Am= operators. Also Dirichlet and Neumann conditions independently of Y∓ give rise to mixed problems, namely,
We now reduce the operator Am to the boundary by means of the operator
Ad. In order to proceed in an analogous manner as in (1.1) we have to employ a relationship between A0 and Ad. Consider
which is an isomorphism, since A0 itself is an isomorphism. Then, according to Lemma 1.2 we have an isomorphism
D0 :=
for s > 1, s − 1 /∈ N. For AD:= D0A˜0 it follows that operator D0A˜0 contains Ad and the extra potential operators d±. Let us write
PD:= A−1D =:P0 C− C+
cf. (3.8), which follows from
A−1D = ˜A−10 D0−1=P0 K0 0 cf. the formula (1.13). In particular, we have continuous operators
C∓= K0e∓d : Hs−1/2(int Y∓) → Hs(int G).
Let us also rephrase Am, cf. (1.19), and form the operator
AM:= formulation, with Z being the edge. It follows that
AMPD= cf. the formula (1.26). Here
RND:= N+C+ : Hs−1/2(int Y+) → Hs−3/2(int Y+) (1.30) corresponds to the Dirichlet-to-Neumann operator on the Neumann-side Y+. Since C+
depends on s also the operator N+C+ depends on s for s > 1, s − 1 /∈ N.
1 THE DIRICHLET-TO-NEUMANN OPERATOR 13 1.4 Elliptic edge conditions
In elliptic boundary or edge problems for a (pseudo-)differential operator A the principal boundary or edge symbol of A induces a family of Fredholm operators
a : H → ˜H (1.31)
between Hilbert spaces with group action. By virtue of twisted homogeneity it suffices to consider (1.31) as an operator function parametrised by the points of the unit cosphere bundle induced by the cotangent bundle of the boundary or the edge. We give here some further comment on the general role of (1.31) for ellipticity of boundary or edge conditions.
For convenience we mainly consider the case of boundary conditions. Analogous structures can be observed in the edge case. We illustrate here some phenomena belonging to the key words Green, trace and potential symbols, and the associated operators.
Assume that the boundary Y is smooth and compact. Then also the unit cosphere bundle S is compact. Let π : S → Y be the canonical projection. Recall that Vect(·), the set of all smooth complex vector bundles on the manifold in parenthesis, generates the K-group K(·) as the additive group of equivalence classes of pairs (E, F ) for E, F ∈ Vect(·). Here (E, F ) ∼ ( ˜E, ˜F ) ⇔ E ⊕ ˜F ⊕ G ∼= ˜E ⊕ F ⊕ G for some G ∈ Vect(·). Equivalently we can ask the condition E ⊕ L ∼= ˜E ⊕ M, F ⊕ L ∼= ˜F ⊕ M for some L, M ∈ Vect(·). The corresponding equivalence class is denoted by [E] − [F ]. The bundle pull back induces a homomorphism
π∗ : K(Y) → K(S). (1.32)
Lemma 1.3. Let (1.31) be a family of Fredholm operators parametrised by S, regarded as a bundle morphism S × H → S × ˜H between the respective trivial Hilbert bundles. Then there are elements G1, G2 ∈ Vect(S) and an isomorphism
a k t q
:
S × H
⊕ G1
−→
S ×H˜
⊕ G2
(1.33)
for suitable bundle morphisms k : G1 → S × ˜H, t : S × H → G2, and q : G1→ G2.
This result is well-known in K-theory; an explicit proof can be found in [16, Subsection 3.3.4] which is also a reference for other material in the present subsection. Clearly there are many choices of G1, G2 and k, t, q such that(1.33) is an isomorphism. However,
indS(a) := [G2] − [G1] ∈ K(S) (1.34) only depends on a. A typical and most elementary example for an operator function a is the Fredholm family a(η) := σ∂(∆)(η) considered in (2.19) for H := Hs(R+), ˜H := Hs−µ(R+) where the covariable η belongs to the cotangent bundle of the boundary (here in local representation). By virtue of twisted homogeneity σ∂(∆)(δη) = δ2κδσ∂(∆)(η)κ−1δ , δ ∈ R+, the operator family a(η) is completely determined by its values on the unit sphere |η| = 1.
