  Let $\mathcal{L}_a$ be a Schr\"odinger operator with inverse square potential
$a|x|^{-2}$ on $\mathbb{R}^d, d\geq 3$. The main aim of this paper is to prove
weighted estimates for fractional powers of $\mathcal{L}_a$. The proof is based
on weighted Hardy inequalities and weighted inequalities for square functions
associated to $\mathcal{L}_a$. As an application, we obtain smoothing estimates
regarding the propagator $e^{it\mathcal{L}_a}$.
