The edge algebra structure of boundary value problems with respect to the

:

Hs−1/2,s−1/2(Y₊)

⊕

H^s−1/2(Z, C^L(s−1/2))

→

Hs−3/2,s−3/2(Y₊)

⊕

H^s−3/2(Z, C^L(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

A_M = ΠM⊕ id₂
A_M ΞD⊕ id₁
:

H^s,s(G)

⊕ H^s(Z, C^{N (s)})

⊕

H^s−1/2(Z, C^L(s−1/2))

⊕ H^s−1/2(Z, C^d1)

→

H^s−2,s−2(G)

⊕

H^s−2(Z, C^{N (s−2)})

⊕

Hs−1/2,s−1/2(Y−)

⊕

H^s−1/2(Z, C^L(s−1/2))

⊕

Hs−3/2,s−3/2(Y₊)

⊕

H^s−3/2(Z, C^L(s−3/2))

⊕ H^s−3/2(Z, C^d2)

.

(2.11) Note that A_M 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 A_D, A_Nas 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

B^µ,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 = (A_ij)_i,j=1,2,3where (A_ij)_i,j=1,2restricts to (2.13) as a subcalculus of the operator spaces B^µ,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

(σ_ψ(A), σ_∂(A))

of interior and boundary symbols where σ_ψ(A) = σ_ψ(A₁₁), A₁₁ ∈ L^µ_cl(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

(σ⁰_ψ(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 L^µ_cl(Z), more precisely, those are block matrices of such operators, according to the above-mentioned dimensions.

The fact that A_D and A_N 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 × R^q,

S¹₊= S¹∩ {x_n≥ 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 σ_ψ(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 (σ_ψ(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⁻¹_D 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 A_NA⁻¹_D 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 σ⁰_ψ, 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⁻¹_D . Moreover, the operator (2.10) is the lower right 2 × 2 submatrix of (2.7).

Since S is σ⁰_ψ-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 A_D(γ), first given for γ = s by (2.3) and then realised as an operator (3.131) in weighted spaces, belongs to B^µD,1(G, g_D; w_D) where µ_D, g_D, are defined by (3.135), (3.136) and wD by (3.137). The operator AD(γ) is (σ_ψ, σ_∂, σ∧)-elliptic, and we have P_D(γ) = A_D(γ)⁻¹∈ B^−µD,0(G, g⁻¹_D ; w⁻¹_D ).

(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 B^µN,2(G, g_N; w_N) where µ_N, g_N, are defined by (3.138), (3.139) and w_N by (3.140). The operator A_N(γ) is (σ_ψ, σ_∂, σ∧ )-elliptic, and for the parametrix we have PN(γ) ∈ B^−µN,0(G, g⁻¹_N ; w⁻¹_N ).

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) = (σ_ψ(A0), σ_∂(A0))

that we express in suitable coordinates. First we have σ_ψ(A0)(ξ) = −|ξ|² every where on G the Riemannian metric is induced by the ambient Euclidean space Rⁿ. Concerning σ_∂(A₀) = _σ^σ∂(∆)

∂(T0)
we refer to coordinates in the half-space Rⁿ+ 3 (y, t) and covariables (η, τ ). Here

σ∂(∆)(η) = −|η|²+ ∂²

∂t² : H^s(R+) → H^s−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) : H^s(R+) →

H^s−2(R+)

⊕ C

(2.20)

for every η 6= 0. Next we look at ˜A₀ which contains A₀ as a submatrix, and we set σ_ψ( ˜A₀)(ξ) = σ_ψ(A₀)(ξ) = σ_ψ(∆)(ξ), σ_∂( ˜A₀)(η) = σ_∂(A₀)(η). (2.21) In the next step we pass to A_D, defined in the formula (1.24). On the level of the symbols we write

σ_ψ(AD) = σ_ψ( ˜A₀) and

σ_∂(A_D) = σ_∂−(A_D), σ_∂+(A_D)

for σ_∂±(A_D)(η) =σ_∂(∆)(η) σ_∂(D±)



where σ_∂(D±) refers to ± side. Here we choose the coordinates (y, x_n) = (z, x_n−1, x_n) 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

σ_ψ(∆)(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) : H^s(R+) → H^s−2(R+), for r > 0, (ρ, ζ) 6= 0, (2.24)

˜

σ_∂±(∆)(ρ, ζ) = −ρ²+ ∂_ϕ²− |ζ|² : H^s(R+) → H^s−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_∂(A_D)(r, ρ, ζ) = σ_∂−(∆) γ₀



,σ_∂+(∆) γ₀

!

: H^s(R+) →

H^s−2(R+)

⊕ C

(2.26)

for r > 0, (ρ, ζ) 6= 0, and

˜

σ_∂(A_D)(r, ρ, ζ) =  ˜σ_∂−(∆) γ0



, ˜σ_∂+(∆) γ0

!

