- Research Article
- Open Access

# Poincaré Inequalities with Luxemburg Norms in -Averaging Domains

- Yuming Xing
^{1}Email author

**2010**:241759

https://doi.org/10.1155/2010/241759

© Yuming Xing. 2010

**Received:**28 November 2009**Accepted:**29 January 2010**Published:**2 February 2010

## Abstract

We prove both local and global Poincaré inequalities with Luxemburg norms for differential forms in -averaging domains, which can be considered as generalizations of the existing versions of Poincaré inequalities.

## Keywords

- Closed Form
- Convex Function
- Bounded Domain
- Integrable Form
- Maximal Operator

## 1. Introduction

The Poincaré-type inequality has been playing a crucial role in analysis and related fields during the last several decades. Many versions of the Poincaré inequality have been developed and used in different areas of mathematics, including PDEs and analysis. For example, in 1989, Staples in [1] proved the following Poincaré inequality for Sobolev functions in -averaging domains. If is an -averaging domain, , then there exists a constant , such that

for each Sobolev function defined in . In [2], a global Poincaré inequality for solutions of the -harmonic equation was proved over the John domains; see [3–7] for more results about the Poincaré inequality. However, most of these inequalities are developed with the -norms. In this paper, we will establish the Poincaré inequalities with the Luxemburg norms in a relative large class of domains, the -averaging domain, so that many existing versions of the Poincaré inequality are special cases of our new results.

Let be a bounded domain in , , and let and be the balls with the same center and throughout this paper. The -dimensional Lebesgue measure of a set is denoted by . For a function , we denote the average of over by . All integrals involved in this paper are the Lebesgue integrals. Differential forms are generalizations of differentiable functions in . For example, the function is called a -form. A differential -form in can be written as , where the coefficient functions , , are differentiable. Similarly, a differential -form can be expressed as

where , . Let be the set of all -forms in , be the space of all differential -forms in , and let be the -forms in satisfying for all ordered -tuples , . We denote the exterior derivative by and the Hodge star operator by . The Hodge codifferential operator is given by , . We consider here the nonlinear partial differential equation

which is called nonhomogeneous -harmonic equation, where and satisfy the conditions , and for almost every and all . Here are constants and is a fixed exponent associated with (1.3). A solution to (1.3) is an element of the Sobolev space such that for all with compact support. If is a function ( -form) in , (1.3) reduces to

If the operator , (1.3) becomes

which is called the (homogeneous) -harmonic equation. Let be defined by with . Then, satisfies the required conditions and (1.5) becomes the -harmonic equation for differential forms. See [8–12] for recent results on the -harmonic equations and related topics.

## 2. Local Poincaré Inequalities

In this section, we establish the local Poincaré inequalities for the differential forms in any bounded domain. A continuously increasing function with is called an Orlicz function. The Orlicz space consists of all measurable functions on such that for some . is equipped with the nonlinear Luxemburg functional

A convex Orlicz function is often called a Young function. If is a Young function, then defines a norm in , which is called the Luxemburg norm.

Definition 2.1 (see [13]).

We say that a Young function lies in the class , , , if (i) and (ii) for all , where is a convex increasing function and is a concave increasing function on .

From [13], each of and in above definition is doubling in the sense that its values at and are uniformly comparable for all , and the consequent fact that

where and are constants. Also, for all and , the function belongs to for some constant . Here is defined by for ; and for . Particularly, if , we see that lies in , . We will need the following Reverse Hölder inequality.

Lemma 2.2 (see [8]).

Let be a solution of the nonhomogeneous -harmonic equation (1.3) in a domain an d let . Then, there exists a constant , independent of , such that for all balls with for some .

We first prove the following generalized Poincaré inequality that will be used to establish the global inequality.

Theorem.

Proof.

We have completed the proof of Theorem 2.3.

Since each of , and in Definition 2.1 is doubling, from the proof of Theorem 2.3 or directly from (2.3), we have

for all balls with and any constant . From (2.1) and (2.11), the following Poincaré inequality with the Luxemburg norm

holds under the conditions described in Theorem 2.3.

Theorem.

Proof.

The proof of Theorem 2.4 has been completed.

Similar to (2.12), from (2.1) and (2.13), the following Luxemburg norm Poincaré inequality

holds if all conditions of Theorem 2.4 are satisfied.

## 3. Global Poincaré Inequalities

In this section, we extend the local Poincaré inequalities into the global cases in the following -averaging domains.

Definition 3.1 (see [14]).

for some ball and all such that , where are constants with , and the supremum is over all balls .

From above definition we see that -averaging domains and -averaging domains are special -averaging domains when in Definition 3.1. Also, uniform domains and John domains are very special -averaging domains; see [15–18] for more results about domains.

Theorem.

where is some fixed ball.

Proof.

We have completed the proof of Theorem 3.2.

Similar to the local case, the following global Poincaré inequality with the Luxemburg norm

holds if all conditions in Theorem 3.2 are satisfied. Also, by the same way, we can extend Theorem 2.4 into the following global result in -averaging domains.

Theorem.

where is some fixed ball.

Note that (3.5) can be written as

It has been proved that any John domain is a special -averaging domain. Hence, we have the following results.

Corollary.

where is some fixed ball.

Choosing in Theorems 3.2 and 3.3, respectively, we obtain the following Poincaré inequalities with the -norms.

Corollary.

for any bounded -averaging domain and is some fixed ball.

Note that (3.8) can be written as the following version with the Luxemburg norm:

provided that the conditions in Corollary 3.5 are satisfied.

Corollary.

where is some fixed ball.

## 4. Applications

Choose to be a -form (a function) in the homogeneous -harmonic equation (1.5). Then, (1.5) reduces to the following -harmonic equation:

for functions. If the above operator is defined by with , then, satisfies the required conditions and (4.1) becomes the usual -harmonic equation

for functions. Thus, from Theorem 3.2, we have the following example.

Example.

where is some fixed ball.

Example.

where is some fixed ball.

- (i)
We know that the -averaging domains are the special -averaging domains. Thus, Theorems 3.2 and 3.3 also hold for the -averaging domain. (ii) In Theorems 2.4 and 3.3, does not need to be a solution of any version of the -harmonic equation.

## Declarations

### Acknowledgments

The author was supported by the NSF of China (no. 10771044) and by the Science Research Foundation at Harbin Institute of Technology (HITC200709).

## Authors’ Affiliations

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