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Transactions of the American Mathematical Society

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On compact subsets of Sobolev spaces on manifolds


Authors: Leszek Skrzypczak and Cyril Tintarev
Journal: Trans. Amer. Math. Soc. 374 (2021), 3761-3777
MSC (2020): Primary 46B50, 46E35; Secondary 46N20
DOI: https://doi.org/10.1090/tran/8322
Published electronically: March 2, 2021
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Abstract: The paper considers compactness of Sobolev embeddings of non-compact manifolds, restricted to subsets (typically subspaces) defined either by conditions of symmetry (or quasisymmetry) relative to actions of compact groups, or by restriction in the number of variables, i.e. consisting of functions of the form $ f\circ \varphi $ with a fixed $ \varphi $. The manifolds are assumed to satisfy general common conditions under which Sobolev embeddings exist. We provide sufficient conditions for compactness of the embeddings, which in many situations are also necessary.


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Additional Information

Leszek Skrzypczak
Affiliation: Faculty of Mathematics & Computer Science, Adam Mickiewicz University, ul. Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland
Email: lskrzyp@amu.edu.pl

Cyril Tintarev
Affiliation: Technion – Israel Institute of Technology, Haifa 32000, Israel
Email: tammouz@gmail.com

DOI: https://doi.org/10.1090/tran/8322
Keywords: Compactness, Sobolev embeddings on manifolds, multiradial spaces
Received by editor(s): August 5, 2020
Received by editor(s) in revised form: October 22, 2020
Published electronically: March 2, 2021
Additional Notes: The first author was supported by National Science Center, Poland, Grant no. 2013/10/A/ST1/00091.
The second author expresses his gratitude to the first author and the Faculty of Mathematics & Computer Science of Adam Mickiewicz University, as well as to Simeon Reich, Yehuda Pinchover and the Faculty of Mathematics at Technion, for their kind hospitality. The latter stay was as a Lady Davis Visiting Professor.
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