: H^s(R+) →

H^s−2(R+)

⊕ C

. (2.27)

up to r = 0, (ρ, ζ) 6= 0.

It remains to recognise the principal edge symbol σ∧(A_D)(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 B^µD,1(M, g_D; w_D). 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 (xn−1, xn) half-plane coming from A0 locally expressed in Rⁿ⁻²z × R²+ 3 (z, x_n−1, xn), namely,

σ(∆)(ζ) := −|ζ|²+ ∆xn−1,xn : H^s(R²+) → H^s−2(R²+), (2.28) together with the trace operator t0 : H^s(R²+) → H^s−1/2(Rxn−1), R²+= {(xn−1, xn) ∈ R² : x_n> 0}.

2 MELLIN-EDGE QUANTISATIONS 29 Remark 2.4. The notation σ in (2.28) as well as in the subsequent considerations has an analogous meaning as edge symbols σ∧ in the operator spaces considered in Defini-tion 3.34 below. The modified notaDefini-tion here is chosen since we did not yet translate the occurring standard Sobolev spaces into weighted spaces as we do later on by using H^s(R±) ∼= K^s,s(R∓) ⊕ C^L(s) coming from Proposition 3.6 and the second relation in (3.10) together with V^s(R∓) ∼= C^L(s), and similarly for H^s(R), and Proposition 3.9 (ii).

The meaning of t₀ is analogous to T₀, indicating the Dirichlet condition, here in the case of the parameter-dependent operator (2.28).

For the proof of the σ∧-ellipticity we need the following result.

Proposition 2.5. The operators

Proof. We first consider the case j = 0 and show that the family of operators

α²− ∂_x² is an isomorphism for s > 3/2. We assume α > 0 and write

`<sub>+</sub>(ξ<sub>n</sub>) := α + iξ<sub>n</sub>, `−(ξ_n) := α − iξ_n, α²− ∂_x²

are isomorphisms. The isomorphism (2.31) is a consequence of the Paley-Wiener Theorem, see, e.g., [10], while the isomorphism (2.32) holds since op⁺(`+) : S(R+) → S(R+) is

is an isomorphism. Let us compute the shape of the inverse. First we have

(op⁺(`−))<sup>−1</sup>= op<sup>+</sup>(`⁻¹₋ ) (2.34) since `−is a minus symbol, see also the terminology in Eskin’s book [10]. Moreover, observe

that op⁺(`+)

The relations (2.33), (2.34) and (2.35) allows us to express the inverse of (2.30), namely

α²− ∂_x² the boundary symbolic calculus of Boutet de Monvel’s algebra, cf. [32, Theorem 4.3.17], characterised by the mapping properties g, g^∗ : L²(R+) → S(R+).

Let us now prove the assertion for j = 1. By virtue of Proposition 2.5 for j = 1 the operator

α²−∂²_xn

γ0
: S(R+) →

S(R+)

⊕ C

is an isomorphism for every real α 6= 0, or, equivalently,

α²−∂²_xn

2 MELLIN-EDGE QUANTISATIONS 31

The next steps in the proof of Theorem 2.3 is that we construct analogues of the iso-morphisms of A_d, ˜A₀, D₀, A_D and A_D, cf. the formulas (1.20), (1.22), (1.23), (1.24) and (2.3), now starting with σ(A₀)(ζ) rather than A₀. The resulting operator functions will be denoted by

σ(A_d)(ζ), σ( ˜A₀)(ζ), σ(D₀)(ζ), σ(A_D)(ζ), σ∧(A_D)(ζ). (2.36) We shall see that the operators (2.36) are also isomorphisms which finally shows the σ∧ -ellipticity of AD. We form

σ(A_d)(ζ) : H^s(R²+) →

Using a corresponding analogue of Lemma 1.2 we have

σ(D₀)(ζ) :=

Concerning the decomposition R = R−∪ R+ we employ the existence of an isomorphisms

r⁻ σ(d−)

σ(D∓) := r^∓σ(T₀), which follows from a_D(ζ) by applying the half-space analogue of Proposition 3.9 (ii) below. Moreover, the half-axis analogue of Lemma 1.1 gives us an isomorphism

The proof of part (ii) of Theorem 2.3 is of analogous structure. Instead of (2.20) we have

σ_∂(A₁)(η) =σ_∂(∆)

The relations (2.21) hold in the same form for ˜A₁ and A₁, respectively. Next we consider A_N defined in (2.4) where

σ_ψ(AN) := σ_ψ( ˜A₁) are isomorphisms, similarly as (2.26), (2.27) where it suffices to replace γ0 by γ1.

