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

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Changxing Miao, Xiaoyan Su and Jiqiang Zheng
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## Abstract:

In this paper, we investigate the $W^{s,p}$-boundedness for stationary wave operators of the Schrödinger operator with inverse-square potential \begin{equation*} \mathcal L_a=-\Delta +\tfrac {a}{|x|^2}, \quad a\geq -\tfrac {(d-2)^2}{4}, \end{equation*} 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.## References

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## Additional Information

**Changxing Miao**- Affiliation: Institute of Applied Physics and Computational Mathematics, Beijing 100088, People’s Republic of China
- Email: miao_changxing@iapcm.ac.cn
**Xiaoyan Su**- Affiliation: Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China
- MR Author ID: 1288686
- Email: suxiaoyan0427@qq.com
**Jiqiang Zheng**- Affiliation: Institute of Applied Physics and Computational Mathematics, Beijing 100088, People’s Republic of China
- MR Author ID: 903431
- Email: zheng_jiqiang@iapcm.ac.cn, zhengjiqiang@gmail.com
- Received by editor(s): March 3, 2022
- Received by editor(s) in revised form: August 1, 2022
- Published electronically: December 8, 2022
- Additional Notes: This work is supported by the National Key Research and Development Program of China (No. 2022YFA1005700). The first author was supported by NSFC Grants 11831004 and 12026407, and the third author was supported by National key R&D program of China: 2021YFA1002500, PFCAEP project No. YZJJLX2019012, NSFC Grant 11901041 and Beijing Natural Science Foundation 1222019.
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**376**(2023), 1739-1797 - MSC (2020): Primary 58J50, 42B20, 35J10, 33C05
- DOI: https://doi.org/10.1090/tran/8823
- MathSciNet review: 4549691