In this particular case the dependence on η even disappears. In any case, as observed for (2.20), we find a block matrix isomorphism of the kind (1.33), here a column matrix, since a(η) is surjective. This example is important also for another reason, namely, in contrast to (1.34) we have
indS(a) ∈ π∗K(Y). (1.35)
For general elliptic operators A, say, pseudo-differential operators with the transmission property at the boundary, the boundary symbol a(y, η) := σ∂(A)(y, η) is not necessarily surjective, and both kernel and cokernel can be non-trivial. If (1.35) holds we also say that the Atiyah-Bott obstruction vanishes, cf. [3] for the case of differential operators, [4] for pseudo-differential operators with the transmission property, [29] for elliptic edge operators. The information is as follows.
Theorem 1.4. An elliptic operator A in one of the above-mentioned pseudo-differential algebras admits a 2 × 2 block matrix Fredholm operator A = {Aij}i,j=1,2 with A (modulo a Green or a smoothing Mellin plus Green operator) as the upper left corner A11, if and only if (1.35) holds.
Let us make the construction more explicit in the case of an elliptic pseudo-differential operator R ∈ Lµ(cl)(int Y+) that is quantised as an element R ∈ Lµ(Y+, g), g = (γ, γ − µ) for some weight γ ∈ R in the edge calculus on Y+, regarded as a manifold with edge Z. In such a quantisation we have also R ∈ Lµ(cl)(int Y+) and
R = R mod L−∞(int Y+).
Moreover, R is σψ-elliptic in the sense of Definition 3.40. This has the consequence that σ∧(R)(z, ζ) : Ks,γ(R+) → Ks−µ,γ−µ(R+), (1.36) ζ 6= 0, is Fredholm for any s ∈ R if and only if the subordinate conormal symbol
σcσ∧(R)(z, w) : C → C
is bijective for all z ∈ Z, w ∈ Γ1/2−γ. It is well-known in this context that this is the case for all w ∈ C \ D(z) for a discrete set D(z) in the complex plane. In other words, (1.36) is Fredholm if and only if Γ1/2−γ ∩ D(z) = ∅ for all z ∈ Z. Now applying Lemma 1.3 to a := σ∧(R)(z, ζ), |ζ| = 1, we obtain a block matrix isomorphism
σ∧(R) k
t q
(z, ζ) :
Ks,γ(R+)
⊕ G1,(z,ζ)
−→
Ks−µ,γ−µ(R+)
⊕ G2,(z,ζ)
(1.37)
for (z, ζ) in the unit cosphere bundle S∗Z. Under the condition (1.35) which is fulfilled here in the case of the Dirichlet-to-Neumann operator (where µ = 1) we find k, t, q as edge symbols in the edge calculus, i.e., G1, G2 are liftings of bundles J1, J2 over Z with respect to the canonical projection π : S∗Z → Z. So far we looked at the symbols for
|ζ| = 1, but extensions by twisted homogeneities of any desired orders to T∗Z \ 0 gives us the homogeneous symbols themselves.
1.5 The edge-quantised Dirichlet-to-Neumann operator
Theorem 1.5. (i) We have
N+C+ = R|int Y+ modulo L−∞(int Y+), (1.38) i.e., N+C+∈ L1cl(int Y+), and its homogeneous principal symbol is equal to c|η|.
1 THE DIRICHLET-TO-NEUMANN OPERATOR 15 (ii) The operator
R11:= Ps−3/2+ N+C+Es−1/2+ : Hs−1/2,s−1/2(Y+) → Hs−3/2,s−3/2(Y+)
cf. (1.16), (1.17), belongs to the edge calculus L1(Y+, g) for the weight data g = (s, s − 1) over Y+ as a manifold with edge Z, cf. the formula (3.126). We have
R11= r+Re+ modulo M + G + G
with r+Re+ being the truncated operator belonging to the Dirichlet-to-Neumann op-erator (1.2) on the plus side of the boundary, moreover, M + G ∈ L1M +G(Y+, g), cf.
(3.127), and G is an irregular Green operator on Y+, i.e., an Green operator refer-ring to other (possibly more singular) weight data than g, cf. the weight shifts in [32, Remark 2.3.67].
Proof. (i) Because of the second relation of (1.5) the restriction of the Dirichlet-to-Neumann operator R to int Y+ is equal to N+K0|int Y+. This allows us to write
R|int Y+ = N+K0e+d + N+K0G,
Here G has the form G = EG+ for a Green operator G+ in Boutet de Monvel’s calculus, G+ : Hs−1/2(int Y+) → Hs−1/2(int Y+), composed with a specific extension operator E : Hs−1/2(int Y+) → Hs−1/2(Y ). Both G+ and E are contained in the construction of e+d, cf. the observations below in Subsection 3.1. This entails N+K0G ∈ L−∞(int Y+) which is just the relation (1.38). Since c|η| is the homogeneous principal symbol of R|int Y+, cf.