In the final step we verify that σ∧(A_N)(z, ζ) is a family of isomorphisms for ζ 6= 0, as is required in the σ∧-ellipticity of A_N. Similarly as in the proof for (i) in Theorem 2.3 we consider operator functions

σ(An)(ζ), σ( ˜A₁)(ζ), σ(D1)(ζ), σ(AN)(ζ), σ∧(AN)(ζ) (2.37) parallel to A_n, ˜A₁, D₁, A_N, A_N. while A_n, ˜A₁, A_N, A_Nare not isomorphisms but only Fred-holm operators, the σ∧-ellipticity of ANrequires that the operators (2.37) are all isomor-phisms for ζ 6= 0. The operators

σ(An)(ζ) : H^s(R²+) →

2 MELLIN-EDGE QUANTISATIONS 33 give rise to the isomorphism

σ(A_N)(ζ) = weighted spaces of the edge symbolic calculus, as in the first part of the proof.

Corollary 2.6. The operator A_D, first given as an isomorphism (2.3) induces isomor-phisms

A similar observation is true of ANin (2.5), cf. also the formulas (3.131) and (3.133) below, however, in terms of Fredholm operators of index zero rather than isomorphisms. In fact, it suffices to use what we know from the above considerations when we replace everywhere s by γ and then employ Remark 3.35 (i) below.

Our next objective is to study the edge symbol of the operator RND in more detail.

Analogously as (1.28) we form the operator functions

σ(A_M)(ζ) :=

where σ(N₊)σ(C₊)(ζ) : H^s−1/2(R+) → H^s−3/2(R+) is the (twisted homogeneous) edge (or boundary) symbol of the operator N+C+, cf. (1.30). In the following theorem we verify the results on the Zaremba problem A_M which is obtained by interpreting A_m in weighted edge Sobolev spaces rather than the spaces in (1.4) and enlarging to a block matrix operator by adding additional interface conditions of trace and potential type. The method in principle reminds of the method of making elliptic operators on a manifold with boundary to a Fredholm operator by adding trace and potential condition on the boundary. The new point here is that we have our Zaremba boundary conditions anyway but add extra edge conditions along the interface. The new achievement is also that everything is done within the calculus of operators on manifolds with edge and boundary.

We now characterise our Zaremba problem with additional edge conditions in terms of the operator spaces of Definition 3.45 below.

Theorem 2.7. The Fredholm operator AM(γ), defined in (2.11) belongs to the space B^µM,2(G, g_M; w_M) where µ_M, g_M are defined by (3.143), (3.144) and w_M by (3.145).

The operator A_M(γ) is (σ_ψ, σ_∂, σ∧)-elliptic, and it has a parametrix A⁽⁻¹⁾_M (γ) ∈ B^−µM,0(G, g⁻¹; w⁻¹).

Proof. The first assertion employs the characterisation of operators of the form (3.128) in Subsection 3.5. This has to be combined with the observation that operators belonging to (3.126) form a substructure of (3.121). In fact, our operator AM(γ) is obtained by means of several algebra operations within (3.121). The ellipticity of AM(γ) follows from the Fredholm property which is obtained by construction, using the Fredholm property of the involved operator (1.43) and the σψ-ellipticity, cf. Theorem 3.44 and Remark 3.46.

3 Boundary value problems on manifolds with edge

3.1 Cutting and pasting, continuation

Let us prove Lemma 1.2 and express the inverse of the cutting operator (1.10) for s − 1/2 instead of s. It will be convenient to replace (1.10) by

r⁻ f−

r⁺ f+

 :

H^s−1/2(Y )

⊕ V^s−1/2(Y )

→

H^s−1/2(int Y−)

⊕ H^s−1/2(int Y₊)

, (3.1)

using the isomorphism

Ks−1/2: H^s−1/2(Z, C^L(s−1/2)) → V^s−1/2(Y ), (3.2) V^s−1/2(Y ) := im Ks−1/2, for the potential operator Ks−1/2 from Lemma 1.1 (ii).

In other words, if we consider (3.1) rather than (1.10) we set d∓:= f∓◦ Ks−1/2. We also observe the trace operator

Ts−1/2: V^s−1/2(Y ) → H^s−1/2(Z, C^L(s−1/2))

3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 35 with the property Ts−1/2Ks−1/2 = id_Hs−1/2(Z,C^L(s−1/2)). It suffices to compute the inverse of the cutting operator (1.10), namely,

e⁻_d e⁺_d

The operator (3.3) plays the role of a pasting of the Sobolev spaces H^s−1/2(int Y∓) to H^s−1/2(int Y ) on the expense of the space H^s−1/2(Z, C^L(s−1/2)) of traces at the interface.

