Double layer potentials on three-dimensional wedges and pseudodifferential operators on Lie groupoids

Let W be a three-dimensional wedge, and K be the double layer potential operator associated to W and the Laplacian. We show that 12±K are isomorphisms between suitable weighted Sobolev spaces, which implies a solvability result in weighted Sobolev spaces for the Dirichlet problem on W. Furthermore, we show that the double layer potential operator K is an element in C⁎(G)⊗M2(C), where G is the action (transformation) groupoid M⋊G, with G={(10ab):a∈R,b∈R+}, which is a Lie group, and M is a kind of compactification of G. This result can be used to prove the Fredholmness of 12+KΩ, where Ω is “a domain with edge singularities” and KΩ the double layer potential operator associated to the Laplacian and Ω.