(1.2), we obtain the same for N+C+.
(ii) From [36, Theorem 3.3.3] we know that the truncated operator r+Re+ belongs to the edge pseudo-differential calculus, modulo a smoothing Mellin plus Green operator and an extra finite-dimensional Green operator referring to “irregular” weights relative to the prescribed ones, contained in g, cf. the lower right 2×2 block matrix structure in Definition 3.29 below. Using (1.15) and (1.17) we now transform (1.30) to the 2 × 2 block matrix operator RND:= Λ+NN+C+(Λ+D)−1= Ps−3/2+ N+C+Es−1/2+ Ps−3/2+ N+C+Ks−1/2+
Ts−3/2+ N+C+Es−1/2+ Ts−3/2+ N+C+Ks−1/2+
! ,
RND :
Hs−1/2,s−1/2(Y+)
⊕
Hs−1/2(Z, CL(s−1/2))
→
Hs−3/2,s−3/2(Y+)
⊕
Hs−3/2(Z, CL(s−3/2))
. (1.39)
In all these constructions we assume s to be so small that L(s − 1/2) and L(s − 3/2) become minimal.
Remark 1.6. As noted before the Dirichlet-to-Neumann operator R has not the trans-mission property with respect to any interface on Y . Therefore, we cannot expect that the truncated operator r+Re+ induces continuous maps Hs(int Y+) → Hs−1(int Y+) (say, for s > −1/2). Theorem 1.5 (i) shows that there are s-dependent quantisations N+C+ such that (1.30) are continuous under natural conditions on s. This choice of quantisation is crucial for the rest of this paper.
Set
Γβ := {w ∈ C : Re w = β}
for any β ∈ R. We will interpret RND as an operator in the edge calculus, cf. the notation (3.126) below. In this framework, as explained in connection with (1.14), we interpret the second upper subscript in the spaces over Y+ as a weight and often write γ instead of s. Then, as soon as we realise RND as a continuous operator in such weighted spaces we write RND(γ). In particular, RND in (1.39) then has the meaning of RND(s). Moreover, N+C+= (N+C+)(s) will be replaced by (N+C+)(γ) when s is used in the meaning of γ.
Theorem 1.7. (i) The Dirichlet-to-Neumann operator R = T1K0 reduces the Neu-mann boundary condition on the plus-side Y+ of the boundary to a continuous oper-ator
RND(γ) :
Hs−1/2,γ−1/2(Y+)
⊕
Hs−1/2(Z, CL(γ−1/2))
→
Hs−3/2,γ−3/2(Y+)
⊕
Hs−3/2(Z, CL(γ−3/2))
, (1.40) for every s, γ ∈ R with
γ > 2 and γ /∈ 1 + N. (1.41)
The operator RND(γ) = (Rij(γ))i,j=1,2 belongs to the edge pseudo-differential calcu-lus over Y+, regarded as a manifold with edge Z, where
R11(γ) = Pγ−3/2+ (N+C+)(γ)Eγ−1/2+ , R12(γ) = Pβ−1+ (N+C+)(γ)Kγ−1/2+ , R21(γ) = Tγ−3/2+ (N+C+)(γ)Eγ−1/2+ , R22(γ) = Tγ−3/2+ (N+C+)(γ)Kγ−1/2+ . (ii) There exists a discrete set D ⊂ C such that the edge symbol σ∧(RND(γ))(z, ζ) defines
a family of Fredholm operators
σ∧(RND(γ))(z, ζ) :
Ks−1/2,γ−1/2(R+)
⊕ CL(γ−1/2)
→
Ks−3/2,γ−3/2(R+)
⊕ CL(γ−3/2)
, (1.42)
for all γ ∈ R with (1.41) and Γ1/2−γ ∩ D = ∅, and RND(γ) itself admits an elliptic operator
RND(γ) :=RND K
T Q
(γ) :
Hs−1/2,γ−1/2(Y+)
⊕
Hs−1/2(Z, CL(γ−1/2))
⊕ Hs−1/2(Z, Cd1)
→
Hs−3/2,γ−3/2(Y+)
⊕
Hs−3/2(Z, CL(γ−3/2))
⊕ Hs−3/2(Z, Cd2)
(1.43)
for certain d1, d2 depending on γ in the edge calculus with additional potential and trace operators and some classical first order pseudo-differential operator Q on Z.
