Equivalence of Sobolev norms involving generalized Hardy operators

TL;DR

This paper establishes the equivalence of Sobolev norms involving fractional Schrödinger operators with Hardy potential, extending classical inequalities and aiding in the analysis of large relativistic atoms.

Contribution

It introduces a comparison between Sobolev spaces generated by fractional Schrödinger operators with Hardy potential and standard Sobolev spaces, including generalized Hardy inequalities.

Findings

Proves equivalence of Sobolev norms for fractional Schrödinger operators with Hardy potential.

Derives generalized Hardy inequalities for these operators.

Extends results to applications in large relativistic atoms.

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.