The Transmission Problem on a Three-Dimensional Wedge

We consider the transmission problem for the Laplace equation on an infinite three-dimensional wedge, determining the complex parameters for which the problem is well-posed, and characterizing the infinite multiplicity nature of the spectrum. This is carried out in two formulations leading to rather different spectral pictures. One formulation is in terms of square integrable boundary data, the other is in terms of finite energy solutions. We use the layer potential method, which requires the harmonic analysis of a non-commutative non-unimodular group associated with the wedge.


Introduction
Let ⊂ R 3 be a surface, dividing R 3 into interior and exterior domains + and − , respectively. Given a spectral parameter 1 = ∈ C and boundary data f and g on , the static transmission problem seeks a potential U : + ∪ − → C, harmonic in + and − , such that Tr + U − Tr − U = f and ∂ + n U − ∂ − n U = g on .
Here Tr ± U and ∂ ± n U denote the limiting boundary values and outward normal derivatives of U on , + indicating an interior limiting approach, − indicating exterior approach. For precise definitions, see equation (19). To discuss well-posedness, that is, the existence and uniqueness of solutions, one has to impose growth and regularity conditions on the potential and the boundary data. We will consider two different sets of conditions which are widely used. One formulation is in terms of square integrable boundary data, the other in terms of finite energy potentials. We refer to these formulations as problems (L) and (E), respectively.
In elecrostatics, the parameter corresponds to the (relative) permittivity of a material and is a positive quantity, > 0. In this case the problems (L) and (E) are very well-studied, and they have been shown to be well-posed for any Lipschitz surface , see [17,18,20,48,59] and [8,9,27], respectively. One approach to prove well-posedness is via the layer potential method, the success of which relies on the development and power of the theory of singular integrals. By means of layer potentials, Problem (L) has even been shown to be well-posed for a wide class of very rough surfaces which are not Lipschitz regular [29].
The transmission problem also appears as a quasi-static problem in electrodynamics, when an electromagnetic wave is scattered from an object that is much smaller than the wavelength. The permittivity is then complex and dependent on the frequency of the wave. In this setting, the properties of the transmission problem are very subtle. Problems (L) and (E) are no longer well-posed for certain ∈ C. When < 0 this corresponds to the possibility of exciting surface plasmon resonances in nanoparticles made out of gold, silver, and other materials [3,4,43,60]. Metamaterials, specifically designed synthetic materials, can also exhibit effective permittivities with negative real part [2,46,49].
The set of ∈ C for which the problem is ill-posed-the spectrum-depends on the shape of the interface . Strikingly, when the surface has singularities, the spectrum also depends heavily on the imposed growth and regularity conditions. For instance, when ⊂ R 2 is a curvilinear polygon in 2D, the spectrum of problem (L) is a union of two-dimensional regions in the complex plane, in addition to a set of real eigenvalues [47,57]. On the other hand, the formulation of problem (E) is more directly grounded in physics. Accordingly, the spectrum of problem (E) is a real interval, plus eigenvalues, when is a curvilinear polygon [6,52]. In three dimensions, similar results hold for surfaces with rotationally symmetric conical points [28].
(3) By convention, we refer to α,− = R 3 \ α,+ as the exterior domain. The two transmission problems (L) and (E) are given by Here M ± (∇U ) is an interior/exterior non-tangential maximal function of ∇U , anḋ H s ( α ) denotes a homogeneous Sobolev space of index s along α , see Sections 3 and 4. Alternatively,Ḣ 1/2 ( α ) can be viewed as the trace space of the space of harmonic functions in α,+ satisfying that α,+ |∇U | 2 dV < ∞. The negative spaceḢ −1/2 ( α ) will be given an intrinsic description in terms of single layer potentials. The incomparability ofḢ s 1 ( α ) andḢ s 2 ( α ), s 1 = s 2 , will cause us some difficulties.
The purpose of this detailed study of the wedge is to have α,+ serve as a model for general domains in R 3 with edges. For domains with corners in 2D, the problems (L) and (E) are now well understood; a successful approach is to first consider the layer potential method on the infinite 2D-wedge [25,30,39], and to then reduce the study of curvilinear polygons to that of infinite wedges via a localization procedure. In 3D, similar approaches can be taken for domains with conical points [28,37,54].
