Geodesic
Weighted Sobolev-type embedding theorems for functions with symmetries
Algebra i analiz, Tome 18 (2006) no. 1, pp. 108-123.

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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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S. V. Ivanov; A. I. Nazarov. Weighted Sobolev-type embedding theorems for functions with symmetries. Algebra i analiz, Tome 18 (2006) no. 1, pp. 108-123. http://geodesic.mathdoc.fr/item/AA_2006_18_1_a3/

[1] Strauss W. A., “Existence of solitary waves in higher dimensions”, Comm. Math. Phys., 55 (1977), 149–162 | DOI | MR | Zbl

[2] Lions P.-L., “The concentration-compactness principle in the calculus of variations. The limit case. II, I”, Rev. Mat. Iberoamericana, 1 (1985), 45–121 ; 145–201 | MR | Zbl | Zbl

[3] Ding W., “On a conformally invariant elliptic equation on $\mathbb R^n$”, Comm. Math. Phys., 107 (1986), 331–335 | DOI | MR | Zbl

[4] Nazarov A. I., “O resheniyakh zadachi Dirikhle dlya uravneniya, vklyuchayuschego $p$-laplasian, v sfericheskom sloe”, Tr. S.-Peterburg. mat. o-va, 10, 2004, 33–62

[5] Bartsch T., Schneider M., Weth T., “Multiple solutions of a critical polyharmonic equation”, J. Reine Angew. Math., 571 (2004), 131–143 | MR | Zbl

[6] Scheglova A. P., “Mnozhestvennost reshenii odnoi kraevoi zadachi s nelineinym usloviem Neimana”, Probl. mat. anal., 30, S.-Peterburg. un-t, SPb., 2005, 121–144 | MR | Zbl

[7] Hebey E., Vaugon M., “Sobolev spaces in the presence of symmetries”, J. Math. Pures Appl. (9), 76 (1997), 859–881 | MR | Zbl

[8] Nazarov A. I., “O simmetrichnosti ekstremali v vesovoi teoreme vlozheniya”, Probl. mat. anal., 23, S.-Peterburg. un-t, SPb., 2001, 50–75 | Zbl

[9] Gilbarg D., Trudinger N. S., Ellipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989 | MR | Zbl

[10] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, 3-e izd., Nauka, M., 1984 | MR | Zbl

[11] Triebel H., “Approximation numbers and entropy numbers of embeddings of fractional Besov – Sobolev spaces in Orlicz spaces”, Proc. London Math. Soc. (3), 66:3 (1993), 589–618 | DOI | MR | Zbl

[12] Kobayasi Sh., Nomidzu K., Osnovy differentsialnoi geometrii, t. 1, Nauka, M., 1981