Single and Double Layer Potentials on Domains with Conical Points I: Straight Cones

Let Ω R ω be an open cone in Rn, n ¥ 3, where ω € Sn 1 is a smooth subdomain of the unit sphere. Denote by K and S the double and single layer potential operators associated to Ω and the Laplace operator ∆. Let r be the distance to the origin. We consider a natural class of operators on BΩ, called Mellin convolution operators with good mapping properties between weighted Sobolev spaces and show that raKr a and r 1 2 Sr b 1 2 are Mellin convolution operators for a P p 1, n 1q and b P p 1 2 , n 3 2 q. This result allows us to prove that 1 2 K and r 1 2 Sr 1 2 are invertible between suitable weighted Sobolev spaces. Classically, it is known that the invertibility of these operators gives solvability results in weighted Sobolev spaces for the Dirichlet problem on Ω.