ˜κ, respec-tively, we define the symbol space
Sµ(U × Rq; H, ˜H) for an open set U ⊆ Rp
, µ ∈ R, as the set of all a(y, η) ∈ C∞(U × Rq, L(H, ˜H))
such that
k˜κ−1hηi{Dα yD
β
ηa(y, η)}κhηikL(H, ˜H)≤ chηi µ−|β|
for all (y, η) ∈ K × Rq
, K b U, α ∈ Np
, β ∈ Nq, for constants c = c(α, β, K) > 0.
Let S(µ)
(U × (Rq \ {0}); H, ˜H) be the set of all a
(µ)(y, η) ∈ C∞(U × (Rq \
{0}), L(H, ˜H)) such that
a(µ)(y, δη) = δµκ˜δa(µ)(y, η)κ−1δ
for all δ ∈ R+. Let χ(η) be an excision function, i.e., χ(η) = 0 for |η| < c0, χ(η) = 1
for |η| > c1, with some constants c0, c1, 0 < c0< c1. Then,
χ(η)a(µ)(y, η) ∈ Sµ(U × Rq; H, ˜H).
By Sµcl(U × Rq; H, ˜H) we denote the set of all a(y, η) ∈ Sµ
(U × Rq; H, ˜H) with a(y, η) ∼ ∞ X j=0 χ(η)a(µ−j)(y, η)
for certain a(µ−j) ∈ S(µ−j)(U × (Rq \ {0}); H, ˜H). Here ∼ means the asymptotic
expansion of symbols which makes sense also in the operator-valued set-up, cf. [19].
Note that for H = ˜H = C and κ = ˜κ = idC we just recover the standard symbol spaces Sclµ(U × Rq). Recall that we write (cl) as subscript if some consideration is true both in the classical and in the general case.
Operator-valued symbol spaces can be also formulated for Fr´echet spaces E and ˜E with group action κ and ˜κ in the above-mentioned sense. In this case, if k : N → N is any fixed function then if E = lim
←−k∈NE
k, ˜E = lim
←−j∈N
˜
Ej, we have the spaces S(cl)µ (U × Rq; Ek(j), ˜Ej
) for every j ∈ N. Then S(cl)µ (U × Rq; E, ˜E)k:= \ j∈N S(cl)µ (U × Rq; Ek(j), ˜Ej), and we set Sµ(cl)(U × Rq; E, ˜E) =S kS µ (cl)(U × R q; E, ˜E) k.
Example. Let us set E = L2
(R+), ˜E = S(R+) := S(R)|R+ = lim←−j∈Nhti−jHj(R+)
and consider (κλu)(t) := λ1/2u(λt), λ ∈ R+. Then the operator-valued symbols
g(η) ∈ Sclµ(Rq; L2(R+), S(R+)) such that g∗(η) ∈ Sclµ(R q; L2
(R+), S(R+)) are
spe-cific Green symbols (of type 0) in Boutet de Monvel’s calculus. Here g∗(η) means the η-wise adjoint in L(L2(R+)).
Remark 2.1. A symbol k(y, η) ∈ Sclµ(U ×Rq; C, ˜E) with idCas the group action on C is the abstract model of a potential symbol. Moreover, t(y, η) ∈ Sclµ(U × Rq; E, C) is modelling an abstract trace symbol in the operator theories that we have in mind. A symbol
k(y, η)t(y, η)
Clearly the choice of κ, ˜κ affects the structure of symbol spaces; if necessary we write S(cl)µ (U × Rq; H, ˜H)
κ,˜κ. Based on the group action (2.5) in Ks,γ(X∧) and
the edge spaces given by (2.6), here for H = Ks,γ(X∧), we can form the spaces Ws
(Rq, Ks,γ(X∧)), s ∈ R. (2.7)
Now the space
Hs,γ(B), s, γ ∈ R, (2.8)
occurring in (2.4) is defined as the subspace of all u ∈ Hs
loc(B \ Y ) such that locally
close to the edge Y in the splitting of variables (r, x, y) ∈ X∧× Rq from (2.2) we
have the property
ωψu ∈ χ∗0Ws
(Rq, Ks,γ(X∧))
for any cut-off function ω and ψ ∈ C0∞(G); here we refer to χ0 in (2.2) and
χ1: G → Rq in (2.3) for G := V ∩ Y , Ω = Rq.
Let us now turn to asymptotics in the space Ws
(Rq, Ks,γ(X∧)) for r → 0, i.e.,
close to the edge Rq.
An example of a function u on the stretched cone X∧ with asymptotics for r → 0 is
ω(r)c(x)r−plogkr ∈ K∞,γ(X∧) for some p ∈ C, Re p < n+12 − γ, k ∈ N, c(x) ∈ C
∞(X), for a cut-off function
ω on the half-axis. Observe that the weighted Mellin transform Mγ−n/2u(z) is
meromorphic with a pole at p of multiplicity k + 1. The pole lies in the weight strip when Re p > n+1
2 − γ + ϑ. Now if finitely many such singular functions play
a role then the asymptotic information is incorporated by the system
P = {(pj, mj)}j=0,...,J⊂ C × N (2.9)
for some J ∈ N where πCP := {pj}j=0,...,J is contained in the above-mentioned
weight strip on the left of Γn+1 2 −γ
.
A sequence (2.9) for J ∈ N ∪ {+∞} will be called a (discrete) asymptotic type associated with weight data (γ, Θ), Θ = (ϑ, 0], −∞ ≤ ϑ < 0, if
πCP ⊂ n + 1
2 − γ + ϑ < Re z < n + 1
2 − γ
where J < ∞ for ϑ > −∞ and Re pj → −∞ as j → ∞ for ϑ = −∞ (when πCP is
infinite).
Definition 2.2. Fix a cut-off function ω, let Θ = (ϑ, 0] be a finite weight strip, P an asymptotic type (2.9) for finite J. We define
EP(X∧) := n ω(r) J X j=0 mj X k=0 cjk(x)r−pjlogkr : cjk(x) ∈ C∞(X) o (2.10) and Ks,γP (X∧) := EP(X∧) + Ks,γΘ (X ∧) (2.11)
where Ks,γΘ (X∧) := lim←− l∈N Ks,γ−ϑ−(1+l)−1(X∧). (2.12) For Θ = (−∞, 0] we set T b∈NK s,γ Pb(X ∧) =: Ks,γ P (X∧) where Pb := {(p, m) ∈ P : Re p > n+12 − (1 + b)} for some b ∈ N.
