The edge calculus of boundary value problems

, (3.62) cf. the formula (3.51).

Theorem 3.22. An elliptic A ∈ B^µ,d(N ; v; R^l) for v = (j₁, j₂) has a properly sup-ported parametrix P ∈ B^{−µ,(d−µ)+}(N ; v⁻¹; R^l) for v⁻¹ = (j2, j1) in the sense PA = I − C_L, AP = I − C_R for remainders C_L ∈ B−∞,max{µ,d}(N ; (j₁, j₁); R^l), C_R ∈ B^{−∞,(d−µ)+}(N ; (j₂, j₂); R^l).

3.4 The edge calculus of boundary value problems

Our next objective is to outline the pseudo-differential edge calculus of boundary value problems. In order to explain the background we first recall the notion of a manifold B with edge Z.

Definition 3.23. (i) A manifold B with edge Z is a disjoint union B = (B \ Z) ∪ Z where both B \ Z and Z are C^∞manifolds, and Z has a neighbourhood V in B with the structure of a locally trivial X^∆-bundle over Z for a closed C^∞ manifold X,

X^∆:= (R+× X)/({0} × X). (3.63)

The space (3.63) is a cone with base X; the bottom {0} × X of the cylinder R+× X is collapsed to a single point, the vertex of the cone. The transition maps X^∆× Ω → X^∆× ˜Ω between different trivialisations of the bundle V referring to open sets Ω, ˜Ω ⊆ R^{dim Z} on Z (that represent charts on Z) are defined as quotient maps of isomorphisms (R+× X) × Ω → (R+× X) × ˜Ω between the trivial bundles with fibre R+× X with respect to the projection R+× X → X^∆.

3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 53 (ii) A manifold M with edge Z and boundary is a disjoint union M = (M \ Z) ∪ Z where M \ Z is a C^∞ manifold with boundary and Z a C^∞ manifold, and Z has a neighbourhood W in M with the structure of an N^∆-bundle over Z for a C^∞manifold N with boundary ∂N . The transition maps N^∆× Ω → N^∆× ˜Ω between different trivialisations are defined as restrictions of trivialisations (2N )^∆× Ω → (2N )^∆× ˜Ω in the sense of (i) where 2N is the double of N , (obtained by gluing together two copies N−, N₊ of N along the common boundary ∂N with N₊ being identified with N ) and the restriction concerns the “plus copy” N₊^∆ of (2N )^∆× Ω.

Example 3.24. (i) Given a C^∞ manifold X then the wedge

B := X^∆× Z (3.64)

is a manifold with edge in the sense of Definition 3.23 (i).

(ii) If N is a C^∞ manifold with boundary ∂N then

M := N^∆× Z (3.65)

is a manifold with edge and boundary in the sense of Definition 3.23 (ii).

Remark 3.25. If M is a manifold with edge Z and boundary then ∂M := ∂(M \ Z) ∪ Z is a manifold with edge as in Definition 3.23 (i), where X plays the role of ∂N .

In our application we have G = M for N = S₊¹ := S¹∩ R²+. The edge Z is embedded in the boundary of codimension 1 and in M of codimension 2, and in this case, since M is a smooth manifold with boundary, we simply write ∂M , regarded as a manifold with edge Z. By gluing together two copies M−, M₊ of M along the common boundary ∂(M∓\ Z), i.e., setting 2M := 2(M \ Z) ∪ Z we obtain a manifold with edge Z in the sense of Definition 3.23 (i). The analysis of operators on a manifold M with edge Z and boundary refers to M \ Z, locally modelled on Cartesian products N^∧× Ω for N^∧ := R+× N 3 (r, x), r ∈ R+, x ∈ N , z ∈ Ω. An empty boundary as in Definition 3.23 (i) is a special case. As soon as we have a non-trivial boundary then for n = dim N we identify N locally near

∂N by Rⁿ+3 x = (x⁰, x_n) and ∂N by x_n= 0. The covariables then split into (ξ⁰, ξ_n).

Another essential notion is the corresponding stretched manifold. The stretched manifold associated with (3.64) is

B := R+× X × Z.

This is a smooth manifold with boundary {0} × X × Z, the latter identified with X × Z.

In other words

∂B = X × Z

is a (in this case trivial) X-bundle over the edge Z. From this we also obtain the stretched manifold B associated with B in Definition 3.23 (i), by invariantly attaching {0} × X × Ω to the Cartesian products X^∧× Ω from the trivialisations X^∆.

We also have the notion of a stretched manifold for (3.65), namely,

M := R+× N × Z (3.66)

which may be regarded as a subspace of the double of M . Note that (3.66) has corners.

Then we also obtain the streched manifold M associated with M in Definition 3.23 (ii) by invariantly attaching {0} × N × Ω to the Cartesian products N^∧ × Ω from the trivialisations N^∆× Ω.

