- Research Article
- Open Access

# Brézis-Wainger Inequality on Riemannian Manifolds

- Przemysław Górka
^{1}Email author

**2008**:715961

https://doi.org/10.1155/2008/715961

© Przemysław Górka. 2008

**Received:**10 November 2007**Accepted:**29 April 2008**Published:**8 May 2008

## Abstract

The Brézis-Wainger inequality on a compact Riemannian manifold without boundary is shown. For this purpose, the Moser-Trudinger inequality and the Sobolev embedding theorem are applied.

## Keywords

- Partial Differential Equation
- Riemannian Manifold
- Sobolev Space
- Green Function
- Integral Representation

## 1. Introduction

There is no doubt that the Brézis-Wainger inequality (see [1]) is a very useful tool in the examination of partial differential equations. Namely, a lot of estimates to a solution of PDE are obtained with the help of the Brézis-Wainger inequality. Especially, the inequality is often applied in the theory of wave maps.

In this paper, we extend the Brézis-Wainger result onto a compact Riemannian manifold. We show the following theorem.

Theorem 1.1.

The proof relies on the application of a Moser-Trudinger inequality (see Theorem 2.2) and the Sobolev embedding theorem (see Theorem 2.1). Moreover, we will use the integral representation of a smooth function via the Green function (see [2]).

## 2. Preliminaries

In order to make this paper more readable, we recall some definitions and facts from the theory of Sobolev spaces on Riemannian manifolds. In particular, we present useful inequalities and embeddings.

*m*th covariant derivative of . Next, for and for a fixed integer and a real , we set

where by we have denoted the Riemannian measure on the manifold .

We define the Sobolev space as a completion of with respect to .

We close this section stating the following results, which will be used in the proof of the main result.

Theorem 2.1 (Sobolev Embedding Theorem [3, 4]).

Let be a smooth, compact Riemannian -manifold. Then, for any real numbers and any integers , if , then . Moreover, there exists a constant such that for all , the following inequality holds:

Theorem 2.2 (Moser-Trudinger Inequality [5]).

Let be a smooth, compact Riemannian -manifold and a positive integer, strictly smaller than . There exist a constant and such that for all with and , the following inequality holds:

Let us stress that this inequality is a generalization of the Moser and Trudinger result (see [6–8]).

## 3. Proof of the Main Result

In this section, we will prove the main result, that is, Theorem 1.1.

Proof.

Let us notice that by the assumptions we have an embedding . First of all, we show the following lemma.

Lemma 3.1.

Proof.

where is a constant from the Moser-Trudinger inequality (see Theorem 2.2).

From this, the proof of Lemma 3.1 follows.

Let us recall (see [2]) that for , a compact Riemannian -manifold, there exists a Green function such that

where is a Riemannian volume of the manifold , and is the Laplace-Beltrami operator on a manifold;

and is a Riemannian distance from to .

Using elementary calculations, one can easily show the lemma.

Lemma 3.2.

where is the exponent from the Sobolev theorem.

This finishes the proof of the inequality in the case such that and . Subsequently, one can easily obtain the inequality for such that .

## Declarations

### Acknowledgments

The author wishes to thank Professor Yuxiang Li for pointing out the paper [5]. Moreover, the author thanks the referees for comments and invaluable suggestions.

## Authors’ Affiliations

## References

- Brézis H, Wainger S:
**A note on limiting cases of Sobolev embeddings and convolution inequalities.***Communications in Partial Differential Equations*1980,**5**(7):773–789. 10.1080/03605308008820154MATHMathSciNetView ArticleGoogle Scholar - Aubin T:
*Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Grundlehren der Mathematischen Wissenschaften*.*Volume 252*. Springer, New York, NY, USA; 1982:xii+204.View ArticleGoogle Scholar - Hebey E:
*Sobolev Spaces on Riemannian Manifolds, Lecture Notes in Mathematics*.*Volume 1635*. Springer, Berlin, Germany; 1996:x+116.MATHGoogle Scholar - Hebey E:
*Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes in Mathematics*.*Volume 5*. American Mathematical Society, Providence, RI, USA; 2000:x+309.Google Scholar - Fontana L:
**Sharp borderline Sobolev inequalities on compact Riemannian manifolds.***Commentarii Mathematici Helvetici*1993,**68**(1):415–454. 10.1007/BF02565828MATHMathSciNetView ArticleGoogle Scholar - Adams DR:
**A sharp inequality of J. Moser for higher order derivatives.***The Annals of Mathematics*1988,**128**(2):385–398. 10.2307/1971445MATHView ArticleMathSciNetGoogle Scholar - Moser J:
**A sharp form of an inequality by N. Trudinger.***Indiana University Mathematics Journal*1971,**20:**1077–1092. 10.1512/iumj.1971.20.20101View ArticleMATHMathSciNetGoogle Scholar - Trudinger NS:
**On imbeddings into Orlicz spaces and some applications.***Journal of Mathematics and Mechanics*1967,**17:**473–483.MATHMathSciNetGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.