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Weighted Sobolev-type embedding theorems for functions with symmetries


Authors: S. V. Ivanov and A. I. Nazarov
Translated by: A. I. Nazarov
Original publication: Algebra i Analiz, tom 18 (2006), nomer 1.
Journal: St. Petersburg Math. J. 18 (2007), 77-88
MSC (2000): Primary 46E35; Secondary 58D99
DOI: https://doi.org/10.1090/S1061-0022-06-00943-5
Published electronically: November 27, 2006
MathSciNet review: 2225214
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Abstract: It is well known that Sobolev embeddings can be refined in the presence of symmetries. Hebey and Vaugon (1997) studied this phenomena in the context of an arbitrary Riemannian manifold $ \mathcal{M}$ and a compact group of isometries $ G$. They showed that the limit Sobolev exponent increases if there are no points in $ \mathcal{M}$ with discrete orbits under the action of $ G$.

In the paper, the situation where $ \mathcal{M}$ contains points with discrete orbits is considered. It is shown that the limit Sobolev exponent for $ W^1_p({\mathcal{M}})$ increases in the case of embeddings into weighted spaces $ L_q({\mathcal{M}},w)$ instead of the usual $ L_q$ spaces, where the weight function $ w(x)$ is a positive power of the distance from $ x$ to the set of points with discrete orbits. Also, embeddings of $ W^1_p({\mathcal{M}})$ into weighted Hölder and Orlicz spaces are treated.


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Additional Information

S. V. Ivanov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: svivanov@pdmi.ras.ru

A. I. Nazarov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Pr. 28, St. Petersburg 198904, Russia
Email: an@AN4751.spb.edu

DOI: https://doi.org/10.1090/S1061-0022-06-00943-5
Keywords: Embedding theorems, Sobolev spaces, symmetries
Received by editor(s): June 28, 2005
Published electronically: November 27, 2006
Additional Notes: The first author was partially supported by RFBR (grant no. 05-01-00939) and by the Russian Science Support Foundation. The second author was partially supported by RFBR (grant no. 05-01-01063).
Article copyright: © Copyright 2006 American Mathematical Society

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