#### Abstract

We first establish the local Poincaré inequality with -averaging domains for the composition of the sharp maximal operator and potential operator, applied to the nonhomogenous -harmonic equation. Then, according to the definition of -averaging domains and relative properties, we demonstrate the global Poincaré inequality with -averaging domains. Finally, we give some illustrations for these theorems.

#### 1. Introduction

Poincaré inequality applied to differential forms has a vital role in PDEs, nonlinear analysis, and other related fields. With the further research conducted, we have established various versions of Poincaré inequality under different conditions. From [1–8], we have obtained the Poincaré inequality for the solution to the -harmonic equation in uniformly bounded domain, John domains, and -averaging domains. Nevertheless, most of these Poincaré inequalities are developed in -averaging domains. In this paper, we will establish the Poincaré inequality for the composition of the sharp maximal operator and potential operator in -averaging domains. As we all know, both the uniformly bounded domain and John domains are special -averaging domains, and the -averaging domains are also particular -averaging domains, so the following results are the generalizations of the Poincaré inequality in -averaging domains.

For convenience, we firstly introduce some notations and terminologies. Except for special instructions, is a bounded domain, denotes the Lebesgue measure of , and . The constant and can be varied at each step of the proof. Suppose that is a ball, with a radius , centered at . For any , and have the same center and satisfy . Let be the space of all -forms in , which is expanded by the exterior product of , where , . is the space of a smooth -form on . We use to denote the space of all differential -forms on ; that is, belongs to if and only if there exist some th-differential functions in such that . is a Banach space with the norm equipped by , where and every coefficient function . In fact, on is the Schwartz distribution. If a.e. and , is called a weight. Let ; then is a weighted Banach space with the norm expressed by . In this notation, the exterior derivative is denoted by and Hodge codifferential operator is expressed by . Search [9] for more details.

Considering our purpose, we intend to give a brief discussion about the -harmonic equation for the differential form. The following equation is called a nonhomogeneous -harmonic equation: where and satisfy the conditions: for almost every and all . Here, are constants and is a fixed exponent associated with (1). If , the equation is called a homogenous -harmonic equation. See [9] for more information.

In order to describe it easily, we first give some definitions in this part.

*Definition 1. *Let be a bounded domain and ; the sharp maximal operator is equipped with
where is the ball of radius , centered at , , .

Especially, if we take , denote .

*Definition 2 (see [10]). *Suppose that is a differential -form; the potential operator is expressed by
where the nonnegative and measurable function , defined on the set , is a kernel function, and the summation is over all ordered -tuple .

*Definition 3. *Take an increasingly continuous function as a convex function with , and is a bounded domain, for any ; the Orlicz norm for differential form is denoted by
where measure satisfies is a weight.

We call an Orlicz function if is an increasingly continuous function and satisfies and . Meanwhile, if the Orlicz function is a convex function, it is called a Young function.

Based on the above definition, we get the notation of -averaging domains.

*Definition 4 (see [3]). *Let be a Young function; the proper domain is called the -averaging domains if and there exists a constant such that for any and , satisfies
where the measure is denoted by , is a weight, and are constants with , , and the supremum is over all balls with .

Notice that if we let , -averaging domains become the -averaging domains, so -averaging domains are the generalization of -averaging domains.

*Definition 5 (see [11]). *We call belongs to the WRH-class, , if for any constants , and any ball with , there exists a constant such that satisfies
where is a constant.

*Remark 6. *If is a solution to the -harmonic equation, we can prove that belongs to the WRH-class.

#### 2. Main Results

Before the main results are given, we need to impose some restrictions on the kernel function and Young function . Firstly, let the kernel function satisfy the standard estimates; it is equal to say that if there exist and a constant such that for any point , the kernel function satisfies that(1), ;(2), ;(3), , where function , .

With regard to the Young function , we let the Young function belong to the -class ; that is, for any , the Young function satisfies that(1);(2),where and are the increasingly convex and concave functions defined on , respectively.

Now, we establish these two important theorems based on the above conditions.

Theorem 7. *Suppose that the Young function belongs to the -class, is a solution to the nonhomogenous -harmonic equation, the sharp maximal operator is noted by , is the potential operator with its kernel function satisfying the standard estimates, , and the bounded subset is the -averaging domains. Then, for any ball , one gets
**
where and and the constant .*

Based on the above theorem, we can establish the following theorem for the global Poincaré inequality in -averaging domains.

