Sobolev spaces for singular perturbation of 2D Laplace operator

We study the perturbed Sobolev space Hα1,r, r∈(1,∞), associated with singular perturbation Δα of Laplace operator in Euclidean space of dimension 2. The main results give the possibility to extend the L2 theory of perturbed Sobolev space to the Lr case. When r∈(2,∞) we have appropriate representation of the functions in Hα1,r in regular and singular part. An application to local well-posedness of the NLS associated with this singular perturbation in the mass critical and mass supercritical cases is established too.