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 $a|x|^{-2}$ 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 $e^{it\mathcal{L}_a}$. 
The Anh Bui, Piero D'Ancona, Xuan Thinh Duong, Ji Li and Fu Ken Ly
math.AP