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

Abstract

We study the perturbed Sobolev spaces $H_{\alpha }^{s,p}({\mathbb{R}}^d)$, associated with singular perturbation ${\Delta }_{\alpha }$ 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.

References in corpus (5)

• Ground states for the planar NLSE with a point defect as minimizers of the constrained energy
• Two dimensional Schr{\" o}dinger operators with point interactions: threshold expansions, zero modes and $L^p$-boundedness of wave operators
• The $L^p$ boundedness of wave operators for the three-dimensional multi-centre point interaction
• Blow-up and instability of standing waves for the NLS with a point interaction in dimension two
• The fractional Bessel equation in Hölder spaces

Cited by in corpus (1)

• Local well-posedness and blow-up in the energy space for the 2D NLS with point interaction