Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 1273–1298 | Cite as

Sobolev spaces adapted to the Schrödinger operator with inverse-square potential

Article

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 \(\mathcal L_a\).

Keywords

Riesz transforms Inverse-square potential Littlewood–Paley theory Mikhlin multiplier theorem Heat kernel estimate 

Mathematics Subject Classification

35P25 35Q55 

Notes

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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Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  • R. Killip
    • 1
  • C. Miao
    • 2
  • M. Visan
    • 1
  • J. Zhang
    • 3
  • J. Zheng
    • 4
  1. 1.Department of MathematicsUCLALos AngelesUSA
  2. 2.Institute of Applied Physics and Computational MathematicsBeijingChina
  3. 3.Department of MathematicsBeijing Institute of TechnologyBeijingChina
  4. 4.Université Nice Sophia-AntipolisNice Cedex 02France

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