# Trudinger-Moser Embedding on the Hyperbolic Space

Authors: Yunyan Yang; Xiaobao Zhu
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

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.


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## Body

## 1. Introduction

LetΩ be a bounded smooth domain in ℝn. The classical Trudinger-Moser inequality [1–3] says
(1)sup⁡u∈W01,n(Ω),∥u∥W01,n(Ω)≤1∫Ω‍eαn|u|n/(n-1)dx≤C|Ω|
for some constant C depending only on n, where W01,n(Ω) is the usual Sobolev space and |Ω| denotes the Lebesgue measure of Ω. In the case where Ω is an unbounded domain of ℝn, the above integral is infinite, but it was shown by Cao [4], Panda [5], and do Ó [6] that for any τ>0 and any α<αn there holds
(2)sup⁡u∈W1,n(ℝn),∫ℝn‍(|∇u|n+τ|u|n)dx≤1∫ℝn‍(∑k=0n-2‍αk|u|nk/(n-1)k!eα|u|n/(n-1)-∑k=0n-2‍αk|u|nk/(n-1)k!)dx<∞.
Later Ruf [7], Li and Ruf [8], and Adimurthi and Yang [9] obtained (2) in the critical case α=αn.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 (M,g) be an n-dimensional compact Riemannian manifold without boundary. Then there holds
(3)sup⁡∫M‍|∇gu|ndvg≤1,∫M‍udvg=0∫M‍eαn|u|n/(n-1)dvg<∞.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 (M,g) be a complete noncompact Riemannian manifold. If the Trudinger-Moser inequality holds on it, then there holds inf⁡x∈Mvolg(B1(x))>0. If the Ricci curvature has lower bound, say Ricg(M)≥-K, the injectivity radius has a positive lower bound i0 then for any α<αn there exists a constant τ>0 depending only on α, n, K, and i0 such that
(4)sup⁡(∫M‍|∇u|ndvg)1/n+τ(∫M‍|u|ndvg)1/n≤1∫M‍(∑k=0n-2‍αk|u|nk/(n-1)k!eα|u|n/(n-1)sup⁡(∫M‍|∇u|ndvg)1/n+τ(∫M‍|u|ndvg)1/n≤1∫M‍-∑k=0n-2‍αk|u|nk/(n-1)k!)dvg<∞.
Since τ depends on α, (4) is weaker than (2) when (M,g) is replaced by ℝn. Moreover, the condition that Ricg(M) 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 ℍn by gluing local uniform estimates. Particularly, we have the following.Theorem 1.
Let(ℍn,g) be an n-dimensional hyperbolic space, αn=nωn-11/(n-1), where ωn-1 is the measure of the unit sphere in ℝn. Then for any α<αn, any τ>0, and any u∈W1,n(ℍn) satisfying ∫ℍn‍(|∇gu|n+τ|u|n)dvg≤1, there exists some constant β depending only on n and τ such that
(5)∫ℍn‍(eα|u|n/(n-1)-∑k=0n-2‍αk|u|nk/(n-1)k!)dvg≤β.The proof of Theorem1 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 Section2 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 anyp∈ℍn, any R>0, and any u∈W01,n(BR(p)) with ∫BR(p)‍|∇gu|ndvg≤1, there exists some constant Cn depending only on n such that