For future references we indicate the local models of M near Z in terms of singular charts. By definition the edge Z ⊂ M has a neighbourhood U in M such that there is an isomorphism in the catogery of manifols with edge an boundary

χ^∆: U → N^∆× R^q (3.67)

which restricts to a diffeomorphism χ0 : U ∩ Z → R^q and to an isomorphism

χ^∧ : U \ Z → N^∧× R^q (3.68)

in the category of manifolds with boundary. There is finite atlas of such maps close to Z and the sets U ∩ Z form an open covering of Z by coordinate neighbourhoods. Choose functions

{ϕ_j}_j=1,...,N, {ϕ⁰_j}_j=1,...,N (3.69) where the ϕ_j form a subordinate partition of unity on Z, and ϕ⁰_j ∈ C₀^∞(G_j), ϕ⁰_j  ϕ_j for all j.

In order to formulate the edge calculus of boundary value problems on a manifold M with edge and boundary we prepare some tools on parameter-dependent boundary value problems over N .

According to the generalities of the edge calculus we start with operator functions p(r, z, ρ, ζ) := ˜p(r, z, rρ, rζ) (3.70) for ˜p(r, z, ˜ρ, ˜ζ) ∈ C^∞(R+× R^q, B^µ,d(N ; R^1+q_{ρ, ˜˜ζ} )), cf. Definition 3.13.

Functions with values in B^µ,d refer to the natural Fr´echet topology in spaces of pseudo-differential boundary value problems with the transmission property at the boundary, see, for instance, [16, Remark 3.1.20], or [32, Theorem 4.3.38].

Let M^µ,d_O (N ; R^l_λ) defined to be the set of all h(w, λ) ∈ A(C, B^µ,d(N ; R^l_λ)) such that h(β + iρ, λ) ∈ B^µ,d(N ; R^1+l_ρ,λ)) for every β ∈ R, uniformly in compact β-intervals. A well-known Mellin quantisation result, cf. [32, Theorem 3.2.7], [16, Theorem 7.2.1] tells us that there is an

˜h(r, z, w, ˜ζ) ∈ C^∞(R+× R^q, M^µ,d_O (N ; R^q_ζ˜)) (3.71) such that for

h(r, z, w, ζ) := ˜h(r, z, w, rζ) we have

Op_r(p)(z, ζ) = op^γ_M(h)(z, ζ) mod C^∞(R^q, B^−∞,d(R+× N ; R^q_ζ)) (3.72) for every γ ∈ R. The operator family ˜p(r, z, ˜ρ, ˜ζ) has a pair of parameter-dependent prin-cipal and boundary symbols, namely,

σ_ψ(˜p)(r, x, z, ˜ρ, ξ, ˜ζ), σ_∂(˜p)(r, x⁰, z, ˜ρ, ξ⁰, ˜ζ), (3.73)

3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 55 the relations (3.55), (3.58), and (3.60). The variables (r, z) comes from the coefficients and the parameters are ( ˜ρ, ˜ζ), treated as additional covariables, cf. also the notation in Subsection 3.2. Then (3.70) itself has corresponding symbols

σψ(p)(r, x, z, ρ, ξ, ζ), σ∂(p)(r, x⁰, z, ˜ρ, ξ⁰, ζ), (3.74) obtained from (3.73) by replacing ( ˜ρ, ˜ζ) by (rρ, rζ).

Let us fix cut-off functions ω⁰⁰ ≺ ω ≺ ω⁰ on the r half-axis, and let φ ≺ φ⁰ be other cut-off functions in r. For a function χ in r ∈ R+ we set χζ(r) := χ(r[ζ]) where ζ → [ζ] is a strictly positive function in R^q such that [ζ] = |ζ| for |ζ| > const for some constant > 0, and χ_|ζ|(r) := χ(r|ζ|).

Set

a(z, ζ) := φr^−µω_ζop^γ−n/2_M (h)(z, ζ)ω_ζ⁰ + (1 − ω_ζ)Op_r(p)(z, ζ)(1 − ω⁰⁰_ζ) φ⁰ (3.75) with p and h being connected via (3.72). According to (3.55) and (3.58), (3.57), we form

σψ(a)(r, x, z, ρ, ξ, ζ) := r^−µφ(r)σψ(p)(r, x, z, ρ, ξ, ζ), (3.76)

σ_∂(a)(r, x⁰, z, ρ, ξ⁰, ζ) := r^−µφ(r)σ_∂(p)(r, x⁰, z, ρ, ξ⁰, ζ). (3.77) Let

p₀(r, z, ρ, ζ) := ˜p(0, z, rρ, rζ), h₀(r, z, w, ζ) := ˜h(0, z, w, rζ). (3.78) Then we define the (twisted) homogeneous principal edge symbol of (3.75) by

σ∧(a)(z, ζ) := r^−µ{ω_|ζ|op^γ−n/2_M (h0)(z, ζ)ω_|ζ|⁰ + (1 − ω|ζ|)Op_r(p0)(z, ζ)(1 − ω_|ζ|⁰⁰ )}. (3.79) For µ =

 µ µ − 1/2

µ + 1/2 µ



we have

a(z, ζ) ∈ S^µ

R^q× R^q;

K^s,γ(N^∧)

⊕

Ks−1/2,γ−1/2((∂N )^∧) ,

K^{s−µ,γ−µ}(N^∧)

⊕

Ks−1/2−µ,γ−1/2−µ((∂N )^∧)



κ⊕κ⁰,κ⊕κ⁰. (3.80) Concerning s we impose the same conditions as in Subsections3.2, 3.3, e.g., s > −1/2, etc.

