Trudinger-Moser Embedding on the Hyperbolic Space
Let (ℍn,g) be the hyperbolic space of dimension n. By our previous work (Theorem 2.3 of (Yang (2012))), for any 0<α<αn, there exists a constant τ>0 depending only on n and α such that supu∈W1,n(ℍn),∥u∥1,τ≤1∫ℍn(eαun/(n-1)-∑k=0n-2αk|u|nk/(n-1)/k!)dvg<∞, where αn=nωn-11/(n-1), ωn-1 is the m...
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/908216 |
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| author | Yunyan Yang Xiaobao Zhu |
| author_facet | Yunyan Yang Xiaobao Zhu |
| author_sort | Yunyan Yang |
| collection | DOAJ |
| description | Let (ℍn,g) be the hyperbolic space of dimension n. By our previous work (Theorem 2.3 of (Yang (2012))), for any 0<α<αn, there exists a constant τ>0 depending only on n and α such that supu∈W1,n(ℍn),∥u∥1,τ≤1∫ℍn(eαun/(n-1)-∑k=0n-2αk|u|nk/(n-1)/k!)dvg<∞, where αn=nωn-11/(n-1), ωn-1 is the measure of the unit sphere in ℝn, and u1,τ=∇guLn(ℍn)+τuLn(ℍn). In this note we shall improve the above mentioned inequality. Particularly, we show that, for any 0<α<αn and any τ>0, the above mentioned inequality holds with the definition of u1,τ replaced by (∫ℍn(|∇gu|n+τ|u|n)dvg)1/n. We solve this problem by gluing local uniform estimates. |
| format | Article |
| id | doaj-art-46993214911a471894f7ad48842cf96b |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-46993214911a471894f7ad48842cf96b2025-02-03T05:58:09ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/908216908216Trudinger-Moser Embedding on the Hyperbolic SpaceYunyan Yang0Xiaobao Zhu1Department of Mathematics, Renmin University of China, Beijing 100872, ChinaDepartment of Mathematics, Renmin University of China, Beijing 100872, ChinaLet (ℍn,g) be the hyperbolic space of dimension n. By our previous work (Theorem 2.3 of (Yang (2012))), for any 0<α<αn, there exists a constant τ>0 depending only on n and α such that supu∈W1,n(ℍn),∥u∥1,τ≤1∫ℍn(eαun/(n-1)-∑k=0n-2αk|u|nk/(n-1)/k!)dvg<∞, where αn=nωn-11/(n-1), ωn-1 is the measure of the unit sphere in ℝn, and u1,τ=∇guLn(ℍn)+τuLn(ℍn). In this note we shall improve the above mentioned inequality. Particularly, we show that, for any 0<α<αn and any τ>0, the above mentioned inequality holds with the definition of u1,τ replaced by (∫ℍn(|∇gu|n+τ|u|n)dvg)1/n. We solve this problem by gluing local uniform estimates.http://dx.doi.org/10.1155/2014/908216 |
| spellingShingle | Yunyan Yang Xiaobao Zhu Trudinger-Moser Embedding on the Hyperbolic Space Abstract and Applied Analysis |
| title | Trudinger-Moser Embedding on the Hyperbolic Space |
| title_full | Trudinger-Moser Embedding on the Hyperbolic Space |
| title_fullStr | Trudinger-Moser Embedding on the Hyperbolic Space |
| title_full_unstemmed | Trudinger-Moser Embedding on the Hyperbolic Space |
| title_short | Trudinger-Moser Embedding on the Hyperbolic Space |
| title_sort | trudinger moser embedding on the hyperbolic space |
| url | http://dx.doi.org/10.1155/2014/908216 |
| work_keys_str_mv | AT yunyanyang trudingermoserembeddingonthehyperbolicspace AT xiaobaozhu trudingermoserembeddingonthehyperbolicspace |