Proof. (i) The continuity of the operator (1.40) for s = γ under the conditions (1.41) follows from (1.39) for s − 1/2 instead of γ − 1/2. Then we can assume the smoothness s to be arbitrary, since operators in the edge calculus are continuous in edge spaces of any smoothness. According to Theorem 1.5 (ii) the operator RND(γ) belongs to the edge pseudo-differential calculus on Y+ which is a smooth manifold with boundary Z, and Z is interpreted as an edge. From Theorem 1.5 we know that the operator RND(s) belongs
1 THE DIRICHLET-TO-NEUMANN OPERATOR 17 to the edge calculus with the weight data g = (s, s − 1) for the involved s, namely, s > 2, s /∈ 1 + N. A such it induces continuous operators (1.40) for weights (1.41) and arbitrary s, and RND(γ) belongs to the edge calculus for the weight data g = (γ, γ − 1).
(ii) From the discussion in the preceding subsection, specialised to the case of the operator RNDin the formula (1.40) we obtain the following result. The operator (1.40) for γ > 2, γ /∈ 1 + N, and Γ1/2−γ∩ D = ∅ for a discrete set D in C admits an elliptic operator (1.43) with additional potential operators K, trace operators T, and pseudo-differential operators Q, on Z. In other words, (1.43) in L1(Y+, (γ, γ − 1); w) for w = (L(γ − 1/2) + d1, L(γ − 3/2) + d2) is a Fredholm operator. In fact, in the case of the operator R = RND the exceptional set D does not depend on z ∈ Z. In addition, since the homogeneous principal symbol of the Dirichlet-to-Neumann operator R only depends on |ζ| and the other ingredients in the quantisation R ⇒ RND are only of smoothing Mellin plus Green type, the topological obstruction for the existence of additional Shapiro-Lopatinskij edge condition vanishes, i.e., we find an isomorphism (1.37) which corresponds to σ∧(RND). Together with the interior ellipticity this is equivalent to the Fredholm property of the operator (1.43).
It will be convenient to consider the operator (1.43) for a while for γ = s and to return to the standard Sobolev spaces over Y+
RND:=RND K
T Q
:
Hs−1/2(int Y+)
⊕ Hs−1/2(Z, Cd1)
→
Hs−3/2(int Y+)
⊕ Hs−3/2(Z, Cd2)
. (1.44)
More precisely, the relationship between RND and RND(s) is given by
RND(s) = diag(Λ+N id2) RNDdiag((Λ+D)−1 id1) (1.45) for
id1 := idHs−1/2(Z,Cd1), id2:= idHs−3/2(Z,Cd2). (1.46) As pointed out before the Dirichlet-to-Neumann operator R over Y+has not the transmis-sion property at Z but the operator RND(s) with the quantisation RND(s) in the upper left corner is an elliptic boundary value problem on the Neumann side of Y . It is of block matrix form, similarly as in Boutet de Monvel’s calculus, with extra trace and potential operators T and K, respectively, and a d2× d1 block matrix over Z. In the following con-sideration we will relate RND(γ) with an elliptic problem of the edge calculus of boundary value problems, applied to the Zaremba problem Am over G, cf. (1.19), interpreted as a manifold with edge Z and boundary Y.We shall see that this interpretation works in both directions, i.e., we can work in the larger block matrix algebra of such edge/boundary problems over Y+. This shows that the calculus of boundary value problems for operators without the transmission property on Y+ is an interesting substructure of mixed elliptic problems. It can be interpreted as a special case of the edge calculus on Y+, regarded as a manifold with edge Z and R+, the inner normal, as the model cone of local wedges.
1.6 Parametrices of the Zaremba problem
In the following considerations when we talk about the Fredholm property we tacitly assume that s is chosen in such a way that in the interpretation as a weight γ as in Theorem 1.7 we just have Γ1/2−s∩ D = ∅.
Write the operator (1.29) in the form cf. the formulas (1.25), (1.46), and pass to
AM:= JMDAD:
which is a Fredholm operator, since both JMDand AD are Fredholm between the respec-tive spaces and a parametrix is obtained by
A(−1)M = PDJMD(−1), (1.50)
cf. formula (1.48). Let R(−1)ND =:S C B E
.
