# Brézis-Wainger Inequality on Riemannian Manifolds

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## 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.

where Open image in new window is a positive constant.

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 Open image in new window . Next, for Open image in new window and for a fixed integer Open image in new window and a real Open image in new window , we set

where by Open image in new window we have denoted the Riemannian measure on the manifold Open image in new window .

We define the Sobolev space Open image in new window as a completion of Open image in new window with respect to Open image in new window .

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

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

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

Let us stress that this inequality is a generalization of the Moser and Trudinger result (see [6, 7, 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 Open image in new window . First of all, we show the following lemma.

Lemma 3.1.

Proof.

where Open image in new window is a constant from the Moser-Trudinger inequality (see Theorem 2.2).

From this, the proof of Lemma 3.1 follows.

for Open image in new window .

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

where Open image in new window is a Riemannian volume of the manifold Open image in new window , and Open image in new window is the Laplace-Beltrami operator on a manifold;

and Open image in new window is a Riemannian distance from Open image in new window to Open image in new window .

and we have that Open image in new window .

Using elementary calculations, one can easily show the lemma.

Lemma 3.2.

where Open image in new window is the exponent from the Sobolev theorem.

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

Finally, we take Open image in new window . This completes the proof.

## Notes

### 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.

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