Considerations of that kind are useful for constructing cutting and pasting isomorphisms also in other contexts. Therefore, we replace the involved spaces by more general vector spaces. The direct sums (1.11) will be abbreviated by

H = H_θ⊕ V, H^∓= H_θ^∓⊕ V^∓

. For our purposes it

suffices to make a specific choice of such isomorphisms and not to characterise the most general form which could be done as well. We set f∓ := α^∓r^∓₁ for some α^∓ ∈ C. Since

is already an isomorphism it suffices to consider

r⁻₁ α⁻r⁻₁

and to determine possible α^∓. We employ the standard algebraic fact that a block matrix operator (3.5) is an isomorphism exactly when

(r⁻₁ α⁻r⁻₁) : V

⊕ V

→ V⁻ (3.6)

is surjective and (3.7) holds if and only if

α⁻6= α⁺.

Next we express the inverse of (3.5). We employ the fact that the inverse of a block matrix isoporphism

between direct sums of Hilbert spaces where also a : H → ˜H is an isomorphism has the form The operator b has the form

b = α⁺r⁺₁ − r⁺₁e⁻₁α⁻r⁻₁ = (α⁺− α⁻)r⁺₁.

We obtain for the abstract analogues of the ingredients of (3.2) a potential operator K : L → H which induces an isomorphism K : L → V and a trace operator T : H → L which restricts to an isomorphism T : V → L. Then our cutting operator (1.10) is translated to

r⁻ d−

3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 37

with the inverse





e⁻_θ 0 e⁺_θ 0 0 γ⁻e⁻₁ 0 γ⁺e⁺₁ 0 β⁻Te⁻₁ 0 β⁺Te⁺₁



. In other words, the operators d∓ have the meaning

d∓=

 0

α^∓r^∓₁K

 . For abbreviation we write

r⁻ d−

r⁺ d₊

−1

=e⁻_d e⁺_d c− c₊



for e^∓_d =e^∓_θ 0 0 γ^∓e^∓₁



, c∓= (0 β^∓Te^∓₁). (3.8) There is an elementary variant of Lemma 1.2 as follows.

Lemma 3.1. Let s ∈ R, s > 1/2, s − 1/2 /∈ N and L(s) = #{l ∈ N : l + 1/2 < s}. Then there is an isomorphism

r⁻ d−

r⁺ d₊

 :

H^s(R)

⊕ C^L(s)

→

H^s(R⁻)

⊕ H^s(R+)

(3.9)

where r^∓ : H^s(R) → H^s(R∓) are the restriction operators and d∓ potential symbols from Boutet de Monvel’s version of transmission problems.

Proof. Let s > 1/2, s − 1/2 /∈ N, and set

H_θ^s(R) := {u(t) ∈ H^s(R) : ∂t^ju(0) = 0 for all j < s − 1/2}, H_θ^s(R^∓) := r^∓H_θ^s(R).

Then H_θ^s(R∓) =u(t) ∈ H^s(R∓) : ∂_t^ju(0) = 0 for all j < s − 1/2 . A cut-off function on R is any real-valued ω ∈ C₀^∞(R) such that ω = 1 in a neighbourhood of t = 0. Analogously we speak about cut-off functions on R∓. For V^s(R) := P

0≤j<L(s)ω(t)_j!¹c_jt^j : c_j ∈ C and V^s(R^∓) = r^∓V^s(R) we have direct decompositions

H^s(R) = H_θ^s(R) ⊕ V^s(R), H^s(R∓) = H_θ^s(R∓) ⊕ V^s(R∓) (3.10) and H_θ^s(R) = H_θ^s(R−) ⊕ H_θ^s(R+). Setting

L(s) := #{j ∈ N : j < s − 1/2} (3.11) we have L(s) = dim V^s(R) = dim V^s(R^∓). Choose an isomorphism

k : C^L(s) → V^s(R), (3.12)

k : (c₀, . . . , c_L(s)−1) →P

0≤j<L(s)ω(t)_j!¹c_jt^j, and form

t := k⁻¹, tv := {γ_jv : 0 ≤ j < s − 1/2}. (3.13) Moreover, write r^∓= diag (r^∓_θ, r^∓₁) for the restriction operators r^∓_θ = r^∓|_Hs

θ(R) : H_θ^s(R) → H_θ^s(R∓) and r^∓₁ = r^∓|_Vs(R) : V^s(R) → V^{∼= s}(R∓). Then k^∓ = r^∓₁k : C^L(s) → V^s(R∓) are potential operators with respect to the respective ∓ half-axis. Because of the considerations at the beginning of this subsection the proof is complete when we replace the involved spaces H, Hθ, V, H^∓, H_θ^∓, V^∓ by H^s(R), H_θ^s(R), V^s(R), H^s(R∓), H_θ^s(R∓), V^s(R∓).