1 THE DIRICHLET-TO-NEUMANN OPERATOR 19 Theorem 1.8. The operator Am which represents the Zaremba problem can be completed by additional conditions Tm of trace type and Km of potential type with respect to the interface Z and a d2× d1 block matrix Qm of classical pseudo-differential operators over Z to a Fredholm operator
AM =Am Km
which coincides with (1.49), and
A(−1)M :=
is a parametrix of (1.51).
Proof. First we identify the operator (1.49) as an operator containing Am as upper left corner. In fact, we have
AM= JMDAD = obtain AM in the form (1.51), where
Tm:= T D+, Km:= Let us now compute the parametrix A(−1)M of AM. Applying (1.48) and (1.50) we obtain (1.52), namely,
2 Mellin-edge quantisations
2.1 The Zaremba problem in edge spaces
In the construction of the Zaremba problem with additional interface conditions (1.51), (1.53) we employ the operator RND, defined in (1.43), with elliptic conditions defined by (1.54) for T , K, Q from (1.44). Now we translate the operator (1.51) into an operator between weighted edge spaces over Y±. Here we employ Lemma 1.1.
Using ΛD in (1.18) we first rephrase the operator AD in (1.24) for s > 1, s − 1 /∈ N, into (1.29) we first write down the transformed operator AM, analogously as (2.1), namely,
idHs−2(int G) 0
This allows us to transform the operator (1.29) to
idHs−2(int G) 0
2 MELLIN-EDGE QUANTISATIONS 21 , and the operator (1.39). Analogously as (1.47) we consider the Fredholm operator
˜JMD:= I 0 (1.53), namely, a Fredholm operator
A˜M:
2.2 Operators with jumping conditions of Dirichlet or Neumann type We now identify operators connected with the Dirichlet and the Neumann problem with edge operators of the spaces studied in Subsections 3.4, 3.5.
Concerning s in the meaning of a weight we make here the same assumptions as at the beginning of the preceding subsection.
We apply Proposition 3.9 (ii) below to s and s − 2 and obtain isomorphisms
Γs: Hs(int G) →
Let us define the isomorphism
ΞD:= Γ−1s 0
The operator (2.3) is an isomorphism.
2 MELLIN-EDGE QUANTISATIONS 23 Similarly as Ad we consider the operator An and the Neumann analogue ˜A1 of (1.22), namely,
which is again Fredholm of index 0 but not an isomorphism as (1.24). Analogously as (2.3) we consider the operator
. Similarly as (1.1) we perform the composition
ANA−1D (2.6)
which is a 6 × 6 block matrix operator. The lower 4 × 4 submatrix is the announced new representation of the Dirichlet-to-Neumann operator between the respective weighted edge
spaces plus Sobolev spaces on the interface Z, namely,
S:
Hs−1/2,s−1/2(Y−)
⊕
Hs−1/2(Z, CL(s−1/2))
⊕
Hs−1/2,s−1/2(Y+)
⊕
Hs−1/2(Z, CL(s−1/2))
→
Hs−3/2,s−3/2(Y−)
⊕
Hs−3/2(Z, CL(s−3/2))
⊕
Hs−3/2,s−3/2(Y+)
⊕
Hs−3/2(Z, CL(s−3/2))
. (2.7)
Note that the upper left 2 × 2 corner of (2.6) just represents the identity in the space Hs−2,s−2(G) ⊕ Hs−2(Z, CN (s−2)), while the lower left 2 × 4 corner is a corresponding reformulation of the operator T1P0 in (1.1).
Let us finally set
AM:= ΠMAMΞD, cf. the formula (1.28), for ΠM:=Γs−2 0
0 ΛM
. Analogously as (1.29) for we now form
AMPD= ΠMAMΞD
Ξ−1D PDΠ−1D = ΠMAMPDΠ−1D (2.8) for PD := A−1D which plays the role of the reduction to the boundary of the Zaremba problem, now in the weighted edge spaces. It is again a 6 × 6 block matrix operator. Again the upper left 2 × 2 corner of (2.8) represents the identity operator in Hs−2,s−2(G) ⊕ Hs−2(Z, CN (s−2)) which is just the reformulated identity 1 = idHs−2(int G)in the upper left corner of (1.29) by means of Γs−2 in (2.2). The remaining 2 × 4 block matrix occurring in the first two lines of AMPDis equal to zero; it corresponds to the zeros in the first line of (1.29). The lower left 4 × 2 corner of (2.8) is not essential for the ellipticity.
