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Published February 2021 | Submitted
Journal Article Open

Equivalence of Sobolev Norms Involving Generalized Hardy Operators

Abstract

We consider the fractional Schrödinger operator with Hardy potential and critical or subcritical coupling constant. This operator generates a natural scale of homogeneous Sobolev spaces, which we compare with the ordinary homogeneous Sobolev spaces. As a byproduct, we obtain generalized and reversed Hardy inequalities for this operator. Our results extend those obtained recently for ordinary (non-fractional) Schrödinger operators and have an important application in the treatment of large relativistic atoms.

Additional Information

© The Author(s) 2019. Published by Oxford University Press. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model). Received: 24 July 2018; Revision received: 12 March 2019; Accepted: 17 May 2019; Published: 09 July 2019. The authors are very grateful to an anonymous referee for many helpful remarks and suggestions. Funding: This work was supported by the Deutsche Forschungsgemeinschaft [grant SI 348/15-1 to H.S. and EXC-2111 390814868 to R.L.F., K.M., H.S.] and the U.S. National Science Foundation [grant DMS-1363432 to R.L.F.].

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Created:
August 20, 2023
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October 20, 2023