The space (2.11) is Fr´echet in the topology of the direct sum of Fr´echet spaces (2.10) and (2.12). It can be easily verified that the group action κ from (2.5) induces a corresponding group action in Ks,γP (X∧) for any asymptotic type P. This allows us to define the local edge spaces
Ws
(Rq, Ks,γP (X∧)) (2.13)
and the global spaces HPs,γ(B) ⊂ Hs
loc(B \ Y ) where we replace (2.7) by (2.13). At
the same time we obtain HΘs,γ(B) where we insert Ks,γΘ (X∧) rather than KPs,γ(X∧). We are now in the position to explain the notion of Green symbols of the edge calculus with asymptotics of type P and Q, respectively.
Set
Ks,γ;e(X∧) := hri−eKs,γ(X∧), Ks,γ;e
P (X∧) := hri−eK s,γ P (X∧)
for any s, γ, e ∈ R. By Rν
G(U × Rq, g) for g = (γ, γ − µ, Θ) we denote the set of all operator functions
g(y, η) ∈ C∞(U × Rq, L(Ks,γ(X∧), K∞,γ−µ(X∧))) such that g(y, η) ∈ \ s,e∈R Sclν(U × Rq; Ks,γ;e(X∧), K∞,γ−µ;∞P (X∧)) (2.14) and g∗(y, η) ∈ \ s,e∈R Sclν(U × Rq; Ks,−γ+µ;e(X∧), K ∞,−γ;∞ Q (X ∧))
for g-dependent asymptotic types P and Q. The (y, η)-wise formal adjoint is taken with respect to the K0,0(X∧)-scalar product.
By L−∞(B, g) for a compact manifold B with edge Y we denote the space of all C ∈ L−∞(B \ Y ) which are continuous as operators
C : Hs,γ(B) → HP∞,γ−µ(B)
such that the formal adjoint C∗with respect to the H0,0(B)-scalar product induces continuous operators
C∗: Hs,−γ+µ(B) → HQ∞,−γ(B)
for all s ∈ R and some C-dependent asymptotic types P, Q. For fixed P, Q the corresponding space L−∞(B, g) P,Q is Fr´echet. We set L−∞(B, g; Rl)P,Q:= S(Rl, L−∞(B, g)P,Q) and L−∞(B, g; Rl) = [ P,Q L−∞(B, g; Rl)P,Q.
The operators in the latter space are called parameter-dependent smoothing op-erators in the edge calculus.
If E is a Fr´echet space by A(C, E) we denote the space of all E-valued holomorphic functions in the topology of uniform convergence on compact sets. We consider, in particular, E := Lµcl(X; Rl) in its natural Fr´echet topology.
Definition 2.3. By MOµ(X; Rl
), µ ∈ R, we denote the space of all h(z, λ) ∈ A(C, Lµcl(X; R
l)) (2.15)
such that h|Γβ ∈ L
µ
cl(X; Γβ×Rl) for every β ∈ R, uniformly in compact β-intervals.
Moreover, we set MO−∞(X; Rl) := \ j∈N MOµ−j(X; Rl). Theorem 2.4. Let p(r, y, ρ, η) = ˜p(r, y, rρ, rη) (2.16) for ˜ p(r, y, ˜ρ, ˜η) ∈ C∞(R+× Ω, Lµcl(X; R1+qρ, ˜˜η )),
Ω ⊆ Rq open. Then there exists an ˜h(r, y, z, ˜η) ∈ C∞
(R+× Ω, M µ O(X; R q ˜ η)) such
that for h(r, y, z, η) := ˜h(r, y, z, rη) we have
Opr(p)(y, η) = opβM(h)(y, η) mod C∞(Ω, L−∞(X∧; Rqη)), (2.17) for any β ∈ R; here (2.17) is interpreted as operator families C0∞(X∧) → C∞(X∧).
For any other h1(r, y, z, η) = ˜h1(r, y, z, rη), ˜h1(r, y, z, ˜η) ∈ C∞(R+×Ω, MOµ(X; Rqη˜))
satisfying an analogue of the relation (2.17) we have ˜
h(r, y, z, ˜η) = ˜h1(r, y, z, ˜η) mod C∞(R+× Ω, MO−∞(X; R q ˜ η)).
In order to formulate edge pseudo-differential operators we establish the es-sential ingredients in terms of specific operator-valued symbols locally along the edge in (y, η) ∈ Ω × Rq.
We write ϕ ≺ ϕ0 for functions ϕ, ϕ0 if ϕ0 is equal to 1 on supp ϕ.
Let us fix a smooth function η → [η] such that [η] > 0 for all η ∈ Rq and [η] = |η|
for |η| > c for some c > 0. For any ψ ∈ C∞(R+) we set ψη(r) := ψ(r[η]).
Let us now choose cut-off functions ω00≺ ω ≺ ω0 on the half-axis and other cut-off
functions , 0. By
Rµ(Ω × Rq, g) for g = (γ, γ − µ, Θ), Θ = (−(d + 1), 0], for any fixed d ∈ N, we denote the space of all operator functions
a(y, η) = r−µωηop γ−n/2
M (h)(y, η)ω 0
η+ (1 − ωη)Opr(p)(y, η)(1 − ωη00) 0
+ (m + g)(y, η) + ϕOpr(pint)(y, η)ϕ0.
Here p and h are as in Theorem 2.4. Moreover, we assume pint(r, y, ρ, η) ∈ C∞(R+×
Ω, Lµcl(X; R1+q
ρ,η )), ϕ, ϕ0∈ C0∞(R+), and g(y, η) ∈ RµG(Ω × Rq, g). Finally m(y, η) is
a smoothing Mellin symbol as follows m(y, η) = r−µωη d X j=0 X |α|≤j rjopγjα−n/2 M (fjα)(y)ηαωη0
where fjα(y, z) ∈ C∞(Ω, MR−∞jα(X)) for spaces MR−∞jα(X) of meromorphic operator
functions to be explained afterwards, and weights γjα satisfying the conditions
γ − j ≤ γjα≤ γ for all j, α. In addition we require fjα to be holomorphic close to
Γn+1 2 −γjα.