To fix the notation and to explain the layer potential approach at this point, we let K : L 2 ( ) → L 2 ( ) denote the harmonic layer potential where n(r ) denotes the unit outward normal to at r , and σ the surface measure on . The adjoint K * (with respect to L 2 ( )) is known as the double layer potential or the Neumann-Poincaré operator. The single layer potential of a charge f is given by When = α we write K = K α and S = S α . Note that S f is harmonic in + ∪ − . Differentiation leads to the jump formulas The ansatz U = Sh + in + and U = Sh − in − hence relates the transmission problems (L) and (E) to spectral problems for the layer potential K. Previous studies of the transmission problem and layer potentials on the infinite three-dimensional wedge are manifold. Eigensolutions to the transmission problem constructed via separation of variables can be found in [13,60]. Grachev and Maz'ya [25] studied problem (E), using their results as a technical tool to describe the Fredholm radius of the double layer potential on certain weighted Hölder spaces for surfaces with edges. Fabes et al. [21] observed, for α = π/2, that the double layer potential K * α on α can be regarded as a block matrix of convolution operators on the matrix group known as the ax + b group. See also [53], where general angles and weighted L 2 -spaces were considered. G is a non-Abelian and non-unimodular group, and therefore does not support standard harmonic analysis. For α = π/2, Fabes et al. proved that K * α ± I has an infinite-dimensional kernel on L p ( α ) whenever 1 < p < 3/2, where I denotes the identity operator. They proved this by constructing eigenfunctions through a rather delicate argument involving the partial Fourier transform in the z-variable. It is a natural idea to study layer potentials in the wedge by applying partial transforms in the z-and x-variables, cf. [50,55], but such a procedure does not completely resolve K * α . An explicit harmonic analysis for the ax + b group was being developed around the same time that [21] was published, leading to the first example of a non-unimodular group equipped with a Plancherel theorem [14,19,32]. The corresponding Fourier transform of G associates K α with four multiplication operators M T , where T : H → H is an operator on an infinite-dimensional Hilbert space H. As such, it does not provide a high level of resolution of the operator K α , and it may seem that we are gaining an unmerited amount of information from the harmonic analysis of G. However, key to our results will be to identify each operator T as a pseudo-differential operator of Mellin type [16,40,41], after which we can apply the symbolic calculus of such operators to understand the spectrum of K α .
Let α ⊂ C denote the simple closed curve and let α denote this curve together with its interior, see Fig. 1. For an operator T : H → H, the spectrum σ (T, H) is defined as usual, and we define the essential spectrum in the sense of Fredholm operators, Plot in the complex plane, z = x + iy, of the essential spectrum − α ∪ α of K α : L 2 ( α ) → L 2 ( α ) for α = π/3. Every point in the interior is an eigenvalue of infinite multiplicity of the double layer potential In Theorem 10 we will characterize the spectrum of K α : L 2,a ( α ) → L 2,a ( α ) the orthogonal sum referring to the decomposition (3) of α . For simplicity, we shall only state the theorem for a = 0 here.
Remark. For an infinite 2D-wedge γ α of angle α, the spectrum of the double layer potential on L 2 (γ α ) is the curve − α ∪ α , without any interior [47]. In this case, neither the double layer potential, nor its adjoint, has any eigenvalues.
In proving Theorem A we will show that any eigenvalue of K α : L 2 ( α ) → L 2 ( α ) is real. Therefore, for non-real λ, the eigenfunctions of item (2) are truly generalized. Whether the same is valid for real λ is left open. From Theorem A we obtain the promised corollary for the transmission problem (L).
Corollary A. Let 1 = ∈ C and f ∈Ḣ 1 ( α ). Then the transmission problem (L) is well posed (modulo constants) for all g ∈ L 2 ( α ) if and only if To treat problem (E), we follow Costabel [8] and Khavinson et al. [33] by introducing the energy space E( α ) with norm This is motivated by Green's formula, which, ignoring technicalities, shows that f ∈ E( α ) if and only if R 3 |∇S α f | 2 dV < ∞, see equation (22). Section 4 is devoted to proving that E( α ) coincides with the homogeneous Sobolev spacė H −1/2 ( α ), The proof proceeds via interpolation, based on Dahlberg and Kenig's result [10] that S α : L 2 ( α ) →Ḣ 1 ( α ) is an isomorphism, where S α is understood as a map on the boundary α . The advantage of working with the energy space E( α ) is that K α : E( α ) → E( α ) is self-adjoint, a consequence of the Plemelj formula which we will motivate in our setting. This explains why the energy formulation (E) of the transmission problem has a real spectrum. The study of the two operators K α : E( α ) → E( α ) and K α : L 2 ( α ) → L 2 ( α ) is reminiscent of Krein's framework of symmetrizable operators [38]. However, a level of caution is necessary, since, unlike to Krein's setting, S α : The main result concerning K α : E( α ) → E( α ) is the following: Remark. Eigensolutions to the transmission problem (1, 2), f = g = 0, are given in [60], for permissible parameters . These eigensolutions U are constructed by separation of variables, and are thus periodic in z. Hence they could not satisfy that ∂ ± n U ∈ L 2,a ( α ) for any a ∈ R. The relationship between the eigenfunctions of Theorems A and B and the eigensolutions to the transmission problem is interesting, but unclear. The qualitative behavior of solutions to problem (1, 2), when Re < 0, is of importance to the study of plasmonics, as it is related to effects of field enhancement and confinement in plasmonic structures [58,61].
Theorem B yields the expected corollary for the transmission problem. The sufficiency of the condition in Corollary B has been shown previously in [25, Theorem 1.6], but we will give a rather different proof.
The paper is laid out as follows: in Section 2 we recall the convolution structure of K α and the harmonic analysis of the ax +b group. Section 3 is devoted to proving Theorem A. In Section 4 we identify the energy space E( ) with a homogeneous Sobolev space, and in Section 5 we prove Theorem B.