The 4 × 4 lower right corner of AMPD is a continuous map
SM:
Hs−1/2,s−1/2(Y−)
⊕
Hs−1/2(Z, CL(s−1/2))
⊕
Hs−1/2,s−1/2(Y+)
⊕
Hs−1/2(Z, CL(s−1/2))
→
Hs−1/2,s−1/2(Y−)
⊕
Hs−1/2(Z, CL(s−1/2))
⊕
Hs−3/2,s−3/2(Y+)
⊕
Hs−3/2(Z, CL(s−3/2)) ,
which has the form
SM=idHs−1/2,s−1/2(Y−)⊕Hs−1/2(Z,CL(s−1/2)) 0
T RND
. (2.9)
The shape of the first 2 × 4 row of (2.9) comes from the transformation of (1 0) in the second row of (1.29). The operator T in (2.9) is again not essential for the algebraic constructions with ellipticity, because of the triangular form of (2.9). The lower right 2 × 2 corner RND of (2.9) just comes from the right lower corner N+C+in (1.29), namely,
2 MELLIN-EDGE QUANTISATIONS 25 RND= Λ+NN+C+(Λ+D)−1for the isomorphisms Λ+D and Λ+N contained in (1.15) and (1.16), respectively. In other words the operator
RND=S33 S34 S43 S44
:
Hs−1/2,s−1/2(Y+)
⊕
Hs−1/2(Z, CL(s−1/2))
→
Hs−3/2,s−3/2(Y+)
⊕
Hs−3/2(Z, CL(s−3/2))
(2.10)
coincides with (1.39), i.e., R11 = S33, R12 = S34 , R21 = S43, R22 = S44, cf. (1.39).
From the operator AMwhich is of the form (1.53) and defines a Fredholm operator (1.49) we now pass to an operator AM by translating the Sobolev spaces over G and Y± into weighted spaces via the isomorphisms (2.2) which gives us a Fredholm operator
AM = ΠM⊕ id2AM ΞD⊕ id1 :
Hs,s(G)
⊕ Hs(Z, CN (s))
⊕
Hs−1/2(Z, CL(s−1/2))
⊕ Hs−1/2(Z, Cd1)
→
Hs−2,s−2(G)
⊕
Hs−2(Z, CN (s−2))
⊕
Hs−1/2,s−1/2(Y−)
⊕
Hs−1/2(Z, CL(s−1/2))
⊕
Hs−3/2,s−3/2(Y+)
⊕
Hs−3/2(Z, CL(s−3/2))
⊕ Hs−3/2(Z, Cd2)
.
(2.11) Note that AM will be interpreted as the Zaremba problem in the edge calculus.
2.3 The edge algebra structure of boundary value problems with respect to the interface
As noted before one of the essential issues of this paper is to identify the operators AD, ANas elements of the edge pseudo-differential calculus of boundary value problems. To be more precise, in Subsection 3.4 we see that
G is a manifold with edge Z and boundary (2.12) where
G \ Z is a non-compact manifold with smooth boundary ∂G \ Z, (2.13) and
∂G is a manifold with edge Z without boundary, (2.14) cf. Definition 3.23 below. The calculus
Bµ,d(G, g; w) (2.15)
over (2.12) is developed in Subsection 3.4 below for M := G, where g are weight data and w a tuple of involved dimensions, cf. Definition 3.34. The operators in (2.15) consist of
3 × 3 block matrices A = (Aij)i,j=1,2,3where (Aij)i,j=1,2restricts to (2.13) as a subcalculus of the operator spaces Bµ,d(G \ Z, v) with induced dimension data v, that we read off from (2.3) or (2.5), cf. Definition 3.16, here for N := G \ Z, and the case without parameters, i.e., l = 0. As such there is the principal symbolic hierarchy
(σψ(A), σ∂(A))
of interior and boundary symbols where σψ(A) = σψ(A11), A11 ∈ Lµcl(int G). In addition for operators in (2.15) the edge Z contributes the principal edge symbol
σ∧(A).