We employ the symbol spaces (3.18) with (z, ζ) variables and covariables, and the involved Hilbert spaces in this case are direct sums in (3.80) with the group action

κ ⊕ κ⁰ for (κ_δu)(r, x) := δⁿ⁺¹² u(δr, x), (κ⁰_δu⁰)(r, x) := δⁿ²u(δr, x⁰), δ ∈ R+, (3.81) n = dim N . Observe that κ_δand κ⁰_δare unitary in K^0,0(N^∧) and K^0,0((∂N )^∧), respectively, which implies (κδ)^∗ = κ_δ⁻¹, (κ⁰_δ)^∗ = κ⁰_δ−1. Here x and x⁰ denote points on N and ∂N , respectively.

Let us now turn to another category of operator-valued symbols in the edge calculus of boundary value problems, namely, the Green symbols with (in the present case discrete) asymptotics. The notion will also include trace and potential symbols, referring to the various strata of our configuration.

The spaces of symbols in the following refer to the group actions

κ = {κ_δ}_δ∈R+, κ_δ:= diag(κ_δ, κ⁰_δ, id_C) (3.82)

for κ_δ, κ⁰_δ as in (3.81). Those may be applied to spaces with weight for r → ∞, namely, K^s,γ;e(N^∧) := hri^−eK^s,γ(N^∧), K^s0,γ0;e0((∂N )^∧) := hri^−e0K^s0,γ0((∂N )^∧)

for any s, s⁰, γ, γ⁰, e, e⁰∈ R. The same is true of subspaces with asymptotics. In the simplest case asymptotics for r → 0, say in the spaces K^s,γ;e(N^∧), will be indicated by a discrete asymptotic P, i.e., a sequence

P = {(p_j, mj)}_j∈J ⊂ C × N

for a subset J ⊂ N∪{∞}, Re pj < ⁿ⁺¹₂ −γ. Modulo a remainder of some flatness at r = 0 dis-crete asymptotics of a function u ∈ K^s,γ(N^∧) means u(r, x) =P

j∈J

Pmj

k=0c_jk(x)r^−pjlog^kr for coefficients cjk ∈ C^∞(N ). By K^s,γ_P (N^∧) we denote the Fr´echet subspace of K^s,γ(N^∧) with such asymptotics, and we also set K_P^s,γ;e(N^∧) := hri^−eK^s,γ_P (N^∧). Details around weighted distributions may be found in [32]. There is also a notion of continuous asymp-totic types. This case is also studied in [18] and [32]. Here we tacitly employ these notions, using that the group action in K^s,γ;e(N^∧) restricts to group actions in the Fr´echet sub-spaces K^s,γ;e_P (N^∧) for discrete or continuous asymptotic type P. The exponents p_j in the discrete asymptotics will be controlled in a weight interval Θ := (ϑ, 0], −∞ ≤ ϑ < 0 and we require π_CP := {p_j}_j∈J to be contained in the strip_n+1

2 − γ + ϑ < Re z < ⁿ⁺¹₂ − γ . We identify π_CP with the carrier of the discrete asymptotic type P and we say in this case that (γ, Θ) is associated with the weight data (γ, Θ). In the case of continuous asymptotics a similar relation is required for the carrier of asymptotics. Similar notions make sense over (∂N )^∧ for r → 0; in this case we replace n by n − 1. If an operator refers to spaces with asymptotics where a weight γ is transported to another weight, say, γ − µ, then carriers of asymptotics are controlled in a strip of width Θ on the left of the respective weight lines. Weight data in the notation of corresponding operator spaces are then denoted by (γ, γ − µ, Θ) =: g. Later on we keep ϑ fixed once and for all and omit it.

The following definition concerns 3 × 3 block matrix-valued symbols with a scheme of orders

ν :=





ν ν − 1/2 ν − 1 ν + 1/2 ν ν − 1/2

ν + 1 ν + 1/2 ν



, ν^∗:=





ν ν + 1/2 ν + 1 ν − 1/2 ν ν + 1/2

ν − 1 ν − 1/2 ν



, ν ∈ R.

Definition 3.26. (i) A Green symbol g(z, ζ) of order ν ∈ R and type 0 is an operator family

g(z, ζ) ∈ S_cl^ν R^q× R^q;

K^s,γ;e(N^∧)

⊕

Ks−1/2,γ−1/2;e((∂N )^∧)

⊕ C

,

K^{∞,γ−µ;∞}_P (N^∧)

⊕ K∞,γ−1/2−µ;∞

P⁰ ((∂N )^∧)

⊕ C

!

κ,κ

(3.83)

3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 57 such that for the (z, ζ)-wise formula adjoints we have

g^∗(z, ζ) ∈ S_cl^ν∗ R^q× R^q;

K^s,−γ+µ;e(N^∧)

⊕

Ks,−γ+1/2+µ;e((∂N )^∧)

⊕ C

,

K^{∞,−γ;∞}_Q (N^∧)

⊕

K^{∞,−γ+1/2;∞}_Q0 ((∂N )^∧)

⊕ C

!

