Fractional Sobolev Spaces for the Singular-perturbed Laplace Operator in the $L^p$ setting

We study the perturbed Sobolev spaces $H_{\alpha }^{s,p}({\mathbb{R}}^d)$, associated with singular perturbation ${\mathrm{\Delta }}_{\alpha }$ of Laplace operator in Euclidean space of dimensions 2 and 3. We extend the $L^2$ theory of perturbed Sobolev space to the $L^p$ case, finding an analogue description in terms of standard Sobolev spaces. This enables us to extend the Strichartz estimates to the energy space and to treat the {local well-posedness} of the {Nonlinear Schrödinger equation} associated with this singular perturbation, with the contraction method.