Both σ∂(A) and σ∧(A) take values in operators; σ∂(A) lives on T∗(∂G\Z)\0 and σ∧(A) on T∗Z \ 0. The three symbolic components will contribute to the ellipticity of A. Moreover, let us announce that the lower right corners (Aij)i,j=2,3 =: A0 belong to the edge pseudo-differential calculus over (2.14). As such we have a principal symbolic hierarchy
(σ0ψ(A0), σ∂0(A0)), (2.16) here in the meaning of transmission symbols from the respective calculus of transmis-sion problems, since Z is of codimentransmis-sion 1 in ∂G. Finally, the operators A33 belong to Lµcl(Z), more precisely, those are block matrices of such operators, according to the above-mentioned dimensions.
The fact that AD and AN are elements of (2.15) is a consequence of the nature of the involved entries. The “upper left” corners simply consists of the Laplace operator, in the respective weighted edge Sobolev spaces, cf. [21, Theorem 4.2.2] or [7, Theorem 2.2.1].
Formally we obtain the latter relations by locally introducing cylindrical coordinates, ac-cording to corresponding splitting of variables locally near Z where the 2-dimensional normal (xn−1, xn)-plane to Z, intersected with G, is identified with R+× S+1 × Rq,
S1+= S1∩ {xn≥ 0}, q = dim Z. (2.17) In a similar manner the boundary conditions can be transformed into the framework, since they are compositions of differential operators with the operator of restriction r0 to the boundary.
These observations are valid not only for the Dirichlet or Neumann problem, rephrased with respect to the interface Z but for arbitrary differential operators A and boundary conditions r0B for a differential operator B. In the language of the symbolic structures for operators in (2.15) where the so-called type d is now equal to ord B + 1 the first two components σψ(A), σ∂(A) are elliptic. What concerns σ∧(A) (both for AD and AN) also the edge symbol is elliptic. This comes from the fact, that the operators (2.3) and (2.5) are Fredholm, and this property entails the ellipticity (i.e., bijectivity) of the three symbolic components. The corresponding general theorem on the necessity of ellipticity for the Fredholm property in (2.15) is not yet published somewhere. However, in the present case we do not need this information; it is a consequence of the already known ellipticity of (σψ(A), σ∂(A)), cf. Theorem 3.44 in Subsection 3.4 below.
Summing up we have the following result.
Theorem 2.1. (i) The operator (2.7) belongs to the edge pseudo-differential calculus over (2.14) and represents in that way an edge quantisation with respect to Z of the Dirichlet-to-Neumann operator R = T1K0 occurring in (1.1).
2 MELLIN-EDGE QUANTISATIONS 27 (ii) The operator (2.7) is elliptic with respect to both symbolic components in (2.16).
Proof. (i) follows from the fact that the operators in the calculus of operators (2.15) can be composed within the structure and that both ANand A−1D belong to the calculus. Then the same is true of the submatrix (2.7).
(ii) The ellipticity of (2.7) is a consequence of the ellipticity of ANA−1D itself and the observation that the upper left 2 × 2 corner is the identity.
Remark 2.2. (i) The operator (2.9) belongs to the edge calculus over B.
(ii) The operator (2.10) is elliptic with respect to σ0ψ, cf. the notation in (2.16).
In fact, the operator (2.9) is simply a submatrix in the calculus of operators in (2.15), namely, AMA−1D . Moreover, the operator (2.10) is the lower right 2 × 2 submatrix of (2.7).
Since S is σ0ψ-elliptic and because the non-smoothing contributions to S come from (Sij)i,j=1,2, (Sij)i,j=3,4, (2.18) while the other elements are smoothing off Z (more precisely, they consists of smoothing Mellin plus Green operators in the transmission algebra over ∂G with respect to Z) the operators (2.18) are both elliptic and hence, also the operator (2.10).
The following result is formulated in terms of boundary value problems in the sense of Definition 3.34 below.
Theorem 2.3. (i) The isomorphism AD(γ), first given for γ = s by (2.3) and then realised as an operator (3.131) in weighted spaces, belongs to BµD,1(G, gD; wD) where µD, gD, are defined by (3.135), (3.136) and wD by (3.137). The operator AD(γ) is (σψ, σ∂, σ∧)-elliptic, and we have PD(γ) = AD(γ)−1∈ B−µD,0(G, g−1D ; w−1D ).
(ii) The Fredholm operator AN(γ), first given for γ = s by (2.5) and then realised as an operator (3.133) in weighted spaces, belongs to BµN,2(G, gN; wN) where µN, gN, are defined by (3.138), (3.139) and wN by (3.140). The operator AN(γ) is (σψ, σ∂, σ∧ )-elliptic, and for the parametrix we have PN(γ) ∈ B−µN,0(G, g−1N ; w−1N ).