Definition 3.2. (i) A Hilbert space H is said to be endowed with a group action κ = {κ_δ}_δ∈R+ if κ_δ: H → H, δ ∈ R+, are isomorphisms, κ_δκυ= κ_δυ for δ, υ ∈ R+, and δ → κ_δh defines an element of C(R+, H) for every h ∈ H.

(ii) A Fr´echet space E written as a projective limit E = lim←−j∈NE^j of Hilbert spaces E^j continuously embedded in E⁰ for all j, is said to be endowed with a group action κ = {κ_δ}_δ∈R+ if κ is a group action on E⁰ as well as on E^j via κ|_Ej for every j.

For instance, we have S(R+) = S(R)|_R+ = lim←−j∈NH^j;j(R+) for H^j;j(R+) := hti^−jH^j(R+), the group action defined by (κ_δu)(t) := δ^1/2u(δt) extends to S(R+).

The following constructions on Hilbert spaces H with group actions and spaces of L(H, ˜ H)-valued symbols have natural generalisations to the case of Fr´echet spaces E, ˜E with group action, see [32, page 95].

By

W^s(R^q, H), (3.14)

s ∈ R, we denote the completion of S(R^q, H) with respect to the norm kuk_Ws(R^q,H)=

nZ

hηi^2skκ⁻¹_hηiu(η)kˆ ²_Hd¯η o1/2

. (3.15)

We have W^s(R^q, H) ⊂ S⁰(R^q, H) = L(S(R^q), H). For an open set Ω ⊆ R^q we define W_comp^s (Ω, H) as the set of all u ∈ W^s(R^q, H) such that supp u is a compact subset of Ω. Moreover, W_loc^s (R^q, H) is defined to be the set of those u ∈ D⁰(Ω, H) such that ϕu ∈ W_comp^s (Ω, H) for every ϕ ∈ C₀^∞(Ω). Clearly the W^s-spaces depend on the choice of the group action κ. If necessary we write W^s(R^q, H)_κ, etc.

We frequently employ the fact that for H := H^s(R^d) and

(κδv)(x) = δ^d/2v(δx), δ ∈ R+, (3.16) we have

W^s(R^q, H^s(R^d)) = H^s(R^q+d).

Another example is the case H := H^s(R+) (= H^s(R)|R+) with the group action

(κδu)(t) := δ^1/2u(δt), δ ∈ R+. (3.17) Then, setting H^s(R^q+1+ ) = H^s(R^q+1)|_R^q_×R+ we have

W^s(R^q, H^s(R+)) = H^s(R^q+1₊ ).

In the case H = C we usually take the trivial group action κδ = id_C, δ ∈ R+. Note that if we take another group action κ = {κ_δ}_δ∈R+ defined by κ_δ : C → C for κδ := δ^σid_C, for some σ ∈ R we obtain W^s(R^q, C)κ = H^s−σ(R^q) for every s ∈ R. In fact, according to (3.15) we have kuk_W^s_(R^q_,C)κ =

nR hηi^2skhηi^−σu(η)kˆ ²_Hd¯η o1/2

.

Let H and ˜H be Hilbert spaces with group action κ = {κ_δ}_δ∈R+ and ˜κ = {˜κ_δ}_δ∈R+, respectively. Then

S^µ(Ω × R^m; H, ˜H) (3.18)

3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 39 for µ ∈ R, Ω ⊆ R^q open, is the set of all a(y, η) ∈ C^∞(Ω × R^m, L(H, ˜H)) such that

k˜κ⁻¹_hηi{D^α_yD^β_ηa(y, η)}κ_hηik_{L(H, ˜H)}≤ chηi^µ−|β|

for all (y, η) ∈ K × R^m, K b Ω, α ∈ N^q, β ∈ N^m, for constants c = c(α, β, K) > 0. If necessary we write

S^µ(Ω × R^m; H, ˜H)_κ,˜κ (3.19) rather than (3.18), since the symbol spaces depend on the group actions. The elements of (3.18) are called symbols of order µ with twisted symbolic estimates. Let

S^(µ)(Ω × (R^m\ {0}); H, ˜H)

denote the space of all a_(µ)(y, η) ∈ C^∞(Ω × (R^m\ {0}), L(H, ˜H)) such that

a_(µ)(y, δη) = δ^µ˜κδa_(µ)(y, η)κ⁻¹_δ (3.20) for all δ ∈ R+, (y, η) ∈ Ω × (R^m \ {0}). Then, if χ(η) is an excision function (i.e., χ ∈ C^∞(R^m), χ(η) = 0 for 0 < |η| < c0, χ(η) = 1 for |η| > c1 for some 0 < c0 < c1) then we have χ(η)a_(µ)(y, η) ∈ S^µ(Ω × R^m; H, ˜H).

