Equivalence of Sobolev norms involving generalized Hardy operators
Analysis of PDEs
2023-04-19 v3 Mathematical Physics
Functional Analysis
math.MP
Abstract
We consider the fractional Schr\"odinger operator with Hardy potential and critical or subcritical coupling constant. This operator generates a natural scale of homogeneous Sobolev spaces which we compare with the ordinary homogeneous Sobolev spaces. As a byproduct, we obtain generalized and reversed Hardy inequalities for this operator. Our results extend those obtained recently for ordinary (non-fractional) Schr\"odinger operators and have an important application in the treatment of large relativistic atoms.
Keywords
Cite
@article{arxiv.1807.09027,
title = {Equivalence of Sobolev norms involving generalized Hardy operators},
author = {Rupert L. Frank and Konstantin Merz and Heinz Siedentop},
journal= {arXiv preprint arXiv:1807.09027},
year = {2023}
}
Comments
21 pages; v3 contains a new appendix, which extends Theorem 1.1 to all functions in the domains of $(|p|^\alpha+a|x|^{-\alpha})^{s/2}$ and $|p|^{\alpha s/2}$, respectively