The 𝑊^{𝑠,𝑝}-boundedness of stationary wave operators for the Schrödinger operator with inverse-square potential

Abstract

In this paper, we investigate the W s , p W^{s,p} -boundedness for stationary wave operators of the Schrödinger operator with inverse-square potential L a = − Δ + a | x | 2 , a ≥ − ( d − 2 ) 2 4 , \begin{equation*} \mathcal L_a=-\Delta +\tfrac {a}{|x|^2}, \quad a\geq -\tfrac {(d-2)^2}{4}, \end{equation*} in dimension d ≥ 2 d\geq 2 . We construct the stationary wave operators in terms of integrals of Bessel functions and spherical harmonics, and prove that they are W s , p W^{s,p} -bounded for certain p p and s s which depend on a a . As corollaries, we solve some open problems associated with the operator L a \mathcal L_a , which include the dispersive estimates and the local smoothing estimates in dimension d ≥ 2 d\geq 2 . We also generalize some known results such as the uniform Sobolev inequalities, the equivalence of Sobolev norms and the Mikhlin multiplier theorem, to a larger range of indices. These results are important in the description of linear and nonlinear dynamics for dispersive equations with inverse-square potential.