In order to describe more explicitly the spaces where we choose the functions fjα we first fix a sequence
R = {(rj, mj)}j∈Z
called a Mellin asymptotic type, such that for πCR := {rj}j∈Z the intersection
πCR ∩ {c ≤ Re z ≤ c0} is finite for every c ≤ c0. Then M−∞
R (X) is defined to
be the space of all f ∈ A(C \ πCR, L−∞(X)) such that f is meromorphic with
poles at rj∈ πCR of multiplicity mj+ 1 and Laurent coefficients at (z − rj)−(l+1),
0 ≤ l ≤ mj, in L−∞(X) being of finite rank, and χf |Γβ ∈ S(Γβ, L
−∞(X)) for every
β ∈ R, uniformly in compact β-intervals where χ is any πCR-excision function,
i.e., χ ∈ C∞(C), χ(z) = 0 for dist(z, πCR) < ε0, χ(z) = 1 for dist(z, πCR) > ε1
for some 0 < ε0< ε1.
Let RµM +G(Ω × Rq, g) be the space of all (m + g)(y, η) for g(y, η) ∈ Rµ G(Ω × R
q, g)
and m(y, η) as mentioned before. Remark 2.5. We have
Rµ(Ω × Rq, g) ⊂ Sµ(Ω × Rq; Ks,γ(X∧), Ks−µ,γ−µ(X∧)),
RµM +G(Ω × Rq, g) ⊂ Sµcl(Ω × Rq; Ks,γ(X∧), K∞,γ−µ(X∧)), s ∈ R. (2.19) In addition every m + g in RµM +G(Ω × Rq, g) defines an element in
Sclµ(Ω × Rq; Ks,γP (X∧), K∞,γ−µQ (X∧)) for any asymptotic type P and some resulting Q.
Similar constructions make sense when we replace η everywhere by (η, λ) ∈ Rq+l where λ ∈ Rl plays the role of a parameter and the corresponding spaces of operator families are denoted by Rµ
(Ω × Rq+l, g).
Definition 2.6. Let B be a manifold with edge Y, g = (γ, γ − µ, Θ), Θ = (−(d + 1), 0)].
(i) Lµ
(B, g; Rl) is defined to be the space of all operator families A(λ) ∈ Lµ cl(B \
Y ; Rl) that are, modulo L−∞
(B, g; Rl), of the form
for any cut-off functions σ, σ0 on B close to Y and arbitrary ϕ, ϕ0 ∈ C∞(Y )
supported in the same coordinate neighbourhood of Y , up to the pull back under a diffeomorphism as in (2.2).
(ii) By LµM +G(B, g; Rl) we denote the subset of operators where a ∈ RµM +G, called smoothing Mellin plus Green edge operators, with parameter λ ∈ Rl. (iii) By LµG(B, g; Rl) we denote the subset of all operators for a ∈ Rµ
G, called
Green edge operators with parameter λ ∈ Rl.
For l = 0 we simply write Lµ(B, g), etc.. Besides the local amplitude functions
of the edge operators we have homogeneous principal symbols, in this case, σ0, σ1,
coming from the strata s0(B) = B \ Y , s1(B) = Y .
First consider the case l = 0. From the inclusion Lµ(B, g) ⊂ Lµcl(B \ Y )
it follows that an A ∈ Lµ(B, g) has its homogeneous principal symbol σ0(A) of
order µ. Locally close to the edge in the variables (r, x, y) the edge-degenerate behaviour (2.16) allows us to write
σ0(A)(r, x, y, ρ, ξ, η) = r−µσ˜0(A)(r, x, y, rρ, ξ, rη)
for some ˜σ0(A)(r, x, y, ˜ρ, ξ, ˜η), homogeneous in ( ˜ρ, ξ, ˜η) 6= 0 of order µ, smooth up
to r = 0. The (twisted) homogeneous edge symbol σ1(A) is defined in terms of the
underlying local amplitude functions a(y, η) ∈ Rµ
(Ω × Rq, g), g = (γ, γ − µ, (−(d +
1), 0]), d ∈ N.
First according to (2.19), the summand (m + g)(y, η) in (2.18) has a homogeneous principal part, also indicated by σ1, namely,
σ1(m + g)(y, η) = σ1(m)(y, η) + σ1(g)(y, η)
with σ1(g)(y, η) ∈ S(µ)(Ω × (Rq\ {0}); Ks,γ(X∧), Ks−µ,γ−µ(X∧)) being the
princi-pal part of g(y, η) of order µ, cf. (2.14). The homogeneous principrinci-pal part of m(y, η) has the form
σ1(m)(y, η) = r−µω|η| d X j=0 X |α|=j rjopγjα−n/2 M (fjα)(y)ηαω0|η| where ω|η|(r) = ω(r|η|), etc.. Next we form h0(r, y, z, η) := ˜h(0, y, z, rη), p0(r, y, ρ, η) := ˜p(0, y, rρ, rη) and σ1 r−µωηop γ−n/2 M (h)(y, η)ω 0 η+ (1 − ωη)Opr(p)(y, η)(1 − ω00η) 0 := r−µ ω|η|op γ−n/2
M (h0)(y, η)ω0|η|+ (1 − ω|η|)Opr(p0)(y, η)(1 − ω00|η|)
which also belongs to S(µ) (Ω × (Rq\ {0}); Ks,γ(X∧), Ks−µ,γ−µ(X∧)) and define σ1(a)(y, η) = r−µ ω|η|op γ−n/2
M (h0)(y, η)ω0|η|+ (1 − ω|η|)Opr(p0)(y, η)(1 − ω00|η|)
+ σ1(m)(y, η) + σ1(g)(y, η).
Note that; there is no contribution from the last summand in (2.18). Summing up we have
σ1(a)(y, δη) = δµκδσ1(a)(y, η)κ−1δ
for all δ ∈ R+, (y, η) ∈ Ω × (Rq\ {0}).
Observe that
σ1(a)(y, η) = lim δ→∞δ
−µκ−1
δ a(y, δη)κδ for η 6= 0.
We defined altogether the principal symbolic structure
σ(A) = (σ0(A), σ1(A)) for A ∈ Lµ(B, g). (2.20)
We also employ lower order edge operators. To this end we write for the moment σi=: σiµ, i = 0, 1, and σ =: σµ, and set
Lµ−1(B, g) := {A ∈ Lµ(B, g) : σµ(A) = 0}, g = (γ, γ − µ, Θ), Θ = (−(d + 1), 0)]. (2.21) Analogously as σµwe can form σµ−1for operators in (2.21) and define Lµ−2(B, g) by the condition σµ−1(A) = 0. Inductively we then obtain the classes Lµ−N(B, g) for any N ∈ N.