Computations for the Wedge
Recall, for 0 < α < 2π , α = π , that the wedge α,+ has boundary The layer potential operator K α : L 2,a ( α ) → L 2,a ( α ) is, with respect to the orthogonal decomposition (6), given by where, for appropriate functions f ∈ L 2,a (dx dz) and x > 0, z ∈ R, (8) As observed in [21,53], through the change of variables we obtain that (9) It turns out that K α : L 2,a ( α ) → L 2,a ( α ) is bounded for −1 < a < 3, see Lemma 6. Thus, by duality, the double layer potential defines a bounded operator K * α : L 2,−a ( α ) → L 2,−a ( α ) for such a. Note here the convention of this paper; unless otherwise indicated, adjoint operations and dual spaces are calculated with respect to the inner product of L 2 = L 2,0 .
In the present situation, as a map of functions on the unbounded graph α , S α : L 2 ( α ) → L 2 ( α ) is not a bounded operator. However, it is densely defined, see Lemma 13. In Lemma 17 we will find that S α can also be understood as a bounded map between certain weighted L p -spaces. As for K α , the single layer potential can be formally written

Convolution Structure and Harmonic Analysis
Consider the matrix group in which multiplication corresponds to the composition of affine maps w → xw+z. That is, and We always equip the group G with its right Haar-measure dx x dz. G is a nonunimodular group; its left-invariant Haar measure is dx x 2 dz and the Haar modulus is therefore = (x, z) = x −1 .
The connection between G and K α is clear; K α can be interpreted as a convolution operator, Although we shall never make use of this, we point out that the convolution of f and g can also be computed with respect to the left structure of G, We will need Young's inequality for non-unimodular groups [34, Lemma 2.1], stated for the right Haar measure.

Lemma 1.
Suppose that 1 p, q, r ∞ satisfy 1 p + 1 q = 1 + 1 r , and that f ∈ L p (G) and g ∈ L q (G). Then where 1 q + 1 q = 1. The group G was the first example of a non-unimodular group carrying a complete, explicit, harmonic analysis [19,32,35,36]. We shall now recall the main features. The reader should be warned that the statements below have been adapted to the right-invariant structure of G, while most of the references given treat the left structure.
The construction is helped by the fact that G = R R + is a semi-direct product of the two abelian groups R and R + , each of which comes with its own standard Fourier analysis. On R we have the usual Fourier transform F, which extends to a unitary map F : L 2 (R) → L 2 (R). On R + , equipped with its Haar measure dx x , the corresponding Fourier transform is known as the Mellin transform M, Up to a constant scaling factor, M extends to a unitary M : The unitary representations yield corresponding transforms is the bounded operator given by However, due to the non-unimodularity of G, it is not possible to immediately obtain a Plancherel theorem in terms of F ± . In fact, there are compactly supported continuous f for which F ± ( f ) is not even compact [32]. However, it is possible to obtain a Plancherel theorem by introducing an operator correction factor [14,23]. In our case, the correction factor is given by δ, where δη(r ) = √ r η(r ). Consider for f ∈ L 2 (G) the pair of operators P ± ( f ) : It is straightforward to verify that

Proposition 2.
[32] The map P : L 2 (G) → S (2) 2 is onto and an isometry, Due to the correction factor, the convolution theorem is slightly asymmetrical.

Proposition 3. [32]
If k ∈ L 1 (G) and f ∈ L 2 (G), then For γ ∈ R, we let Note that f ∈ L 2,a (dx dz) if and only if Va+1 where a ∈ R, By the formula valid at first for f compactly supported in G, we can extend P ± to L 2,a (dx dz), in such a way that P ± ( f ) : Note also that f k ∈ L 2,a (dx dz) in this situation, by Young's inequality. For easy reference, we summarize what has been said in the following lemma: are bounded operators, and the convolution formula is valid.

