Review Article | Open Access

# Advances in Study of Poincaré Inequalities and Related Operators

**Academic Editor:**Peilin Shi

#### Abstract

We will present an up-to-date account of the recent advances made in the study of Poincaré inequalities for differential forms and related operators.

#### 1. Introduction

The Poincaré inequalities have been playing an important role in analysis and related fields during the last several decades. The study and applications of Poincaré inequalities are now ubiquitous in different areas, including PDEs and potential analysis. Some versions of the Poincaré inequality with different conditions for various families of functions or differential forms have been developed in recent years. 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 , where the integral is the Lebesgue integral, and is the Lebesgue measure of ; see [2–13] for more versions of the Poincaré inequality.

Throughout this paper, we assume that is a domain in , , and are the balls with the same center, and , . We do not distinguish the balls from cubes, throughout this paper. We use to denote the Lebesgue measure of the set . Differential forms are extensions of functions in . For example, the function is called a -form. Moreover, if is differentiable, then it is called a differential -form. The -form in can be written as . If the coefficient functions , , are differentiable, then is called a differential -form. Similarly, a differential -form is generated by , , that is, , where , . Let be the set of all -forms in , the space of all differential -forms on , and 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 write and , where , and is the Radon measure. We use to denote the Sobolev space of -forms. For and the Radon measure , the Sobolev norm with Radon measure of over is denoted by We consider here the solutions to 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 (3). A solution to (3) is an element of the Sobolev space such that for all with compact support. If is a function (-form) in , (3) reduces to If the operator , (3) 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. If is a function (-form), the above equation reduces to the usual -harmonic equation for functions. See [8, 12–18] for recent results on the solutions to the different versions of the -harmonic equation.

Let be a bounded and convex domain. The linear operator : was first introduced in [19], and then it was generalized to the following version in [20]. For any , there exists a linear operator defined by and the decomposition holds. The homotopy operator : is defined by , averaging over all points in , where is normalized by . The -form is defined by , , and , , for all , . From [20], we know that, for any bounded and convex domain , we have and . From [21], any open subset in is the union of a sequence of cubes , whose sides are parallel to the axes, whose interiors are mutually disjoint, and whose diameters are approximately proportional to their distances from . More explicitly (i) , (ii) if , (iii) there exist two constants (we can take and ), so that . Hence, the definition of the homotopy operator can be extended to any domain in . For any , for some , let be the homotopy operator defined on (each cube is bounded and convex). Thus, we can define the homotopy operator on any domain by . Hence, for any bounded domain and any differential form , we have where is a constant, independent of , and , .

We begin the discussion with the following definitions and weak reverse Hölder inequality in [22], which will be used repeatedly later.

*Definition 1 (see [2]). *A weight satisfies -condition in a subset , where , and write when
where supremum is over all .

*Definition 2 (see [10]). *A pair of weights satisfy the -condition in a set , and write for some and , if
for any ball .

Lemma 3. *Let be a solution of the nonhomogeneous -harmonic equation (3) in a domain and . Then, there exists a constant , independent of , such that
**
for all balls or cubes with for some .*

#### 2. Poincaré Inequalities for Differential Forms

We first discuss the Poincaré inequality for some differential forms. These forms are not necessary to be the solutions of any version of the -harmonic equation.

*Definition 4 (see [7]). *We call a proper subdomain an -averaging domain, , if and there exists a constant such that
for some ball and all . Here the measure is defined by , where is a weight and a.e., and the supremum is over all balls with .

In 1993, the following Poincaré-Sobolev inequality was proved in [20], which can be used to generalize the theory of Sobolev functions to that of differential forms.

Theorem 5. *Let and . Then, is in and
**
for a cube or a ball in , , and .*

From Corollary 4.1 in [20], we have the following version of Poincaré inequality for differential forms.

Theorem 6. *Let and . Then, is in with and
**
for a cube or a ball in , .*

The above Poincaré-Sobolev inequalities are about differential forms. We know that the -harmonic tensors are differential forms that satisfy the -harmonic equation. Then naturally, one would ask whether the Poincaré-Sobolev inequalities for -harmonic tensors are sharper than those for differential forms. The answer is “yes”. In [8], Ding and Nolder proved the following symmetric Poincaré-Sobolev inequalities for solutions of the nonhomogeneous -harmonic equation (3).

Theorem 7. *Let be a solution of the nonhomogeneous -harmonic equation (3) in a domain and , . Assume that , , and for some . Then
**
for all balls with . Here is a constant independent of and .*

Note that (12) is equivalent to

Theorem 8. *Let be a solution of the nonhomogeneous -harmonic equation (3) in a domain and , . Assume that , , , and for some . Then
**
for all balls with . Here is a constant independent of and .*

Note that (13) can be written as

Next, we will prove the following global weighted Poincaré-Sobolev inequality in -averaging domains.

Theorem 9. *Let with , , where is a constant. Assume that is an -harmonic tensor and ; then
**
for any -averaging domain and some ball with . Here the measure is defined by and is a constant independent of .*

Clearly, we can write (14) as

In [13], we have obtained Poincaré inequalities in which the integral on one side is about Lebesgue measure, but on the other side, the integral is about general measure induced by a weight . We state these results in the following.

