Weighted estimates for powers and Smoothing estimates of Schr\"odinger operators with inverse-square potentials

Let ${\mathcal{L}}_a$ be a Schr"odinger operator with inverse square potential $a\mathrm{\mid }x{\mathrm{\mid }}^{-2}$ on ${\mathbb{R}}^d,d\ge 3$. 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 $e^{it{\mathcal{L}}_a}$.