Abstract and Applied Analysis

Volume 2014 (2014), Article ID 908216, 4 pages

http://dx.doi.org/10.1155/2014/908216

Research Article

Trudinger-Moser Embedding on the Hyperbolic Space

Yunyan Yang and Xiaobao Zhu

Department of Mathematics, Renmin University of China, Beijing 100872, China

Received 6 November 2013; Accepted 26 December 2013; Published 18 February 2014

Academic Editor: Julio Rossi

Copyright © 2014 Yunyan Yang and Xiaobao Zhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let be the hyperbolic space of dimension . By our previous work (Theorem 2.3 of (Yang (2012))), for any , there exists a constant depending only on and such that where , is the measure of the unit sphere in , and . In this note we shall improve the above mentioned inequality. Particularly, we show that, for any and any , the above mentioned inequality holds with the definition of replaced by . We solve this problem by gluing local uniform estimates.

1. Introduction

Let be a bounded smooth domain in . The classical Trudinger-Moser inequality [1–3] says

for some constant depending only on , where is the usual Sobolev space and denotes the Lebesgue measure of . In the case where is an unbounded domain of , the above integral is infinite, but it was shown by Cao [4], Panda [5], and do Ó [6] that for any and any there holds

Later Ruf [7], Li and Ruf [8], and Adimurthi and Yang [9] obtained (2) in the critical case .

The study of Trudinger-Moser inequalities on compact Riemannian manifolds can be traced back to Aubin [10], Cherrier [11, 12], and Fontana [13]. A particular case is as follows. Let be an -dimensional compact Riemannian manifold without boundary. Then there holds

In view of (2), it is natural to consider extension of (3) on complete noncompact Riemannian manifolds. In [14] we obtained the following results. Let be a complete noncompact Riemannian manifold. If the Trudinger-Moser inequality holds on it, then there holds . If the Ricci curvature has lower bound, say , the injectivity radius has a positive lower bound then for any there exists a constant depending only on , , , and such that

Since depends on , (4) is weaker than (2) when is replaced by . Moreover, the condition that has lower bound is not necessary for the validity of the Trudinger-Moser inequality.

In this note, we will continue to study (4) in whole by gluing local uniform estimates. Particularly, we have the following.

Theorem 1. Let be an -dimensional hyperbolic space, , where is the measure of the unit sphere in . Then for any , any , and any satisfying , there exists some constant depending only on and such that

The proof of Theorem 1 is based on local uniform estimates (Lemma 2 below). This idea comes from [14] and can also be used in other cases [15, 16].

We remark that critical case of (5) was studied by Adimurthi and Tintarev [17], Mancini and Sandeep [18], and Mancini et al. ([19]) via different methods.

The remaining part of this note is organized as follows. In Section 2 we derive local uniform Trudinger-Moser inequalities; in Section 3, Theorem 1 is proved.

2. Local Estimates

To get (5), we need the following uniform local estimates which is an analogy of ([15], Lemma 4.1) or ([16], Lemma 1), and it is of its own interest.

Lemma 2. For any , any , and any with , there exists some constant depending only on such that

where denotes the geodesic ball of which is centered at with radius .

Proof. It is well known (see, e.g., [20], II.5, Theorem 1) that there exists a homomorphism such that , that in these coordinates the Riemannian metric can be represented by

where is the standard Euclidean metric on , and that

where denotes a ball centered at with radius . Moreover, the corresponding polar coordinates read

where is the standard metric on .

Denote ; then , , and . Calculating directly, we have

Since , we have . Noting that , we have by (10)

The standard Trudinger-Moser inequality (1) implies

where is a constant depending only on . This together with (10) immediately leads to

This is exactly (6) and thus ends the proof of the lemma.

As a corollary of Lemma 2, the following estimates can be compared with (1).

Corollary 3. For any , any , and any with , there exists some constant depending only on such that

Proof. Since

it follows from (13) that there exists some constant depending only on such that

In particular,

Here and in the sequel we often denote various constants by the same ; the reader can easily distinguish them from the context. Noting that for any , ,

we conclude

Combining (16) and (19), we obtain (14).

3. Proof of Theorem 1

In this section, we will prove Theorem 1 by gluing local estimates (6).

Proof of Theorem 1. Let be a positive real number which will be determined later. By ([21], Lemma 1.6) we can find a sequence of points such that , that for any , and that for any , belongs to at most balls , where depends only on . Let be the cut-off function satisfies the following conditions: (i) ; (ii) on and on ; (iii) . Let be fixed. For any satisfying

we have . For any , using an elementary inequality , we find some constant depending only on and such that

where in the last inequality we choose a sufficiently large to make sure . Let and . Noting that , we have by (21) and Lemma 2

where is a constant depending only on and . By the choice of and (22), we have

for some constant depending only on and . For any , we can choose sufficiently small such that . This ends the proof of Theorem 1.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the NSFC 11171347. The authors thank the referee for pointing out some grammar mistakes and the reference [19].

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