Multiplication Operators
By Proposition 3 we are led to consider multiplication operators on the Hilbert-Schmidt class S 2 = S 2 (H) of an infinite-dimensional Hilbert space H with norm · . For a bounded operator T : H → H we denote by M T : S 2 → S 2 the operator of multiplication by T on the right, The following proposition is surely known: Furthermore, if λ is an eigenvalue of T * , then λ is an eigenvalue of M T of infinite multiplicity.

Proof. It is clear that
where tr denotes the usual trace of an operator in the trace class. It is a standard fact that ST S 2 S S 2 T B(H) . Conversely, consider, for g, h ∈ H, the rank-one operator S g,h = g ⊗ h ∈ S 2 , If λ ∈ σ (T ) and λ is an eigenvalue of T * with non-zero eigenfunction f , then λ is an eigenvalue of infinite multiplicity of M T , since If λ ∈ σ (T ) and T * − λ is injective but not bounded below, choose a sequence which is a contradiction. Hence M T − λ is not Fredholm in this case either.
Finally, suppose that λ ∈ σ (T ) and that T * − λ is bounded below but does not have full range. Then the range is not dense, and thus λ is an eigenvalue of T . As in (12) Adding up the different cases, we have shown that finishing the proof.

The L 2 -Spectrum
For a ∈ R, recall the definition of V γ from (11) and note that By equation (9), we see that where − a+1 The following lemma was first observed in [21,53]: For a = 1 the right-hand side should be interpreted as |1 − α/π|.
Proof. This follows by Young's inequality and as in Proposition 5, let M T ± α,a denote the operator of right multiplication by T ± α,a on S 2 = S 2 (L 2 (R + )). Then, by equation (13) and Proposition 3, is unitarily equivalent to Explicitly, for η ∈ L 2 (R + ) and r > 0, Hence T ± α,a is an integral operator given by Here and K 1 is a modified Bessel function of the second kind [1, p. 376], K 1 has the following asymptotics [1, p. 378]: and where χ (0,1) 2 denotes the characteristic function of the square (0, 1) 2 .
Proof. In fact, so that T ± α,a − I α,a is Hilbert-Schmidt. To see this, let Finally, when 1 < r < ∞ and 1 < x < ∞ we have that B(r, x) x + r , and thus Observe that I α,a is a truncated Mellin convolution operator (convolution on the group R + ) with kernel in the sense that The range of this transform is the closed curve For a = 1 this is a simple closed curve in C, positively oriented if −1 < a < 1 and negatively oriented if 1 < a < 3, in either case satisfying that α,a = α,2−a . If 0 < α < π then α,a lies in the left half-plane of C, in the right half-plane if π < α < 2π . For a = 1, α,1 is the real interval between 0 and α/π − 1. It is clear that α,a is symmetric with respect to complex conjugation. The curves are increasing in 1 a < 3 in the sense that if 1 a < a < 3, then every point of α,a but the origin is contained in the interior of α,a . For precise calculations we refer to [47].
Lemma 7 shows that, with respect to the decomposition we have that where the entries marked * are compact operators, and J α,a is a pseudo-differential operator of Mellin type. There is a fully fledged theory of such operators developed by Elschner, Lewis, and Parenti [16,40,41], together with a symbolic calculus which for λ / ∈ α,a gives the index of J α,a − λ, and thus of T ± α,a − λ, as the winding number W ( α,a , λ) of λ with respect to α,a . In fact, the same operator J α,a appears in computing the spectrum of double layer potentials on curvilinear polygons in 2D, and thus the relevant calculations already appear in [39,47]. We do not give an account of the theory here, but instead summarize the conclusion it yields in the next proposition. a , λ).

Proposition 8. The essential spectrum of T
The classical Kellogg argument shows that any eigenvalue of K : L 2 ( ) → L 2 ( ) must be real, in the case that is a bounded surface. However, this argument fails in the present setting, essentially because L 2 ( α ) is not contained in the energy space E( α ), in the terminology of Section 4. The next lemma offers a replacement of the Kellogg argument. For the statement, observe by (8) that K α : L 2,1 ( α ) → L 2,1 ( α ) is a self-adjoint operator, hence has real spectrum.
Proof. We give the argument for −1 < a 1. The proof of the statement for 1 < a < 3 is similar. If λ is an eigenvalue of K α : L 2,a ( α ) → L 2,a ( α ), then, by (7), either λ or −λ is an eigenvalue of K α : L 2,a (dx dz) → L 2,a (dx dz). Denote this latter eigenvalue by μ. Let f ∈ L 2,a (dx dz) be a non-zero eigenfunction and consider the decomposition Noting that a 1, we have that f 1 ∈ L 2,1 (dx dz), and therefore by Lemma 6 that K α f 1 ∈ L 2,1 (dx dz) as well. From the eigenvalue equation we hence obtain that In other words, , so that formal application of the Fourier transform yields To justify (17), observe that V 1 f 2 ∈ L 2,a−2 and that by the proof of Lemma 6. Hence, by Lemma 4, the components of (17) are initially well-defined as bounded maps .
Remark. By the symmetry α,a = α,2−a and the increasing nature of the curves α,a , 1 a < 3, it is clear that Theorem 10 implies Theorem A.