Let

S_cl^µ(Ω × R^m; H, ˜H) (3.21)

denote the set of all classical symbols, i.e., those a in (3.21) such that there are a_(µ−j) ∈ S^(µ−j)(Ω × (R^m\ {0}); H, ˜H), j ∈ N, with a −PN

j=0χa_(µ−j) ∈ S^{µ−(N +1)}(Ω × R^m; H, ˜H) for all N ∈ N, where χ is any excision function. The dimensions q, m are arbitrary. In particular, we have symbols with parameter λ ∈ R^l when we replace η ∈ R^m by (η, λ) ∈ R^m+l. Also the variables y ∈ Ω can be modified. For Ω × Ω in place of Ω we often write (y, y⁰) rather than y. Symbols a(y, y⁰, η, λ) ∈ S_(cl)^µ (Ω × Ω × R^q+l; H, ˜H) are also called double symbols with parameter. Subscript “(cl)” is used when a consideration is valid in the classical and the general case.

Remark 3.3. Given a(y, η) ∈ S^µ(R^q × R^q; H, ˜H)_κ,˜κ the associated pseudo-differential operator Op_y(a) induces continuous maps Op_y(a) : W^s(R^q, H)_κ → W^s−µ(R^q, ˜H)_κ˜, s ∈ R, provided that a satisfies some conditions for |y| → ∞, for instance, in connection with an analogue of the Calder´on-Vaillancourt theorem, see [16], or [38]. Moreover, if a(y, η) ∈ S^µ(Ω × R^q; H, ˜H)_κ,˜κ, then there is a similar continuity in comp/loc-spaces, i.e., Op_y(a) : W_comp^s (Ω, H)κ→ W_loc^s−µ(Ω, ˜H)˜κ.

In our applications it makes sense to slightly modify the notation of orders, namely, if the symbols are 2 × 2 matrices p(y, η) = a c

b d

 (y, η) :

H

⊕ L

→ H˜

⊕ L˜

for Hilbert spaces H, L, ˜H, ˜L with group actions κ, β, ˜κ, ˜β, respectively. Then, for a ∈ S^µ(Ω × R^m; H, ˜H)_κ,˜κ, c ∈ S^µ−(Ω × R^m; L, ˜H)_β,˜κ, b ∈ S^µ+(Ω × R^m; H, ˜L)_{κ, ˜β}, d ∈ S^µ(Ω × R^m; L, ˜L)_{β, ˜β} for the matrices of orders

µ =

 µ µ −  µ +  µ



, µ^∗ :=

 µ µ +  µ −  µ



(3.22)

we write

and (y, η)-wise formal adjoints will refer to µ^∗. The above-mentioned constructions and results have straightforward generalisations to the case of such matrices. For instance, under suitable conditions on the y-dependence for large |η|, for Ω := R^q, m := q, we have continuous operators

In Subsection 3.4 below we also employ 3 × 3 block matrix symbols with schemes of orders

µ :=

in obvious meaning. Let us now replace the operators in (3.12) and (3.13) by η-dependent families, namely, This gives us operator-valued symbols

k(η) ∈ S_cl⁰(R^qη; C^L(s), H^s(R)) (3.26) with C^L(s) being endowed with id and H^s(R) with κδ: v → δ^1/2v(δt), δ ∈ R+, and

t(η) ∈ S_cl⁰(R^q; H^s(R), C^L(s)) (3.27) for s and L(s) as in (3.11), cf. also (3.21). Here we have t(η)k(η) = id_CL(s) for every η ∈ R^q. Thus k(η)t(η) takes values in projections H^s(R) → H^s(R). This gives us an η-dependent variant of (3.9), namely, (3.12) and (3.24). We then have

c(η) ∈ S_cl⁰

3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 41 for p_∓,(0)(η) = α^∓P

0≤j<L(s) 1

j!c_j|η|^1/2ω(t|η|)(t|η|)^j|_R∓. For q := n − 1 and Op = Op_y we now obtain the following cutting/pasting result for standard Sobolev spaces in Rⁿ. Theorem 3.4. The operator

Op(c) :

H^s(Rⁿ)

⊕ H^s(Rⁿ⁻¹, C^L(s))

→

H^s(Rⁿ−)

⊕ H^s(Rⁿ+)

(3.30)

is an isomorphism for the symbol (3.28) for s and L(s) defined as in Lemma 1.2.

Proof. The symbol (3.28) is η-wise an isomorphism and we have the relation (3.29). Thus Op(c) gives rise to the isomorphism (3.30), and we have Op(c)⁻¹ = Op(c⁻¹).