κ^∗,κ^∗

,

(3.84) for all s > −1/2, e ∈ R, and g-dependent asymptotic types P, Q and P⁰, Q⁰, asso-ciated with the weight γ − µ, including the fixed weight interval Θ in the involved spaces. Here the formal adjoint is defined via

u, g^∗(z, ζ)

K^0,0(N^∧)⊕K^0,0((∂N )^∧)⊕C= g(z, ζ)u, v

K^0,0(N^∧)⊕K^0,0((∂N )^∧)⊕C (3.85) for arbitrary u ∈ C₀^∞(R+ × 2N )|_{R+×int N} ⊕ C₀^∞(R+ × ∂N ) ⊕ C, v ∈ C₀^∞(R+ × 2N )|_R+×int N ⊕ C₀^∞(R+× ∂N ) ⊕ C, cf. the formulas (3.41) and (3.42).

(ii) A Green symbol of order ν ∈ R and type d ∈ N is an operator family of the form g(z, ζ) =Pd

j=0gj(z, ζ)diag(D^j_t, 0, 0) for s > d − 1/2 where gj are Green symbols of order ν − j and type 0 in the sense of (i). Here D_t means a first order differential operator on N which coincides with ∂_t in a collar neighbourhood of ∂N .

Remark 3.27. The meaning of (3.83), (3.84) is as follows. The relations (3.83), (3.84) may be first restricted to smooth functions indicated after (3.85); then g extends by continu-ity to K^s,γ;e(N^∧) ⊕ K^s,γ−1/2;e((∂N )^∧) ⊕ C and g^∗ to K^s,−γ+µ;e(N^∧) ⊕ K^s,−γ+µ;e((∂N )^∧) ⊕ C for all s > −1/2. What concerns the role of s in the spaces in the middle we may insert an arbitrary s⁰∈ R.

From Definition 3.26 applied to ν = µ and generalities on classical operator-valued symbols a Green symbols g(z, ζ) = (g_ij(z, ζ)_i,j=1,2,3has a 3×3 matrix of principal symbols of twisted the homogeneities

µ =





µ µ − 1/2 µ − 1 µ + 1/2 µ µ − 1/2

µ + 1 µ + 1/2 µ





referring to the triple of group actions (3.82), for all (z, ζ) ∈ R^q× (R^q\ {0}). The corre-sponding 3 × 3 matrix of twisted homogeneous functions associated with g(z, ζ) will be denoted by σ_(µ). It is also called the homogeneous edge symbol belonging to g written

σ∧(g)(z, ζ) := g_(µ)(z, ζ). (3.86) From Definition 3.26 (ii) we see that (3.86) induces an operator family

σ∧(g)(z, ζ) :

K^s,γ(N^∧)

⊕

Ks−1/2,γ−1/2((∂N )^∧)

⊕ C

→

K^∞,γ−µ(N^∧)

⊕

K^{∞,γ−1/2−µ}((∂N )^∧)

⊕ C

. (3.87)

for any s > d − 1/2.

Another essential ingredient of the edge calculus of boundary value problems are the smoothing Mellin operators which are generated in the construction of parametrices of elliptic operators and involved in the asymptotics of solutions close to Z. For the formu-lation we need corresponding meromorphic Mellin symbols.

A sequence

A smoothing Mellin edge symbols in our calculus associated with the weight data g = (γ, γ − µ, Θ), Θ := (−(k + 1), 0]), k ∈ N, is a family of operators

In future we always omit the fixed weight interval and simply write g = (γ, γ − µ). In (3.88) we assume arbitrary fjα ∈ C^∞(R^q, M^−∞,d_R

jα (N )) satisfying the relations γ − j ≤ γ_jα ≤ γ, π_CR_jα∩ Γn+1

2 −γjα = ∅ for all j, α. Here ω, ω⁰ are arbitrary cut-off functions.

Recall that when we change γjα under the mentioned conditions we only change m(z, ζ) by a Green symbol. Also a change of the cut-off functions modifies m(z, ζ) by a Green symbol.

It is easy to verify that

m(z, ζ) ∈ S_cl^µ R^q× R^q; some resulting Q, Q⁰. Asymptotic types in parentheses means that the symbol property holds between spaces without and with corresponding asymptotics.

As classical symbols the operators (3.88) have a (twisted) homogeneous principal symbol of order µ, interpreted as an edge symbol,

σ∧(m)(z, ζ) =

The symbol (3.89) will be interpreted as a family of operators

σ∧(m)(z, ζ) :

3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 59 be the space of Green symbols in the sense of Definition 3.26. The notation g = (γ, γ − µ) in (3.91) is an abbreviation for the weight shifts occurring in the entries of (3.83), i.e., the weight shift in upper left is γ to γ − 1/2 − µ, etc., cf. also the weights in the formulas

we denote the space of all

a(z, ζ) :=a(z, ζ) 0 induces a family of continuous operators

σ∧(a)(z, ζ) :

σ∧(a₃₁)(z, δζ) = δ^µ+1id_Cσ∧(a₃₁)(z, ζ)κ⁻¹_δ , σ∧(a32)(z, δζ) = δ^µ+1/2id_Cσ∧(a32)(z, ζ)κ⁰⁻¹_δ , σ∧(a₃₃)(z, δζ) = δ^µid_Cσ∧(a₃₃)(z, ζ)id_C.

