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

Abstract

In this paper, we investigate the $W^{s,p}$-boundedness for stationary wave operators of the Schrödinger operator with inverse-square potential $${\mathcal{L}}_a=-\Delta +\frac{a}{|x{|}^2},a\ge -\frac{(d-2{)}^2}{4},$$ in dimension $d\ge 2$. 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 $d\ge 2$. 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.

arXiv comment57 pages, 1 figure, 1 table, Improved previous result from $L^p$-boundedness to $W^{s,p}$-boundedness

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Cited by in corpus (1)

• On Sobolev norms involving Hardy operators in a half-space