Remark.
For an unbounded Lipschitz graph , Escauriaza and Mitrea [18] showed that σ (K, L 2 ( )) is contained in a certain hyperbola which only depends on the Lipschitz character of . Perhaps unsurprisingly, Theorem 10 shows that their result is sharp for the wedge boundaries α .
The single layer potential is an isomorphism as a map S α : L 2 ( α ) →Ḣ 1 ( α ) [10, Lemma 3.1]. Evaluating S α f instead on either α,+ or α,− , see equation (4), yields isomorphisms S α : L 2 ( α ) →ḣ 1 ( α,± ). Furthermore, by the weak singularity of the kernel, we have that In other words, the interior Dirichlet problem is well posed (modulo constants), and the solution is of the form of a single layer potential, U = S α f , f ∈ L 2 ( α ). The same statement holds for the exterior Dirichlet problem.
To treat the transmission problem we make use of the jump formulas [18, p. 149] Corollary 11. Let 1 = ∈ C and f ∈Ḣ 1 ( α ). Then the transmission problem is well posed (modulo constants) for all g ∈ L 2 ( α ) if and only if Proof. By well-posedness of the Dirichlet problems there are densities h ± ∈ L 2 ( α ) and a constant c such that U = S α h + + c in α,+ and U = S α h − + c in α,− . By the jump formulas (20), the transmission problem is then equivalent to the system on α , where I denotes the identity map. This system is uniquely solvable if and only if (21) holds, by Theorem 10 and the fact that S α : L 2 ( α ) →Ḣ 1 ( α ) is an isomorphism.

