Equivalence of Sobolev norms in Lebesgue spaces for Hardy operators in a half-space

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.