Sobolev spaces for singular perturbation of 2D Laplace operator

We study the perturbed Sobolev space H-alpha(1,r.), r is an element of(1, infinity), associated with singular perturbation Delta(alpha) of Laplace operator in Euclidean space of dimension 2. The main results give the possibility to extend the L-2 theory of perturbed Sobolev space to the L-r case. When r is an element of (2, infinity) we have appropriate representation of the functions in H-alpha(1,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.