  We study the $L^p$-theory for the Schr\"odinger operator $\mathcal L_a$ with
inverse-square potential $a|x|^{-2}$. Our main result describes when
$L^p$-based Sobolev spaces defined in terms of the operator $(\mathcal
L_a)^{s/2}$ agree with those defined via $(-\Delta)^{s/2}$. We consider all
regularities $0<s<2$.
  In order to make the paper self-contained, we also review (with proofs)
multiplier theorems, Littlewood-Paley theory, and Hardy-type inequalities
associated to the operator $\mathcal L_a$.
