Conformally invariant non-local operators

On a conformal manifold with boundary, we construct conformally invariant local boundary conditions B for the conformally invariant power of the Laplacian (square (k), B) with the property that (square (k), B) is formally self-adjoint. These boundary problems are used to construct conformally invariant nonlocal operators on the boundary Sigma, generalizing the conformal Dirichlet-to-Robin operator, with principal parts which are odd powers h (not necessarily positive) of (-Delta (Sigma))(1/2), where Delta (Sigma) is the boundary Laplace operator. The constructions use tools from a conformally invariant calculus.