(6)∫BR(p)‍(eαn|u|n/(n-1)-∑k=0n-2‍αnk|u|nk/(n-1)k!)dvg≤Cn(sinh⁡R)n∫BR(p)‍|∇gu|ndvg,
where BR(p) denotes the geodesic ball of (ℍn,g) which is centered at p with radius R.Proof.
It is well known (see, e.g., [20], II.5, Theorem 1) that there exists a homomorphism φ:ℍn→D={x∈ℝn:|x|<1} such that φ(p)=0, that in these coordinates the Riemannian metric g can be represented by
(7)g(x)=4(1-|x|2)2g0(x),
where g0(x)=∑i=1n‍(dxi)2 is the standard Euclidean metric on ℝn, and that
(8)φ(BR(p))=𝔹tanhR/2(0),
where 𝔹r(0)⊂ℝn denotes a ball centered at 0 with radius r. Moreover, the corresponding polar coordinates (r,θ)∈[0,∞)×𝕊n-1 read
(9)g=dr2+(sinh⁡r)2dθ2,
where dθ2 is the standard metric on 𝕊n-1.
Denotef=2/(1-|x|2); then g=f2g0, |∇gu|=f-1|∇g0(u∘φ-1)|, and dvg=fndvg0. Calculating directly, we have
(10)∫BR(0)‍|∇gu|ndvg=∫𝔹tanhR/2(0)‍|∇g0(u∘φ-1)|ndvg0.
Since u∈W01,n(BR(p)), we have u∘φ-1∈W01,n(𝔹tanhR/2(0)). Noting that ∫BR(p)‍|∇gu|ndvg≤1, we have by (10)
(11)∫𝔹tanhR/2(0)‍|∇g0(u∘φ-1)|ndvg0≤1.
The standard Trudinger-Moser inequality (1) implies
(12)∫𝔹tanhR/2(0)‍(eαn|u∘φ-1|n/(n-1)-∑k=0n-2‍αnk|u∘φ-1|nk/(n-1)k!)dvg0=∫𝔹tanhR/2(0)‍∑k=n-1∞‍αnk|u∘φ-1|nk/(n-1)k!dvg0≤∫𝔹tanhR/2(0)‍∑k=n-1∞‍αnk|(u∘φ-1)/∥∇g0(u∘φ-1)∥Ln|nk/(n-1)k!dvg0×∫𝔹tanhR/2(0)‍|∇g0(u∘φ-1)|ndvg0≤Cn(tanhR2)n∫𝔹tanhR/2(0)‍|∇g0(u∘φ-1)|ndvg0,
where Cn is a constant depending only on n. This together with (10) immediately leads to
(13)∫BR(p)‍(eαn|u|n/(n-1)-∑k=0n-2‍αnk|u|nk/(n-1)k!)dvg=∫𝔹tanhR/2(0)‍(∑k=0n-2‍αnk|u∘φ-1|nk/(n-1)k!eαn|u∘φ-1|n/(n-1)-∑k=0n-2‍αnk|u∘φ-1|nk/(n-1)k!)fndvg0≤Cn(2tanhR/21-(tanhR/2)2)n∫𝔹tanhR/2(0)‍|∇g0(u∘φ-1)|ndvg0=Cn(sinh⁡R)n∫BR(p)‍|∇gu|ndvg.
This is exactly (6) and thus ends the proof of the lemma.As a corollary of Lemma2, the following estimates can be compared with (1).Corollary 3.
For anyp∈ℍn, any R>0, and any u∈W01,n(BR(p)) with ∫BR(p)‍|∇gu|ndvg≤1, there exists some constant C depending only on n such that
(14)1Volg(BR(p))∫BR(p)‍eαn|u|n/(n-1)dvg≤Csinh⁡RR.Proof.
Since(15)lim⁡R→0+Volg(BR(p))R(sinh⁡R)n-1=lim⁡R→∞Volg(BR(p))R(sinh⁡R)n-1=1,
it follows from (13) that there exists some constant C depending only on n such that
(16)1Volg(BR(p))∫BR(p)‍(eαn|u|n/(n-1)-∑k=0n-2‍αnk|u|nk/(n-1)k!)dvg≤Csinh⁡RR.
In particular,
(17)∫BR(p)‍|u|ndvg≤Csinh⁡RRVolg(BR(p)).
Here and in the sequel we often denote various constants by the same C; the reader can easily distinguish them from the context. Noting that for any q, 0≤q≤n,
(18)∫BR(p)‍|u|qdvg≤Volg(BR(p))+∫BR(p)‍|u|ndvg,
we conclude
(19)∫BR(p)‍∑k=0n-2‍αnk|u|nk/(n-1)k!dvg≤Csinh⁡RRVolg(BR(p)).
Combining (16) and (19), we obtain (14).


