Abstract
We study the \(L^p\)-theory for the Schrödinger operator \(\mathcal L_a\) with inverse-square potential \(a|x|^{-2}\). Our main result describes when \(L^p\)-based Sobolev spaces defined in terms of the operator \((\mathcal L_a)^{s/2}\) agree with those defined via \((-\Delta )^{s/2}\). We consider all regularities \(0<s<2\). In order to make the paper self-contained, we also review (with proofs) multiplier theorems, Littlewood–Paley theory, and Hardy-type inequalities associated to the operator \(\mathcal L_a\).
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Acknowledgements
We are grateful to E. M. Ouhabaz and an anonymous referee for references connected with Theorem 1.1. R. Killip was supported by NSF Grant DMS-1265868. He is grateful for the hospitality of the Institute of Applied Physics and Computational Mathematics, Beijing, where this project was initiated. C. Miao was supported by NSFC Grants 11171033 and 11231006. M. Visan was supported by NSF Grant DMS-1161396. J. Zhang was supported by PFMEC (20121101120044), Beijing Natural Science Foundation (1144014), and NSFC Grant 11401024. J. Zheng was partly supported by the ERC Grant SCAPDE.
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Killip, R., Miao, C., Visan, M. et al. Sobolev spaces adapted to the Schrödinger operator with inverse-square potential. Math. Z. 288, 1273–1298 (2018). https://doi.org/10.1007/s00209-017-1934-8
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DOI: https://doi.org/10.1007/s00209-017-1934-8
Keywords
- Riesz transforms
- Inverse-square potential
- Littlewood–Paley theory
- Mikhlin multiplier theorem
- Heat kernel estimate