Weighted estimates for powers and smoothing estimates of Schrödinger operators with inverse-square potentials

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