## 3. Proof of Theorem1

In this section, we will prove Theorem1 by gluing local estimates (6).Proof of Theorem1.
LetR be a positive real number which will be determined later. By ([21], Lemma 1.6) we can find a sequence of points {xi}i=1∞⊂ℍn such that ∪i=1∞BR/2(xi)=ℍn, that BR/4(xi)∩BR/4(xj)=⌀ for any i≠j, and that for any x∈ℍn, x belongs to at most N balls BR(xi), where N depends only on n. Let ϕi be the cut-off function satisfies the following conditions: (i) ϕi∈C0∞(BR(xi)); (ii) 0≤ϕi≤1 on BR(xi) and ϕi≡1 on BR/2(xi); (iii) |∇gϕi(x)|≤4/R. Let τ>0 be fixed. For any u∈W1,n(ℍn) satisfying
(20)∫ℍn‍(|∇gu|n+τ|u|n)dvg≤1,
we have ϕiu∈W01,n(BR(xi)). For any ϵ>0, using an elementary inequality ab≤ϵa2+(1/(4ϵ))b2, we find some constant C depending only on n and ϵ such that
(21)∫BR(xi)‍|∇g(ϕiu)|ndvg≤(1+ϵ)∫BR(xi)‍ϕin|∇gu|ndvg+C∫BR(xi)‍|∇gϕi|n|u|ndvg≤(1+ϵ)∫BR(xi)‍|∇gu|ndvg+4nCRn∫BR(xi)‍|u|ndvg≤(1+ϵ)∫BR(xi)‍(|∇gu|n+τ|u|n)dvg,
where in the last inequality we choose a sufficiently large R to make sure 4nC/Rn≤(1+ϵ)τ. Let αϵ=αn/(1+ϵ)1/(n-1) and ϕiu~=ϕiu/(1+ϵ)1/n. Noting that ϕiu~∈W01,n(BR(xi)), we have by (21) and Lemma 2(22)∫BR/2(xi)‍(eαϵ|u|n/(n-1)-∑k=0n-2‍αϵk|u|nk/(n-1)k!)dvg≤∫BR(xi)‍(eαϵ|ϕiu|n/(n-1)-∑k=0n-2‍αϵk|ϕiu|nk/(n-1)k!)dvg=∫BR(xi)‍(eαn|ϕiu~|n/(n-1)-∑k=0n-2‍αnk|ϕiu~|nk/(n-1)k!)dvg≤Cn(sinh⁡R)n∫BR(xi)‍|∇g(ϕiu~)|ndvg≤C(sinh⁡R)n∫BR(xi)‍(|∇gu|n+τ|u|n)dvg,
where C is a constant depending only on n and τ. By the choice of {xi}i=1∞ and (22), we have
(23)∫ℍn‍(eαϵ|u|n/(n-1)-∑k=0n-2‍αϵk|u|nk/(n-1)k!)dvg≤∫∪i=1∞BR/2(xi)‍(eαϵ|u|n/(n-1)-∑k=0n-2‍αϵk|u|nk/(n-1)k!)dvg≤∑i=1∞‍∫BR/2(xi)‍(eαϵ|u|n/(n-1)-∑k=0n-2‍αϵk|u|nk/(n-1)k!)dvg≤∑i=1∞‍C(sinh⁡R)n∫BR(xi)‍(|∇gu|n+τ|u|n)dvg≤CN(sinh⁡R)n∫ℍn‍(|∇gu|n+τ|u|n)dvg≤CN(sinh⁡R)n
for some constant C depending only on n and τ. For any α<αn, we can choose ϵ>0 sufficiently small such that α<αϵ. This ends the proof of Theorem 1.



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*Source: 908216-2014-02-18.xml*