Theorem 3.28. Let a(z, ζ) ∈ R^µ,d(R^q× R^q, g) and assume (for convenience) that a is independent of z for large |z|. Then Op_z(a) induces continuous operators

Op_z(a) :

W^s(R^q, K^s,γ(N^∧))

⊕

W^s(R^q, Ks−1/2,γ−1/2((∂N )^∧))

⊕ H^s−1(R^q)

→

W^s−µ(R^q, K^{s−µ,γ−µ}(N^∧))

⊕

W^s−µ(R^q, Ks−1/2−µ,γ−1/2−µ((∂N )^∧))

⊕ H^s−1−µ(R^q) for every s > d − 1/2.

Let us now formulate the algebra of pseudo-differential boundary value problems on a manifold M with boundary and edge. First we have the global analogue of weighted Sobolev spaces and subspaces with asymptotics.

First on a compact manifold D with edge Z as at the beginning of this section we have the weighted Sobolev space H^s,γ(D) ⊂ H_loc^s (D \ Z), locally near Z in the splitting of variables of the stretched wedges

R+× 2N × R^q 3 (r, x, z) (3.98)

modelled on W^s(R^q, K^s,γ((2N )^∧). Then for D := 2M we set

H^s,γ(M ) := H^s,γ(D)|_{(int M )\Z}. (3.99) Moreover, since ∂M is a manifold with edge Z (and without boundary), locally near Z modelled on

R+× ∂N × R^q3 (r, x⁰, z), (3.100) we have the spaces H^s,γ(∂M ), locally near Z modelled on W^s(R^q, K^s,γ((∂N )^∧). In addition we have the subspaces H_P^s,γ(M ) and H_P^s,γ0 (∂M ) with discrete asymptotics of type P and P⁰, locally described by W^s(R^q, K^s,γ_P (N^∧)) and W^s(R^q, K^s,γ_P0((∂N )^∧)), respectively.

We now establish the operator space

B^−∞,d(M, g) (3.101)

of smoothing elements of type d ∈ N in the edge calculus of boundary value problems for weight data g = (γ, γ − µ), k ∈ N ∪ {∞}. We first define (3.101) for d = 0. This space consists of all operators C that induce with their formal adjoints C^∗ continuous maps

H^s,γ(M )

⊕

Hs−1/2,γ−1/2(∂M )

⊕ H^s−1(Z)

→

H_P^∞,γ−µ(M )

⊕

H_P^{∞,γ−1/2−µ}0 (∂M )

⊕ H^∞(Z)

,

H^s,−γ+µ(M )

⊕

Hs+1/2,−γ+1/2+µ(∂M )

⊕ H^s+1(Z)

→

H_Q^∞,−γ(M )

⊕

H_Q^{∞,−γ+1/2}0 (∂M )

⊕ H^∞(Z) for all s > −1/2 and asymptotic types P, P⁰ and Q, Q⁰, depending on C.

For arbitrary d ∈ N we define (3.101) as the set of all C =Pd

j=0C_jdiag(D^j, 0, 0) for arbi-trary C_j ∈ B^−∞,0(M, g) where D^j is a differential operator of order j that is locally in the

3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 61 variables (r, x, z) for x = (x⁰, x_n) of the form ∂_x^j_n close to x_n= 0.

For a compact manifold M with edge Z and boundary we fix a finite system of singu-lar charts χ^∆_j : U_j → N^∆× R^q and associated χ^∧_j : U_j \ Z → N^∧ × R^q as in (3.67), (3.68). Moreover, let ω⁰⁰ ≺ ω ≺ ω⁰ be cut-off functions on the r half-axis, and set Φj := diag(ω, ω, ϕj), Φ⁰_j := diag(ω⁰, ω⁰, ϕ⁰_j) with the above-mentioned functions ϕj ≺ ϕ⁰_j, cf. formula (3.69).

Definition 3.29. Let M be a compact manifold with edge Z and boundary. By

B^µ,d(M, g) (3.102)

for µ ∈ Z, the 3 × 3 matrix as above, d ∈ N, and weight data g = (γ, γ − µ) (which is an abbreviation of (3.107) below, later on also indicated by g_nl) we denote the set of all operators

A= (A_ij)_i,j=1,2,3 (3.103)

of the form

A=

N

X

j=1

Φj(χ^∧_j)⁻¹_∗ Op_z(aj)Φ⁰_j+ Aint+ C (3.104)

for arbitrary aj ∈ R^µ,d(R^q×R^q, g), Aint:= diag(1−ω, 1−ω, 0)(A_ij)_i,j=1,2 0

0 0



diag(1−

ω⁰⁰, 1 − ω⁰⁰, 0) for (A_ij)_i,j=1,2∈ B^µ,d(M \ Z), C ∈ B^−∞,d(M, g).

By

B^µ,d_{M +G}(M, g) (B^µ,d_G (M, g)) (3.105) we denote the subspaces defined by (Aij)i,j=1,2= 0 and aj ∈ B^µ,d_{M +G} (B^µ,d_G ).

Theorem 3.30. An operator A ∈ B^µ,d(M, g) induces continuous operators

A= (A_ij)_i,j=1,2,3:

H^s,γ(M )

⊕

Hs−1/2,γ−1/2(∂M )

⊕ H^s−1(Z)

→

H^{s−µ,γ−µ}(M )

⊕

Hs−1/2−µ,γ−1/2−µ(∂M )

⊕ H^s−1−µ(Z)

(3.106)

for all s ∈ R, s > d − 1/2.