Theorem 8. *Suppose that the Young function belongs to the -class, is a solution to the nonhomogenous -harmonic equation, the sharp maximal operator is denoted by , is the potential operator with its kernel function satisfying the standard estimates, , , and the bounded subset is the -averaging domains. Then, one has
**
where is a fixed ball, which appears in Definition 4.*

#### 3. Preliminary Results

For proving the theorems in Section 2, we will show and demonstrate some lemmas in this part.

Lemma 9 (see [9]). *Let , , and , if and are the measurable functions defined on , then
**
for any .*

Lemma 10 (see [5]). *Let be the potential operator applied on a differential form with , , and assume that the weight belongs to with . Then, there exists a constant , independent of such that
**
for any , where is a constant.*

*Remark 11. *If we take , we get

Lemma 12 (see [3]). *Take defined on to be a strictly increasing convex function, , and is a domain. Assume that satisfies and, for any constant ,
**
where is a Radon measure defined by with a weight ; then for any , one obtains
*

Lemma 13. *If , then there exist constants and , not dependent on , such that
**
for all balls contained in .*

Lemma 14. *The sharp maximal operator is denoted by Definition 1, and the potential operator is defined by Definition 2 with the kernel function satisfying the standard estimates, , . Then, there exists a constant , independent of , such that
**
for all balls .*

*Proof . *Let be a ball in , using Lemma 10 on any , we have
From Lemma 3.5 in [7], it follows that
where . Substituting (18) into (17) yields
Taking the supremum for , we get that
That is,
According to the definition of norm and formula (21), it yields
Choosing , , and in Lemma 12, we have
The proof of Lemma 14 has been completed.

Lemma 15. *Suppose that is a solution to the -harmonic equation, is a bounded domain, is a potential operator with the kernel function satisfying the standard estimates, and the sharp maximal operator is expressed by Definition 1, , . Then, there exists a constant , such that
**
where the ball with , constant , the measure is defined by , weight , , for some and a constant .*

*Proof . *Because , for any with contained in , using Lemmas 9 and 14, we have
According to Lemma 14 and Definition 5, letting , we get
Therefore, we know that
Because of , and using generalized Hölder’s inequality, we get
In the light of , finding details in [9], we know
Therefore, we can see that
In addition, considering , so we have that
Combining (27), (28), and (31), we obtain
Therefore, we finish the proof of this lemma.

#### 4. Demonstration of Main Results

According to the above definitions and lemmas, we will prove these two theorems in detail. Firstly, let us prove Theorem 7.

*Proof of Theorem 7. *Let and , and are, respectively, convex and concave increasing function, use Lemma 15, and take ; then
Because , we know that
Furthermore, we obtain
For function , using Jensen's inequality, we get
Using the doubling property of for the above the formula, we have
The proof of Theorem 7 has been finished.

Now, we will use Definition 4 and Theorem 7 to prove Theorem 8.

* Proof of Theorem 8. *According to Definition 4, we can know
Because is independent on the ball , we obtain that
We finish the proof of Theorem 8.

#### 5. Applications

In this part, we firstly use Theorem 8 to do an estimate for a solution to the Laplace equation .

*Example 16. *Let be a differential 2-form in , and
where . It is very easy to obtain that and , so is a solution for the Laplace equation . If we take
then is a Young function and belongs to the -class, with . According to Theorem 8, we get that, for any fixed , there exists a constant such that
where .

Now, our aim is to prove the following corollary by using Theorem 7.

Corollary 17. *Suppose that the Young function belongs to the -class, and is a solution to the nonhomogenous -harmonic equation. The sharp maximal operator is noted by , is the potential operator with its kernel function satisfying the standard estimates (, ), and the bounded is the -averaging domains. Then, there exists a constant , such that
**
where with , and the constant .*

*Proof . *By using Minkowski inequality, we know that
From [12] and formula (22), we have
where . In addition, according to Theorem 7, there exists a constant such that
Substituting (45) and (46) into (44), we conclude that there exists a constant , independent of , such that
The proof of Corollary 17 has been completed.

Virtually, we can obtain a global estimate about the composition operator by using Definition 4.

Corollary 18. *Suppose that the Young function belongs to the -class, and is a solution to the nonhomogenous -harmonic equation. Let the sharp maximal operator be noted by , is the potential operator with its kernel function satisfying standard estimates (, ), and the bounded is the -averaging domains. Then, there exists a constant such that
**
where is a fixed ball.*

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.