#### Abstract

We establish the Poincaré-type inequalities for the composition of the homotopy operator, exterior derivative operator, and the projection operator with norm applied to the nonhomogeneous -harmonic equation in -averaging domains.

#### 1. Introduction

The purpose of the paper is to develop the Poincaré-type inequalities for the composition of the homotopy operator , exterior derivative operator , and the projection operator with -norm. These operators play critical roles in investigating the properties of the solutions to PDEs and in controlling oscillatory behavior of the solutions in domains [1–6]. We first establish the local Poincaré inequalities for the composition in -averaging domains. Then, we prove the global Poincaré inequalities for the composition of in -averaging domains.

In this paper, we assume is a bounded and convex domain in and is a ball that is centred at with as its radius. For any , we use to denote the ball with centred at with radius . We do not distinguish the balls from the cubes in this paper. We use to denote the Lebesgue measure of a set . We call a weight if and a.e. For a function , we denote the average of over by . Differential forms are extensions of functions in . For example, the function is called a -form. Moreover, if is differentiable, it is called a differential -form. The -form in can be written as . If the coefficient functions , are differentiable, is called a differential -form. Similarly, a differential -form is generated by , , that is, = , where , . Let be the set of all -forms in , be the space of all differential -forms on and be the -forms on satisfying for all ordered -tuples , . We denote the exterior derivative by for , and define the Hodge star operator as follows: if , , is a differential -form, then , where , and . The Hodge codifferential operator is given by on .

We use to denote a bounded and convex domain on . Let be the th exterior power of the cotangent bundle, let be the space of smooth -forms on , and . The harmonic -fields are defined by . The orthogonal complement of in is defined by . Then, the Green's operator is defined as by assigning as the unique element of satisfying Poisson's equation , where is the harmonic projection operator that maps onto so that is the harmonic part of . See [7, 8] for more properties of these operators. The differential forms can be used to describe various systems of PDEs and to express different geometric structures on manifolds. See [9, 10].

The operator with the case was first introduced by Cartan in [11]. Then, it was extended to the following version in [12]. To each there corresponds a linear operator defined by − and the decomposition . A homotopy operator is defined by averaging over all points , where is normalized so that .

We are particularly interested in a class of differential forms satisfying the well-known nonhomogeneous -harmonic equation where and satisfy the conditions for almost every and all . Here and are constants and is a fixed exponent associated with (1). A solution to (1) is an element of the Sobolev space such that for all with compact support. If is a function (form) in , (1) reduces to If the operator , (1) becomes which is called the homogeneous -harmonic equation. Let be defined by with . Then, satisfies the required conditions and becomes the -harmonic equation for differential forms. Some results have been obtained in recent years about different versions of the -harmonic equation; see [1, 2, 8, 9, 13–15].

#### 2. Main Results and Proofs

*Definition 1. *Let be a continuously increasing convex function on with , and let be a domain with . If is a measurable function in , then we define the Orlicz norm of by
A continuously increasing function with is called an Orlicz function, and a convex Orlicz function is often called a Young function.

From Definition 1, it is easy to see that for any domain if is finite.

*Definition 2. *Let be an increasing convex function on with . We call a proper subdomain an Orlicz space , if and there exists a constant such that
for some ball and all integrable functions in , where the supremum is over all balls with .

*Definition 3 (see [15]). *We say that a Young function lies in the class , if and for all , where is a convex increasing function and is a concave increasing function on .

From [15], we know that the class contains some very interesting functions, such as and , and each of and is doubling in the sense that its values at and are uniformly comparable for all , and the consequent fact that where and are constants. We will need the following reverse Hlder inequality.

Lemma 4 (see [4]). *Let be a solution of the nonhomogeneous -harmonic equation (1) in a bounded and convex domain and . Then, there exists a constant , independent of , such that
**
for all balls with for some .*

Lemma 5 (see [1]). *Let be a solution of the nonhomogeneous -harmonic equation (1) in a bounded and convex domain . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that
**
for all balls with .*

Lemma 6 (see [1]). *Let be a solution of the nonhomogeneous -harmonic equation (1) in a bounded and convex domain . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that
**
for all balls with , where is a constant.*

Theorem 7. *Let be a Young function in the class , , , and let be a bounded convex domain. Assume that and is a solution of the nonhomogeneous -harmonic (1) in . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that
**
for some and all balls with .*

*Proof. *For any constant , from Lemmas 4 and 5, (i) in Definition 3, using the fact that is an increasing function, Jensen's inequality, and noticing that and are doubling, we have
Since , then,. Hence, we have . From (i) in Definition 3, we find that . Thus,
Combining (13) and (14) yields
Using Jensen's inequality for , (8), and noticing that and are doubling, we obtain
Substituting (15) into (16) and noticing that is doubling, we have
From Definition 2 and (17), we have the following version of Poincaré inequality with the Orlicz norm:
We have completed the proof of Theorem 7.

Theorem 8. *Let be a Young function in the class , and let be a bounded convex domain. Assume that and is a solution of the non-homogeneous -harmonic (1) in . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that
**
for some and all balls with .*

*Proof. *For any constant , from Lemma 5, (i) in Definition 3, using the fact that is an increasing function, Jensen's inequality, and noticing that and are doubling, we have
Since , then . From (i) in Definition 3, we find that . Thus,
Combining (20) and (21) yields
Using Jensen's inequality for , (8), and noticing that and are doubling, we obtain
Substituting (22) into (23) and noticing that is doubling, we have
From Definition 2 and (24), we have the following version of Poincaré inequality with the Orlicz norm:
We have completed the proof of Theorem 8.

Using a similar method to the proof of Theorem 8, we can establish the following version of Poincaré inequality with the Orlicz norm.

Theorem 9. *Let be a Young function in the class , and let be a bounded convex domain. Assume that and is a solution of the non-homogeneous -harmonic (1) in . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that
**
for some and all balls with .*

Theorem 10. *Let be a Young function in the class , and let be a bounded convex domain. Assume that and is a solution of the non-homogeneous -harmonic (1) in . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that**
where is some fixed ball.*

*Proof. *From definition of the and (12), we have
We have completed the proof of Theorem 10.

Using a similar method to the proof of Theorem 8, we obtain Theorem 11.

Theorem 11. *
where is some fixed ball.*

It has been proved in [5] that any John domain is special -averaging domain. Hence, we have the following results.

Corollary 12. *Let be a Young function in the class , and let be a bounded John domain. Assume that and is a solution of the non-homogeneous -harmonic (3) in . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that
**
where is some fixed ball.*

For some special convex function, we have the following theorems.

Theorem 13. *Let or a Young function, and a bounded convex domain. Assume that and is a solution of the nonhomogeneous -harmonic (1) in . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that
**
for some and all balls with .*

Theorem 14. *Let or ,, a Young function, and a bounded convex domain. Assume that and is a solution of the nonhomogeneous -harmonic (1) in . Let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that
**
where is some fixed ball.*

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of the paper.

#### Acknowledgment

The author would like to thank Professor Shusen Ding for his precious and thoughtful suggestions.