The proof of this result relies on the fact that the local amplitude functions a_j in (3.104) are operator-valued symbols in (3.95) and that the local operators Op_z(·) induce continuous maps in the corresponding edge spaces W^s(R^q, · ), cf. the above definition of (3.99), cf.

Remark 3.3.

Remark 3.31. We already commented the meaning of µ as a 3 × 3 matrix of orders.

Writing A in the form (3.103) the entries A_ij have corresponding orders, associated weight data g_ij and a specific meaning. The 9 appearing operator spaces are denoted by

B^µ,d(M, g₁₁)int, B^µ−1/2,0(M, g₁₂)potential on ∂M, B^µ−1,0(M, g₁₃)potential on Z, B^µ+1/2,d(M, g₂₁)trace to ∂M, L^µ(∂M, g₂₂), L^µ−1/2(∂M, g₂₃)potential on Z, B^µ+1,d(M, g₃₁)_{trace to Z}, L^µ+1/2(∂M, g₃₂)_{trace to Z}, L^µ_cl(Z).

(3.107)

The weight data g_ij correspond to the weights in (3.106), namely, g₁₁:= g = (γ, γ − µ), g₁₂:= (γ − 1/2, γ − µ), g₁₃:= (−, γ − µ)

g₂₁:= (γ, γ − 1/2 − µ), g₂₂:= (γ − 1/2, γ − 1/2 − µ), g₂₃:= (−, γ − 1/2 − µ), g₃₁:= (γ, −), g₃₂:= (γ − 1/2, −), g₃₃:= (−, −)

(3.108) where “ − ” indicates that the corresponding space has no weight.

The following definition of a space of BVPs on a compact manifold M with boundary ∂M and edge Z employs tuples of order reducing operators both on ∂M and on Z. Beginning with Z we refer to the following well-known fact, cf. for instance, [32, Remark 1.2.27].

Lemma 3.32. For every ν ∈ R there is an R^ν ∈ L^ν_cl(Z) which induces isomorphisms R^ν : H^s(Z) → H^s−ν(Z) for all s ∈ R.

In the following we fix such operators in such a way that (R^ν)⁻¹= R^−ν for all ν.

Concerning ∂M we have the space

L^µ(∂M, e), µ ∈ R, e := (γ − 1/2, γ − 1/2 − µ) of edge pseudo-differential operators of order µ, with the weight data e.

Ellipticity, etc., are usually connected with additional Shapiro-Lopatinskij elliptic edge conditions of trace and potential type. However, there are order reducing elements without conditions of that kind.

Theorem 3.33. [20, Theorem 3.1.4] For every µ, γ ∈ R there exists a P^µ ∈ L^µ(∂M, e) which induces isomorphisms P^µ: H^s,γ−1/2(∂M ) → Hs−µ,γ−1/2−µ(∂M ) for all s ∈ R, and for the inverse we have P^−µ:= (P^µ)⁻¹∈ L^−µ(∂M, e⁻¹), e⁻¹:= (γ − 1/2 − µ, γ − 1/2).

Next we define spaces of boundary value problems on a compact manifold M with bound-ary ∂M and edge Z. To this end we extend the meaning of the matrix µ from the nor-malised form in (3.91) to the case

µ :=









µ (µ − 1/2 − αm)m=1,...,j1 (µ − 1 − πi)i=1,...,d1

t(µ + 1/2 + β_l)_l=1,...,j2 (µ + β_l− α_m)m=1,...,j1

l=1,...,j2

(µ−1/2+β_l−π_i)_i=1,...,d1

l=1,...,j2

t(µ + 1 + τ_k)_k=1,...,d2 (µ+1/2+τ_k−α_m)_m=1,...,j1

k=1,...,d2

(µ + τ_k− π_i)_i=1,...,d1

k=1,...,j2







 (3.109) for systems of orders (αm)m=1,...,j1, (βl)l=1,...,j2, (πi)i=1,...,d1, (τk)k=1,...,d2, and dimensions v := (j1, j2), d := (d1, d2). Moreover, we need a notation for the system g of weight shifts of 3 × 3 block matrix operators (3.112) below, consisting of

g₁₁= (γ, γ − µ), g₁₂= (γ − 1/2 − α_m, γ − µ)_m=1,...,j1, g₁₃= (−, γ − µ),

g₂₁= (γ, γ − 1/2 − µ − β_l)_l=1,...,j2, g₂₂= (γ − 1/2 − α_m, γ − 1/2 − µ − β_l)_m=1,...,j1

l=1,...,j2

, g₂₃= (−, γ − 1/2 − µ − β_l)_l=1,...,j2,

g₃₁= (γ, −), g₃₂= (γ − 1/2 − α_m, −)_m=1,...,j1, g₃₃= (−, −).

