Poincaré Inequalities with Luxemburg Norms in -Averaging Domains

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.

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

(1.1)

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

(1.2)

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

(1.3)

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

(1.4)

If the operator , (1.3) becomes

(1.5)

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

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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

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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.

Let be a Young function in the class , , , and be a bounded domain. Assume that and is a solution of the nonhomogeneous -harmonic equation (1.3) in , . Then for any ball with , there exists a constant , independent of , such that

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Proof.

From [7, ()], we have

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for any . Note that if is a solution of the nonhomogeneous -harmonic equation (1.3), then is also a solution of (1.3) since is a closed form. From Lemma 2.2, it follows that

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for any positive numbers and . From (2.5), (i) in Definition 2.1, and using the fact that is an increasing function, Jensen's inequality, and noticing that and are doubling, we have

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Since , then . Hence, we have From (i) in Definition 2.1, we find that . Thus,

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Combining (2.6) and (2.7) yields

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Using Jensen's inequality for , (2.2), and noticing that and are doubling, we obtain

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Substituting (2.8) into (2.9) and noticing that is doubling, we have

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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

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

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holds under the conditions described in Theorem 2.3.

Theorem.

Let be a Young function in the class , , , let be a bounded domain and . Assume that is any differential -form, , and . Then for any ball , there exists a constant , independent of , such that

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Proof.

From (2.9), it follows that

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If , by assumption, we have . Using the Poincaré-type inequality for differential forms

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we find that

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Note that the -norm of increases with and as , and it follows that (2.16) still holds when . Since is increasing, from (2.14) and (2.16), we obtain

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Applying (2.17), (i) in Definition 2.1, Jensen's inequality, and noticing that and are doubling, we have

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Using (i) in Definition 2.1 again yields

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Combining (2.18) and (2.19), we obtain

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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

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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]).

Let be an increasing convex function on with . We call a proper subdomain an -averaging domain, if and there exists a constant C such that

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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.

Let be a Young function in the class , , , an d let be any bounded -averaging domain. Assume that and is a solution of the nonhomogeneous -harmonic equation (1.4) in , . Then, there exists a constant , independent of , such that

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where is some fixed ball.

Proof.

From Definition 3.1, (2.3) and noticing that is doubling, we have

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We have completed the proof of Theorem 3.2.

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

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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.

Let be a Young function in the class , , , be a bounded -averaging domain and . Assume that and and . Then, there exists a constant , independent of , such that

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where is some fixed ball.

Note that (3.5) can be written as

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

Corollary.

Let be a Young function in the class , , , and be a bounded John domain. Assume that and is a solution of the nonhomogeneous -harmonic equation (1.4) in , . Then, there exists a constant , independent of , such that

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where is some fixed ball.

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

Corollary.

Let , and . Assume that and is a solution of the nonhomogeneous -harmonic equation (1.4), . Then, there exists a constant , independent of , such that

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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:

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provided that the conditions in Corollary 3.5 are satisfied.

Corollary.

Let , and and be a bounded -averaging domain and . Assume that , , and Then, there exists a constant , independent of , such that

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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:

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

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

Example.

Let be a solution of the usual -harmonic equation (4.1) or the -harmonic equation (4.2), let be a Young function in the class , , , and be any bounded -averaging domain. If and , then there exists a constant , independent of , such that

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where is some fixed ball.

Example.

For any locally -integrable form , , the Hardy-Littlewood maximal operator is defined by and the sharp maximal operator by where is the ball of radius , centered at . Under the conditions of Theorem 3.3, we have

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where is some fixed ball.

Remark.

• (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.