  We consider Hardy operators, i.e., homogeneous Schr\"odinger operators
consisting of the ordinary or fractional Laplacian in a half-space plus a
potential, which only depends on the appropriate power of the distance to the
boundary of the half-space. We compare the scales of homogeneous $L^p$-Sobolev
spaces generated by these Hardy operators with and without potential with each
other. To that end, we prove and use new square function estimates for
operators with slowly decaying heat kernels. Our results hold for all
admissible coupling constants in the local case and for repulsive potentials in
the fractional case, and extend those obtained recently in $L^2$. They also
cover attractive potentials in the fractional case, once expected heat kernel
estimates are available.
