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arXiv · 2504.19732

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

Vladimir Georgiev·Mario Rastrelli
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Abstract

We study the perturbed Sobolev spaces ${H^{s,p}_\alpha(\mathbb{R}^d)}$, 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 Schr\"odinger equation} associated with this singular perturbation, with the contraction method.

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Vladimir Georgiev, Mario Rastrelli. 2025-04-28. Fractional Sobolev Spaces for the Singular-perturbed Laplace Operator in the $L^p$ setting. https://doi.org/10.1007/s42985-025-00336-z

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