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.Comment: 16 pages; v2 contains improved results for positive coupling
constant