Similar constructions make sense for parameter-dependent edge operators. Here it suffices to replace η everywhere by (η, λ), and 0 means (η, λ) = 0.
Remark 2.7. We have Lµ
(B, g; Rl) ⊂ C∞
(Rl, Lµ(B, g)) and
Dλα: Lµ(B, g; Rl) → Lµ−|α|(B, g; Rl)
for every α ∈ Nl.
Theorem 2.8. Let Aj∈ Lµ−j(B, g; Rl), j ∈ N, be an arbitrary sequence where the
asymptotic types in the involved Green operators are independent of j. Then there is an A ∈ Lµ
(B, g; Rl), unique modulo L−∞(B, g; Rl), such that A −PN
j=0Aj ∈
Lµ−(N +1)(B, g; Rl) for every N ∈ N, and we write A ∼P∞
j=0Aj.
Theorem 2.9. Let B be a compact manifold with edge Y and A ∈ Lµ
(B, g; Rl),
g = (γ, γ − µ, Θ). Then
A : C0∞(B \ Y ) → C∞(B \ Y ) for any fixed λ ∈ Rl extends to continuous operators
A : Hs,γ(B) → Hs−µ,γ−µ(B) (2.22)
and
A : HPs,γ(B) → HQs−µ,γ−µ(B) for every s ∈ R, any asymptotic type P and some resulting Q.
Observe that A ∈ Lµ−1
(B, g; Rl) induces continuous operators
A : Hs,γ(B) → Hs−µ+1,γ−µ+ε(B)
for some ε > 0. Therefore, because of the compact embeddings Hs,γ(B) ,→ H˜s,˜γ(B) for s > ˜s, γ > ˜γ, the operator (2.22) is compact for A ∈ Lµ−1
(B, g; Rl).
Let us finally recall that the spaces of edge operators form a calculus with compo-sitions and formal adjoints. Let us content ourselves with compocompo-sitions; concerning formal adjoints we refer to [19, Theorem 3.4.38].
Theorem 2.10. For A ∈ Lµ (B, g; Rl), g = (γ −ν, γ −(µ+ν), Θ), A0∈ Lν (B, h; Rl), h = (γ, γ − ν, Θ) we have AA0∈ Lµ+ν (B, g ◦ h; Rl)
for g ◦ h = (γ, γ − (µ + ν), Θ), and we have σi(AA0) = σi(A)σi(A0), i = 0, 1. If A
or A0are smoothing Mellin plus Green (Green) then so is the composition. Finally we may replace µ and ν by µ − l and ν − m, respectively, for some l, m ∈ N; then the resulting orders are µ + ν − (l + m).
3. Mellin quantisation for degenerate families of edge operators
According to (1.1), (1.2) the edge operator-valued symbols in the case k = 1 have the form p(t, y, τ, η) = ˜p(t, y, tτ, tη) (3.1) for ˜ p(t, y, ˜τ , ˜η) ∈ C∞(R+× Ω, Lµ(B, g; R 1+q ˜ τ ,˜η )), (3.2)
g = (γ, γ − µ, Θ), Ω ⊆ Rq open. Compared with (1.1), (1.2), for simplicity in this section we write y := y1, q := q1, η := η1. The following considerations explain
what we understand by Mellin quantisation for first order singularities.
Definition 3.1. The space MOµ(B, g; Rl) for g = (γ, γ − µ, Θ), Θ = (−(d + 1), 0],
d ∈ N, is defined to be the set of all
h(v, λ) ∈ A(C, Lµ(B, g; Rl)) (3.3)
such that h(v, λ)|Γβ ∈ L
µ(B, g; Γ
β × Rl) for every β ∈ R, uniformly in
com-pact β-intervals. In an analogous manner we define the subspaces MOµ−j(B, g; Rl), MO,M +Gµ−j (B, g; Rl) and MO,Gµ−j(B, g; Rl), based on Lµ−j, Lµ−jM +G and Lµ−jG , respec-tively, j ∈ N.
Moreover, we set
MO−∞(B, g; Rl) := \
j∈N
MOµ−j(B, g; Rl). Remark 3.2. Definition 3.1 refers to the representation of Lµ
(B, g; Rl) as a union
of Fr´echet spaces, cf. Section 2. Then MOµ(B, g; Rl) is a also a union of Fr´echet
spaces in a natural way and MO,M +Gµ and MO,Gµ are corresponding subspaces in stronger topologies.
Remark 3.3. From Definition 3.1 it follows that h(v, λ) ∈ MOµ(B, g; Rl) has the
parameter-dependent principal symbols
σi(h(v, λ)|Γβ), i = 0, 1 (3.4)
from Lµ(B, g; Γ
β× Rl), cf. (2.20). Then (3.4) is independent of the choice of β.
In fact, for any δ, β ∈ R we have an asymptotic expansion h(v, λ)|Γδ∼
∞
X
j=0
cj∂vjh(v, λ)|Γβ (3.5)
for constants cj = cj(β, δ) where c0 = 1. Remark 2.7 shows that ∂vjh(v, λ)|Γβ
belongs to Lµ−j(B, g; Rl), σi vanishes on Lµ−j for j > 0, i.e.,
σi(h(v, λ)|Γδ) = σi(h(v, λ)|Γβ), i = 0, 1.
A consequence of (3.5) is the following observation: From h(v, λ) ∈ MOµ(B, g; Rl), h(v, λ)|Γβ ∈ L
µ−1(B, g; Γ β× Rl)
for some fixed β ∈ R it follows that h(v, λ) ∈ MOµ−1(B, g; Rl). By iterating this
conclusion we obtain
h(v, λ) ∈ MOµ(B, g; Rl), h(v, λ)|Γβ ∈ L
−∞(B, g; Γ β× Rl)
⇒ h(v, λ) ∈ MO−∞(B, g; Rl). (3.6)
We will employ the latter relation below as follows. Assume h(t, v, ˜η), h1(t, v, ˜η) ∈
C∞(R+, MOµ(B, g; Rqη˜)) and let op 1 2 M(h − h1)(˜η) ∈ L−∞(R+× s0(B); Rqη˜). (3.7) Then we have h(t, v, ˜η) = h1(t, v, ˜η) mod C∞(R+, MO−∞(B, g; Rqη˜)). (3.8)
In fact, (3.7) implies that (h − h1)|Γ0(t, iτ, ˜η) ∈ C
∞
(R+, L−∞(B, g; Γ0× Rqη˜))
and then we can apply (3.6) in the variant of t-depending operator-valued Mellin symbols. By a translation in the complex plane we obtain similar relations as (3.7) ⇒ (3.8) for opγ(·) for any weight γ.
