Sobolev spaces for singular perturbation of Laplace operato

We study the perturbed Sobolev space $H_{\alpha }^{1,r}$, $r\in (1,\mathrm{\infty }),$ associated with singular perturbation ${\mathrm{\Delta }}_{\alpha }$ 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 $r\in (2,\mathrm{\infty })$ 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.