#### 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].