Remark 3.5. The operator (3.30) has the formr⁻ d−

r⁺ d+



where r^∓ are now the restric-tion operators to Rⁿ∓ and d∓= Op(p∓).

The space H^s(Rⁿ) can be written as a direct sum

H^s(Rⁿ) = H_θ^s(Rⁿ) ⊕ V^s(Rⁿ). (3.31) Here H_θ^s(Rⁿ) := ker Op(t) for the trace operator Op(t) : H^s(Rⁿ) → H^s(Rⁿ⁻¹, C^L(s)), with the symbol (3.27). Moreover, we have V^s(Rⁿ) := im Op(k) for the potential operator Op(k) : H^s(Rⁿ⁻¹, C^L(s)) → H^s(Rⁿ) with the symbol (3.26). In a similar manner we have direct decompositions

H^s(Rⁿ∓) = H_θ^s(Rⁿ∓) ⊕ V^s(Rⁿ∓) for H_θ^s(Rⁿ∓) := ker Op(t∓) (3.32) for the trace operator

Op(t∓) : H^s(Rⁿ∓) → H^s(Rⁿ⁻¹, C^L(s))

with the symbol t∓(η) ∈ S_cl⁰(R^q; H^s(Rⁿ∓), C^L(s)) obtained from (3.25) acting on functions in H^s(Rⁿ∓). Moreover,

V^s(Rⁿ∓) := im Op(k∓) for the potential operators

Op(k∓) : H^s(Rⁿ⁻¹, C^L(s)) → H^s(Rⁿ∓) with the symbols k∓(η) : (c₀, . . . , c_L(s)−1) → P

0≤j<L(s) 1

j!c_j[η]^1/2ω(t[η])(t[η])^j|_R∓. From the construction it follows that t(η)k(η) = id_CL(s)and hence Op(t)Op(k) = id_Hs(Rⁿ⁻¹,C^L(s)). Thus Op(k)Op(t) : H^s(Rⁿ) → H^s(Rⁿ) is a projection, namely,

Op(k)Op(t) : H^s(Rⁿ) → V^s(Rⁿ) (3.33) and ker(Op(k)Op(t)) = H_θ^s(Rⁿ), and 1 − Op(k)Op(t) : H^s(Rⁿ) → H_θ^s(Rⁿ) is the comple-mentary projection of (3.33), cf. also the decomposition (3.31).

Analogous relations hold with respect to Rⁿ∓. In particular, (3.32) can be interpreted as an isomorphism

(e^∓_θ Op(k∓)) :

H_θ^s(Rⁿ∓)

⊕

H^s(Rⁿ⁻¹, C^L(s))

→ H^s(Rⁿ∓) (3.34)

with e^∓_θ : H_θ^s(Rⁿ∓) → H^s(Rⁿ∓) being the canonical embedding.

Let H^s,γ(R+) for s, γ ∈ R denote the completion of C0^∞(R+) with respect to the norm kuk_Hs,γ(R+)=nZ

Γ1 2−γ

hwi^2s|M u(w)|²d¯wo1/2

,

d¯w = (2πi)⁻¹dw. Here Γβ := {w ∈ C : Re w = β}, M is the Mellin transform M u(w) =

Z ∞ 0

t^w−1u(t)dt

which is an entire function for u ∈ C₀^∞(R+) where M u|Γβ ∈ S(Γ_β) for every β ∈ R. We then define the space

K^s,γ(R+) :=ωu + (1 − ω)v : u ∈ H^s,γ(R+), v ∈ H^s(R+) ; (3.35) here ω is a cut-off function on the half-axis, i.e., ω is smooth and real-valued, and ω ≡ 1 close to t = 0, ω ≡ 0 for t > C for some C > 0.

Proposition 3.6. [31, page 264] We have an isomorphism H_θ^s(R+) ∼= K^s,s(R+) for every s > −1/2.

Note that H_θ^s(R+) for s > 1/2, s − 1/2 /∈ N coincides with the corresponding space in Lemma 1.1 (in the variant of R+ in place of M ).

The space K^s,γ(R+) is also a Hilbert space with group action (3.17) and, according to the general definition of W^s(R^q, H)-spaces, we have the spaces

W^s(R^q, K^s,γ(R+)) (3.36)

in R^q×R+, called edge spaces of smoothness s and weight γ. We identify Y+locally near Z with R^q× R+, with Z being interpreted as an edge. A partition of unity construction gives us the global edge spaces H^s,γ(Y₊) of smoothness s and weight γ. Recall that those are subspaces of H_loc^s (int Y+) that locally near Z coincide with (3.36). We have, in particular, H_θ^s(R^q× R+) = W^s(R^q, K^s,s(R+)) (3.37) whenever s > 1/2, s − 1/2 /∈ N, and

H_θ^s(R^q× R+) = {u ∈ H^s(R^q× R)|_R^q×R+ : D^α_y,tu|_t=0for all |α| < s − 1/2}, cf. also the notation in Lemma 1.1 (i).