Identification with a Fractional Homogeneous Sobolev Space
In this section only, we will consider the more general situation where is an unbounded Lipschitz graph, where ϕ : R 2 → R is Lipschitz continuous. We think of the region above as the interior domain + , the region below it as the exterior − . The energy space E( ) in the case when is an infinite cone was important in [28], but was not shown to coincide with a Sobolev space. We therefore prove this identification for general Lipschitz graphs here. The considerations of this section apply equally well to the case of an unbounded Lipschitz graph embedded in R n , n 3, but we restrict ourselves to n = 3 for simplicity of notation.
Denote the space of compactly supported functions f ∈ L 2 ( ) by L 2 c ( ). Then This is a standard identity which follows from Green's formula and the jump formulas (20) for the interior and exterior normal derivatives of S f on . When is smooth, bounded, and connected, equation (22) initially for functions f, g ∈ L 2 c ( ). Equation (22) shows positive definiteness; if However, this implies that ∂ + n S f = 0, see equation (19), which, unless f = 0, is incompatible with the estimate [31]. We define the energy space E( ) as the completion of L 2 c ( ) under this inner product.
When is a connected bounded Lipschitz surface, the energy space E( ) consists precisely of the distributions f on in the inhomogeneous Sobolev space H −1/2 ( ) [8]. We will show that for an unbounded Lipschitz graph this remains true upon replacing H −1/2 ( ) by a homogeneous Sobolev space.
Let F : L 2 (R 2 ) → L 2 (R 2 ) denote the usual two-dimensional Fourier transform. For 0 s 1, we define the homogeneous Sobolev spaceḢ s (R 2 ) as the completion of C ∞ c (R 2 ) under the norm We refer to [5,Ch. 1] for the basics of homogeneous Sobolev spaces. When 0 < s < 1, the norm can also be computed as a Slobodeckij norm, see for example [11,Proposition 3.4], where c s is a constant depending on s. For 0 s < 1, we emphasize that the completionḢ s (R 2 ) is a space of functions. In fact, there is an injective embedding is the quotient of a semi-Hilbert space of functions with the subspace of constant functions. More precisely,Ḣ 1 (R 2 ) is the Hilbert space of L 2 loc (R 2 )-functions f modulo constants such that ∇ f ∈ L 2 (R 2 ). We define the negative index spacesḢ −s (R 2 ) as the dual spaces ofḢ s (R 2 ) with respect to the L 2 (R 2 )-pairing. Note that (24) remains valid for −1 s < 0, in the sense that the Fourier transform extends to a unitary Alternatively, homogeneous Sobolev spaces may be understood in terms of the Riesz potential [11,Section 3]. For 0 < s 1, the Riesz potential is given by where c s is a constant depending on s. Clearly, I s : L 2 (R 2 ) →Ḣ s (R 2 ) is a unitary map, and by duality, so is I s :Ḣ −s (R 2 ) → L 2 (R 2 ).
We naturally interpret functions f on as functions on R 2 , by letting For 0 . We definė H −s ( ) as the dual ofḢ s ( ) with respect to the L 2 ( )-pairing.
We interpret Lemma 12 by saying that L 2 c ( ) is densely contained inḢ −1/2 ( ), and we do not notationally distinguish between g and g from this point on.
By the group property I s 1 I s 2 = I s 1 +s 2 , 0 < s 1 , s 2 < 1, and the unitarity of I 1/2 :Ḣ −1/2 (R 2 ) → L 2 (R 2 ), we find that Furthermore, if f ∈ L 2 c ( ) is a nonnegative function, then I s ρ f S f , since the kernels of I s ρ and S are comparable. Comparing (23), (27), and (29) thus yields that To extend this estimate to general functions, we appeal to an interpolation argument, beginning with
Then g n dσ = 1, and 0 Sg n (r ) 1 n , r ∈ , by a straightforward estimate. Hence Similarly we see from (29) that Of course, g n L 2 ( ) 1/n. Now suppose that f ∈ L 2 c ( ) and let d = f dσ . Then f − dg n ∈ L 2 c,0 ( ) and f − dg n → f inḢ −1/2 ( ), E( ), and L 2 ( ) as n → ∞. This proves that L 2 c,0 ( ) is dense in these three spaces, since L 2 c ( ) is. Finally, suppose again that f ∈ L 2 c,0 ( ). If K is any bounded subgraph of containing supp f , then S f ∈ L 2 (K ) by the usual mapping properties of S for connected bounded Lipschitz surfaces [59]. Hence we only need to check the behavior of S f at infinity to finish the proof. Letting S(r, r ) be the kernel of S, note for r ∈ K that S(r, r ) = 1 4π Therefore, since f dσ = 0, It follows that S f ∈ L 2 ( ).
We are ready to state and prove the main theorem of this section. For the proof, note that the J -method, the K -method, and the complex method are all equivalent for interpolation of Hilbert spaces [7,44]. We hence simply refer to the interpolation space (H 0 , H 1 ) θ of exponent 0 < θ < 1 between two compatible Hilbert spaces H 0 and H 1 .
Proof. The starting point is that S : L 2 ( ) →Ḣ 1 ( ) is an isomorphism [10, Lemma 3.1]. Let : L 2 ( ) → L 2 (R 2 ) denote the unitary given by where ρ, as before, is given by ρ(x, y) = (1 + |∇ϕ(x, y)| 2 ) 1/2 . Then is an isomorphism, since multiplication by ρ −1/2 on L 2 (R 2 ) and S : L 2 ( ) → H 1 ( ) are both isomorphisms. It follows from (22) that, M is symmetric with respect to the L 2 (R 2 , dξ)-pairing. Therefore, by duality, we can reformulate (30) by saying that M continuously extends to an isomorphism By Lemma 13, M is initially densely defined on and the meaning of (31) is that M extends continuously to an isomorphism. Interpolation between (30) and (31) also gives that is bounded. It is not, however, possible at this stage to conclude that this operator is an isomorphism. As a consequence of (22) and (32) we conclude that We also want to consider M as an unbounded operator on L 2 (R 2 , dξ). To avoid confusion we call this operator R, In view of (31), we can let the domain of R be The positivity of R on dom(M) extends to dom(R). To see this, given f ∈ dom(R) ⊂ L 2 (R 2 , |ξ | −1 dξ), we may by Lemma 13 choose a sequence in dom(M), approximating f in L 2 (R 2 , |ξ | −1 dξ). By (32) and (33) we conclude that The same argument shows that R is a symmetric operator, Since the operator of (30) is an isomorphism, the domain of R * is given by The range of R being dense in L 2 (R 2 , |ξ | 2 dξ), it follows that dom(R * ) = dom(R).
We conclude that R is a positive self-adjoint operator.
Consider now the Hilbert space H 1 = L 2 (R 2 , dξ) with its usual norm and H 0 = L 2 (R 2 , |ξ | −2 dξ) with the alternative norm We apply the characterization of the interpolation spaces (H 0 , H 1 ) θ , 0 < θ < 1, given by [7,Theorem 3.3]. It extends the usual characterization given in [42,Theorem 15.1] to the present situation in which H 0 and H 1 are incomparable. The conclusion is that the relationship defines an unbounded, self-adjoint, positive operator T : H 1 → H 1 whose square root has domain 1 Furthermore, the norm of the interpolation space (H 0 , H 1 ) 1/2 is given by By (34) and (35) we have that Since R is also positive and self-adjoint it must be that R = T 1/2 , see for example [56,Proposition 10.4].
On the other hand, if we equip H 0 = L 2 (R 2 , |ξ | −2 dξ) with the usual norm, we know that the interpolation space is L 2 (R 2 , |ξ | −1 dξ), and thus Unraveling the definitions, this means that where the last equality is given by (27). Since L 2 c,0 ( ) is dense in E( ) and H −1/2 ( ) by Lemma 13, this proves the statement.