(3.110)

3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 63 Definition 3.34. Let M be a compact manifold with boundary ∂M and edge Z. Then

B^µ,d(M, g; w) (3.111)

for µ ∈ Z, d ∈ N, w := (j1, j2; d1, d2), is defined to be the space of all continuous block matrix operators

A:

H^s,γ(M )

⊕

⊕^j_m=1¹ H^{s−1/2−αm,γ−1/2−αm}(∂M )

⊕

⊕^d_i=1¹ H^s−1−πi(Z)

→

H^{s−µ,γ−µ}(M )

⊕

⊕^j_l=1² H^{s−1/2−µ−βl,γ−1/2−µ−βl}(∂M )

⊕

⊕^d_k=1² H^{s−1−µ−τk}(Z)

(3.112)

for s > d − 1/2, such that the entries of

A_nl:= diag (1, P^−β, R^−τ)A diag (1, P^α, R^π) (3.113) are (according to the position in the block matrix ) operators as in Remark 3.31. Here P^α :=

diag (P^α1, . . . , P^αj1), R^π := diag (R^π1, . . . , R^πd1), etc., are furnished by order reducing operators R^πd1 in the sense of Lemma 3.32 and Theorem 3.33.

By B^µ,d_{M +G}(M, g; w) (B^µ,d_G (M, g; w)) we denote the subspaces defined by the condition that the entries of (3.113) in the upper left 2 × 2 matrices belong to (3.105).

Remark 3.35. (i) The continuity (3.112) appears as a requirement in Definition 3.34.

However, this is an automatic consequence of the nature of the involved symbols. The reason for our formulation is to make the system of orders visible. In particular, if an element of B^µ,d(M, g; w) is continuous in the sense of (3.112) for some weights and particular smoothness, say, s₀, then we have continuity between the respective spaces for all s that are only restricted by the type d.

(ii) In Definition 3.34 we could avoid reductions of orders completely by looking at the individual entries that automatically belong to the normalised operator class, however with orders depending on the entry, according to (3.109).

The (j₁+ d₁) × (j₂+ d₂) lower right corners of operators in (3.111) live on ∂M with edge Z, and they belong to the corresponding edge calculus consisting of spaces

L^µ0(∂M, g⁰; w) (3.114)

where µ⁰indicates the lower right corner of the matrix (3.109) of orders. In similar notation, i.e., for unified orders µ and weight data g the operator classes (3.114) are well-known, cf.

[9], [32]. Here we freely employ the general technique from manifolds with edge without boundary, but we developed the elements anyway by explaining the spaces (3.93).

Remark 3.36. Also if Z is of codimension 1 in ∂M the operators (3.112) have a very rich structure. The upper left (1 + j2) × (1 + j1) corners restricted to Y∓\ Z belong to B^µ,d(Y∓, v) in the sense of Definition 3.16, modulo some transmission operators that map distributions from int Y∓to int Y±. Moreover, the lower right (j₁+d₁)×(j₂+d₂) corner may be interpreted as transmission problems over ∂M with transmission conditions referring to the interface Z. those are operators in the edge algebra over ∂M as a manifold with edge Z, however with different orders in the entries, similarly as Douglis-Nirenberg orders. The lower right d1× d₂ corner is a matrix of classical pseudo-differential operators on Z with the system (µ + τ_k− π_i)_i=1,...,d1

k=1,...,d2

of Douglis-Nirenberg orders..

Remark 3.37. (i) By notation we have B^µ,d(M, g) = B^µ,d(M, g_nl; w_nl) for the nor-malised order µ as in (3.91), g_nl as in (3.107), and dimension data w_nl = (v_nl, d_nl), v_nl= (1, 1), d_nl= (1, 1).

(ii) In explicit computations we will also admit operator matrices that can be rephrased by interchanging rows and columns into the form (3.112). For instance, the operator (2.11) turns to the form (3.112) by rearranging the components in such a way that the spaces over Z are at the end the respective direct sum.

Let us now formulate the principal symbolic structure of operators A in the normalised class B^µ,d(M, g), cf. Definition 3.29,

σ(A) = (σ_ψ(A), σ_∂(A), σ∧(A)). (3.115) First we have A11 ∈ L^µ_cl(int(M \ ∂M )) which gives us σ_ψ(A) = σ_ψ(A11), the standard homogeneous principal symbol of order µ of the respective classical operator. Moreover, (A_ij)_i,j=1,2 ∈ B^µ,d(M \ Z) gives rise to σ_∂(A) = σ_∂((A_ij)_i,j=1,2), the twisted homogeneous principal boundary symbol of order µ, explained before in Subsection 3.2.

Remark 3.38. By virtue of the edge-degeneration of the involved operators the symbol σψ(A) close to Z in the splitting of variables (3.98) with covariables (ρ, ξ, ζ) has the form

σ_ψ(A)(r, x, z, ρ, ξ, ζ) = r^−µσ˜_ψ(A)(r, x, z, rρ, ξ, rζ)

for a function ˜σ_ψ(A)(r, x, z, ˜ρ, ξ, ˜ζ) that is homogeneous in ( ˜ρ, ξ, ˜ζ) 6= 0 of order µ, smooth in r up to 0 and smooth in x up to t = 0 in local coordinates x = (x⁰, t) close to the boundary ∂N , cf. the formulas (3.76) and (3.73).