Remark 3.4. Theorem 2.10 has the consequence that also holomorphic families of Definition 3.1 can be composed in the sense
MOµ(B, g; Rl) × MOν(B, h; Rl) → MOµ+ν(B, g ◦ h; Rl)
where the result is Mellin plus Green or Green as soon as one of the factors has this property. Also the observation for lower order operators remains true in analogous form.
Theorem 3.5. For every p(t, y, τ, η) given by (3.1), (3.2), there exists an ˜ h(t, y, v, ˜η) ∈ C∞(R+× Ω, M µ O(B, g; R q ˜ η))
such that for h(t, y, v, η) := ˜h(t, y, v, tη) we have Opt(p)(y, η) = opγM
t(h)(y, η) mod C
∞(Ω, L−∞
(R+× s0(B); Rqη)) (3.9)
for every γ ∈ R. For any other ˜h1(t, y, v, ˜η) ∈ C∞(R+× Ω, MOµ(B, g; Rqη˜)) such
that h1(t, y, v, η) = ˜h1(t, y, v, tη) satisfies an analogue of the relation (3.9) we have
˜
h(t, y, v, ˜η) = ˜h1(t, y, v, ˜η) mod C∞(R+× Ω, MO−∞(B, g; Rqη˜)). (3.10)
The proof may be obtained by applying some oscillatory integral arguments analogously as in [8, Theorem 4.9] for the case of a smooth compact manifold X. We give in this article an alternative proof, based on the technique of [19, Theorem 3.2.7], see also [5, Lemma 2.2]. In this way we see more details on substructures of smoothing Mellin and Green type that we separately formulate as a consequence. In the higher corner case the iterative approach of [19] seems to be necessary anyway. For the proof of Theorem 3.5 we prepare the following result.
Proposition 3.6. Let p(t, y, τ, ˜η) := ˜p(t, y, tτ, ˜η) for ˜p(t, y, ˜τ , ˜η) as in (3.2). Set f0(t, y, iτ , ˜η) := ˜p(t, y, −τ , ˜η)
which belongs to C∞(R+× Ω, Lµ(B, g; Γ0× Rqη˜)). Then we have
Opt(p)(y, ˜η) = op1/2M
t(f0)(y, ˜η) + Opt(p1)(y, ˜η) mod L
−∞
(R+× Ω × s0(B); Rqη˜)
(3.11) for p1(t, y, τ, ˜η) = ˜p1(t, y, tτ, ˜η), ˜p1(t, y, ˜τ , ˜η) ∈ C∞(R+× Ω, Lµ−1(B, g; R1+qτ ,˜˜η )).
Proof. The method will not depend on y ∈ Ω in an essential way. Therefore, from now on we omit y. First recall that
op1/2M t(f0)(˜η)u(t) = Z ∞ −∞ Z ∞ 0 t t0 −iτ f0(t, iτ , ˜η)u(t0) dt0 t0 d¯τ .
We employ the fact that the push forward of op1/2M (f0) under χ−1: R+,t→ Rt for
χ(t) := e−t = t has the form (χ−1)∗op
1/2
Mt(f0)(˜η) = Opt(a)(˜η)
for an amplitude function
a(t, τ , ˜η) := f0(e−t, iτ , ˜η), (3.12)
with Opt referring to the Fourier transform, i.e., Opt(a)(˜η)v(t) =
Z Z
ei(t−t0)τf0(e−t, iτ , ˜η)v(t0)dt0d¯τ .
The standard rules of the pseudo-differential calculus with respect to push forward under a diffeomorphism apply in our situation in analogous form, i.e., we can write
χ∗Opt(a)(˜η) = Opt(c)(˜η) mod L−∞(R+× s0(B); R q ˜ η)
for an amplitude function c(t, τ, ˜η) ∈ C∞(R+, Lµ(B, g; R1+qτ, ˜η )) with an asymptotic expansion c(t, τ, ˜η)|t=χ(t)∼ ∞ X j=0 1 j!(∂ j τa)(t, dχ(t)τ, ˜η)Φj(t, τ ) (3.13) for dχ(t) = −e−t, Φj(t, τ ) := D j t0eiδ(t,t 0)τ t0=t, where δ(t, t 0) = χ(t0) − χ(t) −
dχ(t)(t0− t), and Φ0= 1. In the present case we have Φj(t, τ )
t=− log t=: Ψ(t, tτ )
where Ψ(t, ˜τ ) is a polynomial in ˜τ of degree ≤ j/2 with coefficients in C∞(R+).
The asymptotic sum within the edge calculus is guaranteed because of Remark 2.7 and Theorem 2.8. We now obtain
c(t, τ, ˜η) = ˜p(t, tτ, ˜η) + ˜p1(t, tτ, ˜η) mod C∞(R+, L−∞(B, g; R 1+q
τ, ˜η )) (3.14)
where ˜p1(t, ˜τ , ˜η) ∈ C∞(R+, Lµ−1(B, g; R1+qτ ,˜˜η)). In fact, the first summand on the
right of (3.14) just corresponds to the term for j = 0 in the asymptotic expansion (3.13), namely, according to (3.12)
a(t, dχ(t)τ, ˜η) t=χ(t)= f0(t, −itτ, ˜η) = ˜p(t, tτ, ˜η).
The second summand in (3.14) is equal to the asymptotic sum
∞ X j=1 1 j!(∂ j τf0)(t, −itτ, ˜η)Ψj(t, tτ )
which is carried out as X∞ j=1 1 j!(∂ j τf0)(t, −i˜τ , ˜η)Ψj(t, ˜τ ) ˜τ =tτ. The summands in parentheses belong to
C∞(R+,t, Lµ−j(B, g; R1+q˜τ ,˜η))
and the asymptotic sum exists in C∞(R+,t, Lµ−1(B, g; R1+q˜τ ,˜η )), cf. Theorem 2.8.
