*L*^{p}-Theory of Differential Forms and Related Operators with Applications

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Research Article | Open Access

# Poincaré-Type Inequalities for the Composite Operator in -Averaging Domains

**Academic Editor:**Shusen Ding

#### 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.

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#### Copyright

Copyright © 2014 Guannan Shi et al. 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.