On the spectrum of the reflection operator on conical surfaces (Q814363)

The radiation of light on a surface \(S\) is determined by an integral equation of the second kind, involving the reflection integral operator \(K_S\). In this paper, acting properties of this operator in appropriate fractional Sobolev spaces and solvability of the corresponding equation in such spaces are studied on the cone and on a conical surface. On the cone, \(K_S\) is a convolution operator, and it is shown that \(K_S: H^\mu(S)\to H^{\mu+2}(S)\) and that \(I-cK_S\) is invertible in \(H^\mu(S)\) for \(c<1\). On a conical surface, \(K_S\) is a convolution operator with respect to the Mellin transform in one of the variables. Some identities of hypergeometric functions are used in this case to obtain results for appropriate weighted Sobolev spaces which are analogous to the above results on the cone.