Inserting then ˜τ = tτ gives us just ˜p1(t, tτ, ˜η) as asserted (taking into account that
the assumptions are satisfied in the present summation).
Similarly as the kernel cut-off constructions in Section 2 we now formulate kernel cut-off operators on the level of Lµ(B, g; Γ
0× Rl)-valued amplitude
func-tions. For any f (t, y, v, λ) ∈ C∞(R+× Ω, Lµ(B, g; Γ0× Rl)) we form
k(f )(t, y, θ, λ) := Z
Γ0
θ−vf (t, y, v, λ)d¯v = (M1/2,v→θ−1 f )(t, y, θ, λ).
This represents a C∞(R+× Ω, Lµ(B, g; Rl))-valued distribution on Rθ,+. Then for
any ψ(θ) ∈ C0∞(R+), ψ(θ) = 1 close to θ = 1, we set
Vψ(f )(t, y, v, λ) =
Z
R+
It is useful to admit in (3.15) more general functions ϕ ∈ C0∞(R+) than a cut-off
function ψ and then even ϕ ∈ CB∞(R+). The latter space is defined as the set of
all ϕ(θ) ∈ C∞(R+) such that
sup
θ∈R+
|(θ∂θ)jϕ(θ)| < ∞
for all j ∈ N. Moreover, let Cb∞(R) defined to be the set of all u(θ) ∈ C∞(R) such
that
sup
θ∈R
|∂θju(θ)| < ∞ for all j ∈ N. For ϕ ∈ C0∞(R+) we have
Vϕ(f )(t, y, iτ, λ) = ZZ θi(τ −τ0)ϕ(θ)f (t, y, iτ0, λ)dθ θ d¯τ 0 = ZZ θi˜τϕ(θ)f (t, y, i(τ − ˜τ ), λ)dθ θ d¯˜τ , (3.16)
interpreted in the oscillatory integral sense regularised with respect to the co-variable ˜τ . In order to extend Vϕ to ϕ ∈ CB∞(R+) we interpret (3.16) in terms
of Kumano-go’s reqularisation in (θ, ˜τ ), here in the variant of the Mellin trans-form rather than the Fourier transtrans-form and for operator-valued symbols. Con-cerning the oscillatory integral technique for scalar symbols, see [9]. Clearly, in order to make it work, we refer to the internal symbolic structure of elements of C∞(R+× Ω, Lµ(B, g; Γ0× Rl)). This gives us the following result.
Proposition 3.7. The operator (ϕ, f ) → Vϕ(f ) defines a separately bilinear
contin-uous map
CB∞(R+)×C∞(R+×Ω, Lµ(B, g; Γ0×Rl)) → C∞(R+×Ω, Lµ(B, g; Γ0×Rl)) (3.17)
with the following properties.
(i) For any cut-off function ψ(θ) ∈ C0∞(R+), ψ(θ) = 1 close to θ = 1 we have
Vψ(f ) = f mod C∞(R+× Ω, L−∞(B, g; Γ0× Rl)).
(ii) For any ϕ(θ) ∈ CB∞(R+) we have
Vϕ(f )(t, y, iτ, λ) ∼ ∞ X j=0 1 j!((θ∂θ) jϕ)(1)(∂j τf )(t, y, iτ, λ) (3.18)
in the sense of Theorem 2.8.
The separate bilinear continuity of (3.17) refers to the Fr´echet topology of CB∞(R+) and the representation of C∞(R+× Ω, Lµ(B, g; Γ0× Rl)) as a union of
Fr´echet spaces, given in Section 2. For any fixed f we remain in such a Fr´echet subspace after applying the kernel cut-off operator (3.17). In those subspaces we may talk about bilinear rather than separate bilinear continuity. Moreover, the terms in the asymptotic summation (3.18) satisfy the assumption of Theorem 2.8. It is also interesting to consider subspaces of MOµ(B, g; Rl) where the functions (3.3) take values in Lµ−j(· · · ) or Lµ−jM +G(· · · ), Lµ−jG (· · · ), j ∈ N, cf. Definition 3.1.
Under the kernel cut-off operation those give rise to subspaces MOµ−j(B, g; Rl),
MO,M +Gµ−j (B, g; Rl), and Mµ−j
O,G(B, g; Rl), respectively.
Theorem 3.8. The operator Vϕ for ϕ ∈ C0∞(R+) induces continuous operators
Vϕ: Lµ−j(B, g; Γ0× Rl) → M µ−j
O (B, g; R l)
for all j ∈ N and Vϕ: L µ M +G(B, g; Γ0× Rl) → M µ O,M +G(B, g; R l), Vϕ: L µ G(B, g; Γ0× Rl) → M µ O,G(B, g; R l).
In an analogous manner we have
Vϕ: C∞(R+× Ω, Lµ−j(B, g; Γ0× Rl)) → C∞(R+× Ω, MOµ−j(B, g; Rl))
for all j, etc..
Proof of Theorem 3.5. We employ notation of Proposition 3.6 and iterate the re-sult. For brevity we omit the dependence on y ∈ Ω which is not essential for the ar-guments. From p1(t, τ, ˜η) we obtain an f1(t, iτ, ˜η) ∈ C∞(R+, Lµ−1(B, g; Γ0× Rlη˜))
and p2(t, τ, ˜η) = ˜p2(t, tτ, ˜η) for some ˜p2(t, ˜τ , ˜η) ∈ C∞(R+, Lµ−2(B, g; R l+q ˜
τ ,˜η)) such
that analogously as (3.11) we have Opt(p1)(˜η) = op 1/2 Mt(f1)(˜η) + Opt(p2)(˜η) mod L −∞ (R+× s0(B); Rqη˜). (3.19) Thus (3.11), (3.19) yield Opt(p)(˜η) = op1/2M t(f0+ f1)(˜η) + Opt(p2)(˜η).
By iterating the process we obtain Mellin symbols
fj(t, iτ, ˜η) ∈ C∞(R+, Lµ−j(B, g; Γ0× Rqη˜)),
j = 0, . . . , N , and
pN +1(t, τ, ˜η) = ˜pN +1(t, tτ, ˜η)
for some ˜pN +1(t, ˜τ , ˜η) ∈ C∞(R+, Lµ−(N +1)(B, g; R1+qτ ,˜˜η )) such that
Opt(p)(˜η) = op1/2M t XN j=0 fj (˜η) + Opt(pN +1)(˜η) mod L−∞(R+× s0(B); Rqη˜).