Proof. (i) We start with the symbolic levels for the Dirichlet problem in Boutet de Mon-vel’s calculus. As noted in Subsection 3.2 the operator has a principal symbolic structure, namely,
σ(A0) = (σψ(A0), σ∂(A0))
that we express in suitable coordinates. First we have σψ(A0)(ξ) = −|ξ|2 every where on G the Riemannian metric is induced by the ambient Euclidean space Rn. Concerning σ∂(A0) = σσ∂(∆)
∂(T0) we refer to coordinates in the half-space Rn+ 3 (y, t) and covariables (η, τ ). Here
σ∂(∆)(η) = −|η|2+ ∂2
∂t2 : Hs(R+) → Hs−2(R+) (2.19) which we observe for s > −3/2; the choice of s unimportant in this case. Later on we specify s according to the steps in the construction of AD. Together with σ∂(T0) = γ0 with γ0 being the restriction to t = 0 we then have an isomorphism
σ∂(A0)(η) =σ∂(∆) γ0
(η) : Hs(R+) →
Hs−2(R+)
⊕ C
(2.20)
for every η 6= 0. Next we look at ˜A0 which contains A0 as a submatrix, and we set σψ( ˜A0)(ξ) = σψ(A0)(ξ) = σψ(∆)(ξ), σ∂( ˜A0)(η) = σ∂(A0)(η). (2.21) In the next step we pass to AD, defined in the formula (1.24). On the level of the symbols we write
σψ(AD) = σψ( ˜A0) and
σ∂(AD) = σ∂−(AD), σ∂+(AD)
for σ∂±(AD)(η) =σ∂(∆)(η) σ∂(D±)
where σ∂(D±) refers to ± side. Here we choose the coordinates (y, xn) = (z, xn−1, xn) where the interface Z (the edge) is locally described by xn= 0, xn−1= 0, z ∈ Rn−2. From the representation ∆ = r−2
(r∂r)2 + ∂ϕ2 + r2∆z
of the Laplacian in cylindrical coordinates we obtain
σψ(∆)(r, ρ, ς, ζ) = r−2(−r2ρ2− ς2− r2|ζ|2), for r > 0, (ρ, ς, ζ) 6= 0, (2.22)
˜
σψ(∆)(r, ρ, ς, ζ) = (−ρ2− ς2− |ζ|2) up to r = 0, (ρ, ς, ζ) 6= 0. (2.23) Moreover,
σ∂±(∆)(ρ, ζ) = r−2(−r2ρ2+ ∂ϕ2− r2|ζ|2) : Hs(R+) → Hs−2(R+), for r > 0, (ρ, ζ) 6= 0, (2.24)
˜
σ∂±(∆)(ρ, ζ) = −ρ2+ ∂ϕ2− |ζ|2 : Hs(R+) → Hs−2(R+), up to r = 0, (ρ, ζ) 6= 0. (2.25) Analogously as before ∂± indicates the boundary symbols over int Y±. This yields isomor-phisms
σ∂(AD)(r, ρ, ζ) = σ∂−(∆) γ0
,σ∂+(∆) γ0
!
: Hs(R+) →
Hs−2(R+)
⊕ C
(2.26)
for r > 0, (ρ, ζ) 6= 0, and
˜
σ∂(AD)(r, ρ, ζ) = ˜σ∂−(∆) γ0
, ˜σ∂+(∆) γ0
!
: Hs(R+) →
Hs−2(R+)
⊕ C
. (2.27)
up to r = 0, (ρ, ζ) 6= 0.
It remains to recognise the principal edge symbol σ∧(AD)(z, ζ) and to verify that it defines also isomorphisms. The orders in the Sobolev spaces over Z occurring in (2.3) show that our operator belongs to BµD,1(M, gD; wD). In the present concrete case it is not advisable to change rows and columns in the 6 × 3 block matrix. The source of the edge symbol is nothing else than the parameter-dependent family of Dirichlet problems in the upper
It remains to recognise the principal edge symbol σ∧(AD)(z, ζ) and to verify that it defines also isomorphisms. The orders in the Sobolev spaces over Z occurring in (2.3) show that our operator belongs to BµD,1(M, gD; wD). In the present concrete case it is not advisable to change rows and columns in the 6 × 3 block matrix. The source of the edge symbol is nothing else than the parameter-dependent family of Dirichlet problems in the upper