Summing up the spaces H^s(int Y±) are locally near Z in the variables (z, x_n−1) ∈ R^q× R±,x_n−1 identified with W^s(R^q, H^s(R±)) where R^qcorresponds to a chart on Z. The spaces H_θ^s(int Y±) are locally near Z identified with W^s(R^q, K^s,s(R±)) for s > 1/2, s − 1/2 /∈ N.

In other words we have canonical identifications

H^s,s(Y±) = H_θ^s(int Y±), (3.38) cf. the formula (1.8). Analogously we set

3 BOUNDARY VALUE PROBLEMS ON MANIFOLDS WITH EDGE 43

H^s,s(Y ) = H_θ^s(Y ), for H_θ^s(Y ) := {u ∈ H^s(Y ) : D_x^αu|Z = 0 for all |α| < s − 1/2}

for Y = Y−∪ Y₊, Z = Y−∩ Y₊.

Let us now obtain similar local identifications for the spaces H^s(int G) = H^s(2G)|_{int G}. In this case Z is embedded in ∂G of codimension 2. Locally near Z we identify G with the half-space Rⁿ+ = {(x₁, . . . , x_n) ∈ Rⁿ : x_n > 0} and 2G with Rⁿ, and without loss of generality we identify Z with {(x₁, . . . , x_n) ∈ Rⁿ : x_n−1 = x_n = 0}. We also write (x1, . . . , xn) = (z, xn−1, xn) for z ∈ Rⁿ⁻². Let s ≥ 0, s − 1 /∈ N, and set

H_θ^s(R²) := {u ∈ H^s(R²) : D^α_xn−1,xnu|_{(xn−1,xn)=0} = 0 for all |α| < s − 1}, H_θ^s(R²+) := {u|_R²

+ : u ∈ H_θ^s(R²)}, R²+= {(xn−1, xn) ∈ R²: xn> 0}, endowed with the group action {κ_λ}_λ∈R+ given by (3.16) for d = 2.

In order to rephrase the latter spaces in a similar manner as in Proposition 3.6 we first formulate an analogue of the spaces K^s,γ(R+).

Define H^s,γ(R+× Rⁿ) for s, γ ∈ R as the completion of C₀^∞(R+× Rⁿ) with respect to the norm kuk_H^s,γ_(R+×Rⁿ₎ = n

R

Γn+1 2 −γ

R

Rⁿhw, ξi^2s|F_x→ξM_r→wu(w, ξ)|²d¯wd¯ξo1/2

, for d¯ξ = (2π)⁻¹dξ. Then, if X is any smooth closed manifold of dimension n a simple partition of unity construction with respect to a system of charts χ : U → Rⁿ on X allows us to pass to spaces H^s,γ(X^∧) for X^∧ = R+× X. More details may be found in [32, Section 2.1.4].

Moreover, for the unit sphere Sⁿ⊂ R¹⁺ⁿ_x˜ we set

K^s,γ((Sⁿ)^∧) := {ωu + (1 − ω)v : u ∈ H^s,γ((Sⁿ)^∧), v ∈ H^s(R¹⁺ⁿ)}.

where ω is a cut-off function in r = |˜x| ∈ R+ and 1 − ω as a function in 1 − ω(˜x) is defined as R¹⁺ⁿ.

Finally if X is a smooth closed manifold we introduce K^s,γ(X^∧) by using restrictions to R+× U for coordinate neighourhoods U on X and charts χ : U → V for open subsets of Sⁿ, more precisely, u ∈ K^s,γ(X^∧) is defined by the relations ϕu = χ^∗(ϕ1u1) for every ϕ ∈ C₀^∞(U ), ϕ = χ^∗ϕ₁, u₁∈ K^s,γ((Sⁿ)^∧), and any such χ.

If N is a smooth compact manifold with boundary embedded in X, dim N = dim X, then we define H^s,γ(N^∧) := {u|_R+×int N : u ∈ H^s,γ(X^∧)},

K^s,γ(N^∧) := {u|_R+×int N : u ∈ K^s,γ(X^∧)}. (3.39) Observe that

K^0,0(X^∧) = H^0,0(X^∧) = r^−n/2L²(R+× X) (3.40) with L²(R+× X) referring to drdx and dx defined in terms of a fixed Riemannian metrix

K^0,0(X^∧) = H^0,0(X^∧) = r^−n/2L²(R+× X) (3.40) with L²(R+× X) referring to drdx and dx defined in terms of a fixed Riemannian metrix

在文檔中 The edge algebra structure of the Zaremba problem (頁 25-44)