By virtue of Theorem 2.8 we can form the asymptotic sum f (t, iτ, ˜η) ∼
∞
X
j=0
fj(t, iτ, ˜η)
in the space C∞(R+, Lµ(B, g; Rq˜η)), and then
Opt(p)(˜η) = op1/2M t(f )(˜η) mod L −∞ (R+× s0(B); R q ˜ η).
Next we apply the kernel cut-off operator Vψ: C∞(R+, Lµ(B, g; R q ˜ η)) → C ∞ (R+, M µ O(B, g; R q ˜ η))
for a cut-off function ψ as in Proposition 3.7 (i) and set h(t, v, ˜η) := Vψf(t, v, ˜η). Using h(t, v, ˜η)|Γ0 = f (t, iτ, ˜η) mod C ∞ (R+, L−∞(B, g; Rqη˜)) we obtain Opt(p)(˜η) = op1/2M (h)(˜η) mod L−∞(R+× s0(B); R q ˜ η). (3.20)
In order to verify (3.9) for arbitrary γ it suffices to apply Cauchy’s theorem, using that h is holomorphic in v with the required behaviour as |Im v| → ∞. In the final step of the proof we pass to dependence on tη rather than ˜η. The replacement ˜
η tη is admitted, since the variable t acts as a multiplication by t from the left. Formally we could replace t by any fixed constant c > 0. Our result then holds for every c, and then we may insert c = t. The uniqueness of (3.10) modulo a smoothing remainder can be obtained as follows. If ˜h1 is another Mellin symbol
satisfying an analogue of relation (3.9) then we obtain opγM(h − h1)(˜η) ∈ L−∞(R+× s0(B); Rqη˜).
As noted before this entails (3.8). In other words we proved the claimed uniqueness. Note that the arguments at the end of the latter proof give rise to the following isomorphism, formulated in the y-dependent case,
C∞(R+× Ω, Lµ(B, g; R 1+q ˜ τ ,˜η))/C ∞ (R+× Ω, L−∞(B, g; R 1+q ˜ τ ,˜η )) ∼ = → C∞(R+× Ω, MOµ(B, g; Rqη˜))/C ∞ (R+× Ω, MO−∞(B, g; Rqη˜)). (3.21) The isomorphism (3.21) is induced by the correspondence
˜
p(t, y, ˜τ , ˜η) → p(t, y, tτ, ˜η) → h(t, y, v, ˜η)
in the sense of (3.1), (3.2) and (3.20). If ˜p in (3.1) is of order −∞ the corre-sponding Mellin symbol is smoothing. The uniqueness of h modulo a smoothing Mellin symbol together with the latter observation gives us a map (3.21) between the respective quotient spaces. It remains the surjectivity of (3.21), but this is straightforward insofar the iterative process established in Proposition 3.6 can be inverted, i.e., from f0we find ˜p such that (3.11) holds, etc.. In other words there is
an inverse Mellin quantisation which recovers from a Mellin symbol the degenerate operator function with the action referring to the Fourier transform.
4. Mellin quantisation of higher order
We now study iterated Mellin quantisations for corner-degenerate edge operator-valued families (1.1), (1.2) in the case k ≥ 2. We illustrate the method for k = 2 and then formulate the general case which may be treated in an analogous manner.
Recall that symbols of the kind (1.1) for k = 2 have the form
for t = (t1, t2) ∈ (R+)2, y = (y1, y2) ∈ Ω1× Ω2, Ωj⊆ Rqj open, ˜ p(t, y, ˜τ1, ˜τ˜2, ˜η1, ˜η˜2) ∈ C∞((R+)2× Ω1× Ω2, Lµ(B, g; R2+qτ˜ 1+q2 1,˜τ˜2, ˜η1, ˜η˜2)). (4.2) Definition 4.1. Let MOµ v1Ov2(B, g; R q η), g = (γ, γ − µ, Θ), Θ = (−(d + 1), 0], d ∈ N,
be defined as the space of all h(v1, v2, η) ∈ A(Cv2, M
µ Ov1(B, g; R q η)) such that h(v1, v2, η)|Γδ ∈ M µ Ov1(B, g; Γδ× Rqη)
for every δ ∈ R, uniformly in compact δ-intervals. In an analogous manner we define the subspaces MOµ−j
v1Ov2, M µ−j Ov1Ov2,M +G and M µ−j Ov1Ov2,G based on M µ−j Ov1, MOµ−j v1,M +Gand M µ−j Ov1,G, j ∈ N, respectively. We set MO−∞ v1Ov2(B, g; R q η) := \ j∈N MOµ−j v1Ov2(B, g; R q η).
Remark 4.2. (i) Similarly as Remark 3.2, the space MOµ
v1Ov2(B, g; R
q
η) is a union
of Fr´echet spaces.
(ii) We have a natural isomorphism MOµ v1Ov2(B, g; R q η) ∼= M µ Ov2Ov1(B, g; R q η),
induced by interchanging the role of v1 and v2. However, below we will give
up the symmetry since the Mellin quantisation gives rise to a degenerate behaviour in v2.
In the following we set
Ω := Ω1× Ω2 and q := q1+ q2.
Theorem 4.3. For every p of the form (4.1), (4.2) there exists an ˜ h(t, y, v1, ˜v2, ˜η1, ˜η˜2) ∈ C∞((R+)2× Ω, M µ Ov1Ov2˜ (B, g; R q ˜ η1, ˜η˜2))
such that for h(t, y, v1, v2, η1, η2) := ˜h(t, y, v1, t1v2, t1η1, t1t2η2) we have
Opt 2Opt1(p)(y, η) = op β2 Mt2op β1 Mt1(h)(y, η) mod C ∞(Ω, L−∞ ((R+)2× s0(B); Rqη))
for any reals β1, β2. Similarly as in Theorem 3.5 the function ˜h is unique modulo
MO−∞
v1Ov2˜ -valued symbols.