Single Layer Potentials and the Dirichlet Problem
It is implicit in the proof of Theorem 14 that the isomorphism property of S : L 2 ( ) →Ḣ 1 ( ) extends to the scale of homogeneous Sobolev spaces. When is a bounded Lipschitz surface the corresponding result is well known, see for example [22,Theorem 8.1].

Corollary 15.
Let be an unbounded Lipschitz graph. For every 0 s 1, Proof. Following the proof of Theorem 14, we see for every 0 < s < 1 that R : L 2 (R 2 , dξ) → L 2 (R 2 , dξ) has dense range by the isomorphism property of (31). It follows that R s extends to an isomorphism R s : L 2 (R 2 , |ξ | −2s dξ) → L 2 (R 2 , dξ), and, by duality, to an isomorphism R s : Thus R extends to an isomorphism This is equivalent to the statement of the corollary.
Consider the homogeneous Sobolev spaces on + and − , These are Hilbert spaces as quotient spaces over the constant functions. The subspaces of harmonic functions are given bẏ It follows from equations (22) and (23) that, evaluating S f in either + or − for a charge f , S extends to bounded maps By the trace inequality [15,Theorem 2.4] and the method of [12], there are (unique) continuous traces Tr ± :Ḣ 1 ( ± ) →Ḣ 1/2 ( ). By the corresponding result for bounded Lipschitz surfaces [8], and by considering smooth cut-off functions, we see that Tr ± S f = S f for f ∈ L 2 c ( ). By Corollary 15, both sides of this equation extend continuously toḢ −1/2 ( ) E( ), and we conclude that This leads to the following result on the interior Dirichlet problem (of course, we could equally well make the analogous statement for the exterior Dirichlet problem):

Corollary 16. The trace Tr
is well-posed. The unique solution U is given by a single layer potential, Proof. Given g ∈Ḣ 1/2 ( ), there is by Corollary 15 an f ∈ E( ) such that S f = g inḢ 1/2 ( ). Then U = S f inḢ 1 h ( + ) solves the Dirichlet problem. Uniqueness is given by [45,Theorem 7.1.2]. Corollary 15 shows that S : E( ) →Ḣ 1 h ( + ) is injective, while surjectivity is given by well-posedness of the Dirichlet problem.

