The $W^{s,p}$-boundedness of stationary wave operators for the Schr\"odinger operator with inverse-square potential

TL;DR

This paper establishes the $W^{s,p}$-boundedness of stationary wave operators for Schrödinger operators with inverse-square potentials, solving open problems and extending key inequalities relevant to dispersive PDEs.

Contribution

It constructs stationary wave operators using Bessel functions and spherical harmonics, proving their boundedness and generalizing important inequalities for a broader range of parameters.

Findings

Proved $W^{s,p}$-boundedness of stationary wave operators for inverse-square potentials.

Solved open problems related to dispersive and local smoothing estimates.

Extended Sobolev inequalities and multiplier theorems to larger index ranges.

Abstract

In this paper, we investigate the $W^{s,p}$-boundedness for stationary wave operators of the Schr\"odinger operator with inverse-square potential $$\mathcal L_a=-\Delta+\tfrac{a}{|x|^2}, \quad a\geq -\tfrac{(d-2)^2}{4},$$ in dimension $d\geq 2$. We construct the stationary wave operators in terms of integrals of Bessel functions and spherical harmonics, and prove that they are $W^{s,p}$-bounded for certain $p$ and $s$ which depend on $a$. As corollaries, we solve some open problems associated with the operator $\mathcal L_a$, which include the dispersive estimates and the local smoothing estimates in dimension $d\geq 2$. 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.