Proof. Let us first observe that Theorem 3.5 and Proposition 3.6 may be obtained in a parameter-dependent version. When we modify the notation and now denote the former t, y, τ, η by t1, y1, τ1, η1 with y1 varying in an open set Ω1⊆ Rq1 then
we may introduce more variables, namely, t2, y2, τ2, η2 for t2∈ R+and y2varying
in an open set Ω2 ⊆ Rq2. The modified functions are smooth on R+× Ω2. The
covariables τ2, η2 in the above tilde notation are also multiplied by t1, and the
relationship between p and ˜p now is
for some ˜p(t, y, ˜τ1, ˜τ2, ˜η1, ˜η2) ∈ C∞((R+)2× Ω, Lµ(B, g; R2+qτ˜
1,˜τ2, ˜η1, ˜η2)). Apart from
the extra parameters we can proceed as in Theorem 3.5 and Proposition 3.6, and define a function ˜ h(t, y, v1, ˜τ2, ˜η1, ˜η2) ∈ C∞((R+)2× Ω, MOµ v1(B, g; R 1+q ˜ τ2, ˜η1, ˜η2)),
such that h(t, y, v1, ˜τ2, ˜η1, ˜η2) satisfies the relation
Opt1(p)(t2, y, ˜τ2, ˜η 1, ˜η2) = opγ Mt1(h)(t2, y, ˜τ2, ˜η1, ˜η2) modulo C∞(R+ × Ω, L−∞(R+ × s0(B); R 1+q ˜
τ2, ˜η1, ˜η2)). Next we apply the method
of Theorem 3.5 and Proposition 3.6, now with ˜τ2 and ˜η2 everywhere replaced
by c˜τ˜2 and c˜η˜2 where ˜τ˜2 and ˜η˜2 stand for t2τ2 and t2η2, respectively. Here c
is a positive constant, and we obtain a ˜p˜c(t, y, v1, c˜τ˜2, ˜η1, c˜η˜2) ∈ C∞((R+)2 ×
Ω, MOµ v1(B, g; R 1+q c˜˜τ2, ˜η1,c ˜η˜2)) and an ˜ ˜ hc(t, y, v1, cv2, ˜η1, c˜η˜2) ∈ C∞((R+)2× Ω, MOµ v1Ocv2(B, g; R q ˜ η1,c ˜η˜2))
such that for
pc(t, y, v1, cτ2, η1, cη2) := ˜p˜c(t, y, v1, ct2τ2, t1η1, ct2η2), hc(t, y, v1, cv2, η1, cη2) :=˜˜hc(t, y, v1, cv2, t1η1, ct2η2) we have Opt 2Opt1(pc)(y, η 1, cη2) = opβ2 Mt2op β1 Mt1(hc)(y, η1, cη2)
modulo C∞(Ω, L−∞((R+)2× s0(B); Rq)). Clearly the constant c substitutes the
variable t1, first fixed, which is involved as an action from the left. Now it is
admitted to leave it variable and to insert t1. Then, in modified notation for p and
h we just obtain the claimed result.
In an iterative manner we can define spaces of the kind MOµ
v(B, g; R
q η)
for v := (v1, . . . , vk), η := (η1, . . . , ηk), as the set of all
h(v0, vk, η) ∈ A(Cvk, M
µ
Ov0(B, g; R q η))
for v0= (v1, . . . , vk−1) such that
h(v0, vk, η)|Γδ∈ M
µ
Ov0(B, g; Γδ× Rqη)
for every δ ∈ R, uniformly in compact δ-intervals. Applying the above-mentioned scheme of organising smoothing Mellin plus Green versions of our spaces we can also introduce MOµ−j
v , M
µ−j
Ov,M +Gand M
µ−j
Ov,G, j ∈ N, respectively. Then we set
MO−∞ v (B, g; R q η) := \ j∈N MOµ−j v (B, g; R q η).
Theorem 4.4. For every p(t, y, τ, η) of the form ˜ p(t, y, t1τ1, t1t2τ2, . . . , t1t2· · · tkτk, t1η1, t1t2η2, . . . , t1t2· · · tkηk) for ˜p(t, y, ˜τ1, ˜τ2, . . . , ˜τk, ˜η1, ˜η2, . . . , ˜ηk) ∈ C∞((R+)k × Ω, Lµ(B, g; R k+q ˜ τ ,˜η )), Ω ⊆ R q open, q =Pk j=1qj, there exists an ˜h(t, y, ˜v, ˜η) ∈ C∞((R+)k× Ω, MOµv˜(B, g; Rqη)), ˜
v = (˜v1, . . . , ˜vk), such that for
h(t, y, v1, . . . ,vk, η1, . . . , ηk) = ˜h(t, y, v1, t1v2, . . . , t1· · · tk−1vk, t1η1, t1t2η2, . . . , t1· · · tkηk) (4.3) we have Optk· · · Opt1(p)(y, η) = opβk Mtk· · · op β1 Mt1(h)(y, η) (4.4)
modulo C∞(Ω, L−∞((R+)k× s0(B); Rqη)) for any reals β1, . . . , βk. The function ˜h
is unique modulo MO−∞-valued ones, more precisely, for any other h1 related to
˜
h1 via an expression (4.3), ˜h1 ∈ C∞((R+)k × Ω, MOµv˜(B, g; Rqη)), satisfying an
analogue of the relation (4.4) we have ˜
h(t, y, ˜v, ˜η) = ˜h1(t, y, ˜v, ˜η) mod C∞((R+)k× Ω, MO−∞(B, g; R q ˜ η)).
Acknowledgment. The final version of this paper has been finished when the authors visited the National Center for Theoretical Sciences, Hsinchu, Tai-wan during January, 2013. They would like to express their profound gratitude to the Director of NCTS, Professor Winnie Li for her invitation and for the warm hospitality extended to them during their stay in Taiwan. The first author is par-tially supported by an NSF grant DMS-1203845 and Hong Kong RGC competitive earmarked research grant #601410.
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Der-Chen Chang
Department of Mathematics and Department of Computer Science, Georgetown Uni-versity, Washington D.C. 20057, USA. Department of Mathematics, Fu Jen Catholic University, Taipei 242, Taiwan, ROC
e-mail: [email protected] N. Habal
Institute of Mathematics, University of Potsdam, 14469 Potsdam, Germany e-mail: [email protected]
B.-W. Schulze
Institute of Mathematics, University of Potsdam, 14469 Potsdam, Germany e-mail: [email protected]