The Energy Space Spectrum
We now return to the situation where = α is the boundary of a wedge of opening angle α. Recall from Section 2 that where K α f = f k α and, by equation (10), S β f = V 1 f s β , with convolution kernels given by Here, as before, V γ , γ ∈ R, is the operator of multiplication by −γ = x γ .
For technical purposes, we begin by establishing some mapping properties of S α and K α , refraining from working out the much more general statement that could be given. For 1 p < ∞ we write L p,a (dx dz) = L p (x a dx dz) and L p,a ( α ) = L p,a (dx dz) ⊕ L p,a (dx dz). Proof. Note first that V a s β ∈ L q (G) for β ∈ {0, α} if and only if 1 < q < 2 and 0 < aq < q − 1. To see this, note that 0 s α s 0 and that if q > 1, then where c q > 0 is a constant. We now let q = 4/3. By Young's inequality, Lemma 1, whenever β ∈ {0, α} and 0 < a < 1/4. This yields that S α : L 2,3/4 ( α ) → L 4,−1/2 ( α ) is bounded upon choosing a = 1/8. That K α : L 2,3/4 ( α ) → L 2,3/4 ( α ) is bounded is part of Lemma 6. The boundedness of K α : L 4/3,1/8 ( α ) → L 4/3,1/8 ( α ) also follows from (the proof of) that lemma, since it shows that V 27/32 k α ∈ L 1 (G), and by Young's inequality This last estimate also proves that K * α : The lemma allows us to motivate the Plemelj formula SK = K * S for the unbounded Lipschitz graph α . Lemma 18. The Plemelj formula is valid for α when either side of the equation is interpreted as a bounded operator from L 2,3/4 ( α ) into L 4,−1/2 ( α ). That is, Proof. Choose a sequence ( j ) ∞ j=1 of bounded connected Lipschitz surfaces such that ( j ∩ α ) ∞ j=1 is an increasing exhaustion of α . The choice of sequence can be made so that for any compact set K ⊂ α it holds for sufficiently large j that where dσ j denotes the surface measure of j . Suppose that f, g ∈ L 2 c ( α ). For sufficiently large j we can understand f and g as functions on j , and then, by the Plemelj formula for bounded domains [59,Theorem 3.3], where K j and S j denote the layer potentials of j . Note that where Since, by Lemma 17, K α f ∈ L 4/3,1/8 ( α ) and S α g ∈ L 4,−1/2 ( α ), we deduce from (37) that Similarly, S j f, K j g L 2 ( j ) → S α f, K α g L 2 ( α ) . By (36) we conclude that The operators S α K α , K * α S α : L 2,3/4 ( α ) → L 4,−1/2 ( α ) are bounded, by Lemma 17. Hence we infer from (38) that they are equal, S α K α = K * α S α .
Lemmas 17 and 18 let us define K α as an unbounded symmetric operator on E( α ). We will later see that K α : E( α ) → E( α ) is bounded (and hence selfadjoint).

Lemma 19. Let
, and this operator is symmetric.
is a consequence of Lemma 17, and the symmetry of Our next goal is to prove that K α : E( α ) → E( α ) is actually bounded and to give the correct estimate for its norm. For γ ∈ R, let and consider for 0 < a 1 the space E a ( α ) = V a E( α ), the completion of V a D( α ) under the scalar product The following is obvious by definition and Lemma 19: , the latter operator having domain V a D( α ).
The space of such functions is included in V a D( α ) and dense in E a ( α ), which follows from the fact that V a L 2 c ( ) is dense in E a ( α ), together with a small modification of the proof of Lemma 13. Let K ⊂ α \ {(0, 0, z) : z ∈ R} be a compact set such that f is compactly supported in the interior of K . Then by the usual mapping properties of the single layer potential on a bounded connected Lipschitz surface. For r ∈ α \ K , we have by Lemma 13 that S α V −a f (r ) = O((1 + |r | 2 ) −1 ). Hence since 0 < a < 1. We conclude that Since f ∈ V a D( α ) and the symmetric (at this stage possibly unbounded) operator V a K α V −a : E a ( α ) → E a ( α ) preserves V a D( α ), we find that Repeating the estimate inductively gives us that Estimating the right-most norm with the help of (41) yields that ) .

Remark.
The reason for not directly considering a = 1 in the proof is that it appears difficult to find an appropriate dense class of functions f for which V 1−a S α V −a f = S α V −1 f ∈ L 2,−1 ( α ).
We are finally in a position to determine the spectrum of K α : E( α ) → E( α ). Let us begin by describing an unsuccessful approach, which nonetheless is illuminating. By inspection of (8) we see that Hence Lemma 20 for a = 1 says that K α : E( α ) → E( α ) is unitarily equivalent to K * α : E 1 ( α ) → E 1 ( α ). The scalar product of E 1 ( α ) is given by where S α = S α V −1 . Note that S α is a block matrix of convolution operators on the group G. Plemelj's formula, Lemma 18, says that S α and K * α commute, Suppose that we could construct a suitable square root of S α which commutes with K * α . Then, in view of (44), it should be possible to conclude that is a unitary map. It would hence follow that K α : E( α ) → E( α ) is unitarily equivalent to K * α : L 2,−1 ( α ) → L 2,−1 ( α ), which in turn, by (43), is unitarily equivalent to K α : L 2,1 ( α ) → L 2,1 ( α ). We have already computed the spectrum of this latter operator in Theorem 10.
Unfortunately, while the scalar product (44) is a positive definite form, it is not clear to the author how to construct the desired square root. Therefore we will compare K * α : E 1 ( α ) → E 1 ( α ) with K * α : L 2,−1 ( α ) → L 2,−1 ( α ) in an indirect way, yielding slightly less information.
Since the spectrum of the self-adjoint operator K α : E( α ) → E( α ) is an interval, a set without isolated points, it is of course essential.
Remark. The statement of Theorem B follows by combining Theorems 10 and 22.