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 Algebra i Analiz: Year: Volume: Issue: Page: Find

 Algebra i Analiz, 2006, Volume 18, Issue 1, Pages 108–123 (Mi aa61)

Research Papers

Weighted Sobolev-type embedding theorems for functions with symmetries

S. V. Ivanova, A. I. Nazarovb

a St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
b Saint-Petersburg State University

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 $W_p^1(\mathcal M)$ 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 $W_p^1(\mathcal M)$ into weighted Hölder and Orlicz spaces are treated.

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English version:
St. Petersburg Mathematical Journal, 2007, 18:1, 77–88

Bibliographic databases:

MSC: Primary 46E35; Secondary 58D99

Citation: S. V. Ivanov, A. I. Nazarov, “Weighted Sobolev-type embedding theorems for functions with symmetries”, Algebra i Analiz, 18:1 (2006), 108–123; St. Petersburg Math. J., 18:1 (2007), 77–88

Citation in format AMSBIB
\Bibitem{IvaNaz06} \by S.~V.~Ivanov, A.~I.~Nazarov \paper Weighted Sobolev-type embedding theorems for functions with symmetries \jour Algebra i Analiz \yr 2006 \vol 18 \issue 1 \pages 108--123 \mathnet{http://mi.mathnet.ru/aa61} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2225214} \zmath{https://zbmath.org/?q=an:1126.46022} \transl \jour St. Petersburg Math. J. \yr 2007 \vol 18 \issue 1 \pages 77--88 \crossref{https://doi.org/10.1090/S1061-0022-06-00943-5} 

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This publication is cited in the following articles:
1. Cabre X., Ros-Oton X., “Sobolev and Isoperimetric Inequalities with Monomial Weights”, J. Differ. Equ., 255:11 (2013), 4312–4336
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