We call

˜

σ_ψ(A)(r, x, z, ρ, ξ, ζ) (3.116)

the reduced interior symbol of A. Moreover, the boundary symbol σ∂(A) close to Z in the splitting of variables (3.100) with the covariables (ρ, ξ⁰, ζ) has the form

σ_∂(A)(r, x⁰, z, ρ, ξ⁰, ζ) = r^−µσ˜_∂(A)(r, x⁰, z, rρ, ξ⁰, rζ)

for an operator function ˜σ_∂(A)(r, x⁰, z, ˜ρ, ξ⁰, ˜ζ) that is twisted homogeneous in ( ˜ρ, ξ⁰, ˜ζ) 6= 0 of order µ and smooth in r up to 0, cf. the formulas (3.74) and (3.77). We call

˜

σ_∂(A)(r, x⁰, z, ρ, ξ⁰, ζ) (3.117) the reduced boundary symbol of A.

Finally we have the twisted homogeneous principal edge symbol σ∧(A) which is deter-mined as a sum over the corresponding edge symbols σ∧(a_j)(z, ζ), cf. the formula (3.96), interpreted as operator functions (3.97) parametrised by (z, ζ) ∈ T^∗Z \ 0. The edge symbol σ∧(A) is a family of continuous operators

σ∧(A)(z, ζ) :

K^s,γ(N^∧)

⊕

Ks−1/2,γ−1/2((∂N )^∧)

⊕ C

→

K^{s−µ,γ−µ}(N^∧)

⊕

Ks−1/2−µ,γ−1/2−µ((∂N )^∧)

⊕ C

3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 65 for s > d − 1/2, ζ 6= 0. The upper left 2 × 2 corner represents a family of operators in the cone algebra of (pseudo-differential) boundary value problems over N^∧, cf. [27], [28]. As such every entry has the symbolic structure from the cone calculus, namely, the interior symbol, only depending on the upper left corner, the boundary symbol, depending on the full 2 × 2 matrix, moreover, the associated reduced versions, both of the interior and the boundary symbols, and finally the 2 × 2 matrix of conormal symbols. These symbols will be referred to as the subordinate symbols, determined by σ∧(A)(z, ζ). In particular, we have the subordinate conormal symbols

σ_cσ∧(A_ij)(z, w)

for i, j = 1, 2, cf. [28, Subsection 3.1.37]. The conormal symbols are considered on weight lines determined both by the dimension of the respective cone as well as on the weights in the involved spaces. The latter weights and dimensions form a matrix of pairs where the relevant entries in the first row are (γ, n) and γ − 1/2, n − 1. The rule is to let w vary on Γdim{base}+1

2 −weight, which means on Γⁿ⁺¹

2 −γ for the first entry and Γⁿ⁻¹⁺¹

2 −(γ−1/2) = Γⁿ⁺¹

2 −γ for the second one. In other words, our dimension/weight convention gives rise to σcσ∧(A)(z, w) as a family of continuous operators

σ_cσ∧(A)(z, w) :

H^s(int N )

⊕ H^s−1/2(∂N )

→

H^s−µ(int N )

⊕ H^s−1/2−µ(∂N ) for all w ∈ Γⁿ⁺¹

2 −γ.

The following theorem will state the composition behaviour of operators in the spaces of Definition 3.16. To this end we fix the systems of involved orders and weights. Those form matrices like (3.109) and (3.110), and in compositions we assume that the weight data g and h are fixed in such a way that the weights h from the first factor indicating weights in the image fit to those of g from the second factor, indicating weights in the domain.

For instance, if the first operator has the system ν of orders, the second one µ, for upper left corners ν and µ, respectively, we assume

h00:= (γ, γ − ν), g₀₀:= (γ − ν, γ − (µ + ν)),

cf. the notation in (3.110). Then the upper left corner of the resulting weight data matrix g ◦ h is just (γ, γ − (µ + ν)). The other entries (g ◦ h)_ij are defined in an analogous manner.

In addition, if the factors of a multiplication have the systems of orders ν and µ then the multiplication gives rise to a new system of orders that we denote by µ ◦ ν.

The algebraic structure is analogous to the composition of matrices A := Λ₁A⁰Λ⁻¹₂ and B := Λ₂B⁰Λ⁻¹₃ where A⁰ is an N × K- matrix and B⁰ a K × M matrix and

Λ1 = diag (L^λ1, . . . , L^λN), Λ2= diag (R^ρ1, . . . , R^ρK), Λ3= diag (T^τ1, . . . , T^τM) for invertible elements L^λj, R^ρk, T^τm. Then we obtain AB = Λ₁A⁰B⁰Λ₃. Now if A⁰ is interpreted as a matrix of operators of order µ and B⁰operator of order ν and the diagonal matrices as operators of order λj, ρk, τm, respectively. If the lk th entry alkof A is of order

Λ1 = diag (L^λ1, . . . , L^λN), Λ2= diag (R^ρ1, . . . , R^ρK), Λ3= diag (T^τ1, . . . , T^τM) for invertible elements L^λj, R^ρk, T^τm. Then we obtain AB = Λ₁A⁰B⁰Λ₃. Now if A⁰ is interpreted as a matrix of operators of order µ and B⁰operator of order ν and the diagonal matrices as operators of order λj, ρk, τm, respectively. If the lk th entry alkof A is of order

在文檔中 The edge algebra structure of the Zaremba problem (頁 52-67)