Theorem 10. *Let be an -harmonic tensor in a domain and , . Assume that , , and for some . Then, there exists a constant , independent of , such that
**
for all balls with . Here, the measure is defined by .*

Theorem 11. *Let be an -harmonic tensor in a domain and , . Assume that , , and for some . Then
**
for all balls with . Here, the measure is defined by and is a constant independent of and .*

Theorem 12. *Let for some , , and . If , then
**
for any -averaging domain with and some ball with . Here, the measure is defined by and is a constant independent of and .*

Theorem 13. *Let with , , , and . If , then
**
for any -averaging domain and some ball with . Here, the measure is defined by and is a constant independent of .*

In recent years, several versions of the two weight Poincaré inequalities have been developed; see [13, 23–25] for example.

Theorem 14. *Let and , . Then, there exists a constant such that if and , where , , and , we have
**
for all balls . Here is a constant independent of and .*

Theorem 15. *Let and , . If and , then there exists a constant , independent of and , such that
**
for any ball or cube .*

We remark that the exponents and on the right hand sides of (20) and (21) can be improved. In fact, the following result is with the sharper right-hand side.

Theorem 16. *Let and , . Then, there exists a constant such that if and , where , and , we have
**
for all balls and any constant . Here is a constant independent of and .*

Clearly, in this result if , then .

Theorem 17. *Let and . Then there exists a constant such that if and , where , , , and , we have
**
for any -averaging domain and some ball with . Here the measures and are defined by , and is a constant independent of and .*

*Remark 18. *(1) Theorems 15 and 16 can be extended to the global versions. (2) From [26], we know that John domains are -averaging domains. Thus, the global results and Theorem 17 also hold if is a John domain.

Next, we discuss the following version of two-weight Poincaré inequality for differential forms.

Theorem 19. *Let be a differential form satisfying the -harmonic equation (3) in a domain and , . Suppose that for some and . If and , then there exists a constant , independent of , such that
**
for all balls with . Here is a closed form.*

If we choose in Theorem 19, we get the following version of the -weighted Poincaré inequality.

Corollary 20. *Let be a differential form satisfying the -harmonic equation (3) in a domain and , . Suppose that for some and . If , then there exists a constant , independent of , such that
**
for all balls with . Here is a closed form.*

Selecting in Theorem 19, we have the following two-weighted Poincaré inequality.

Corollary 21. *Let be a differential form satisfying the -harmonic equation (3) in a domain and , . Suppose that for some and . If and , then there exists a constant , independent of , such that
**
for all balls with . Here is a closed form.*

When in Corollary 21, we obtain the following symmetric two-weighted inequality.

Corollary 22. *Let be a differential form satisfying the -harmonic equation (3) in a domain and , . Suppose that for some . If and , then there exists a constant , independent of , such that
**
for all balls with . Here is a closed form.*

#### 3. Poincaré Inequalities with the Radon Measure

Normally, most of these inequalities are developed with the Lebesgue measure. It is noticeable that the following results from [27] established the Poincaré inequalities with Radon measure. The Radon measure is induced by , where may be an unbounded function. For example, it is allowed that contains a singular factor ; here is a constant and is some fixed point in the integral domain. We are interested in the singular factor case because normally we have to deal with the singular factor in applications, such as in the estimating of the Cauchy operator.

We first introduce the following lemmas that will be used to prove the local Poincaré inequality with the Radon measure.

Lemma 23. *Let , , and . If and are measurable functions on , then
**
for any .*

Now, we prove the following local Poincaré inequality with the Radon measure which will be used to establish the global inequality.

Theorem 24. *Let be a solution of the nonhomogeneous -harmonic equation (3) in a bounded domain , , , and . Then, for any ball with , there exists a constant , independent of , such that
**
where the Radon measures and are induced by and , respectively, with , , and . Here , and are some constants with , , , , and ; is the center of ball .*

*Proof. *Assume that is small enough so that and is any ball with center and radius . Also, let be small enough, and . For any differential forms , we have , where is the exterior differential operator and is the homotopy operator. From (5), we obtain
Since , it follows that
Choose ; then . Select . By the Hölder inequality, (30) and (31), we obtain
We may suppose that . Otherwise, we can move the center to the origin by a simple transformation. Thus, for any , . Using the polar coordinate substitution, we find that
Set ; then . From Lemma 3, we have
where is a constant. Using the Hölder inequality again, we obtain
By a simple calculation, we find that . Substituting (33), (34), and (35) into (32) yields
that is,
Notice that . Letting in (37), we obtain (29). The proof of Theorem 24 has been completed.

Let and in Theorem 24, where and are constants with . We have the following version of the Poincaré inequality with the Radon measures.

Corollary 25. *Let be a solution of the nonhomogeneous -harmonic equation (3) in a bounded domain , , , and . Then, there exists a constant , independent of , such that
**
for all balls with , , where the Radon measures and are induced by and , respectively, with and . Here and are some constants with , and ; is the center of ball .*

Let be a solution of (4). From (2), we have for any ball . Note that and . Hence, Substituting (40) into (39) and using (29) and the fact that for some constant , we have Using the same method as we developed in the proof of Theorem 24, we have Combining (41) and (42) and noticing that is bounded and , we find that where . Hence, we obtain the following Sobolev-Poincaré imbedding inequality with the Radon measure.

Corollary 26. *Let be a solution of (4) and all other conditions in Theorem 24 are satisfied. Then, there exists a constant , independent of , such that
**
for all balls with , .*

Then, we will prove the global Poincaré inequalities with the Radon measures in the following statement. We firstly introduce the definition of John domains and the Lemma.

*Definition 27. *A proper subdomain is called a -John domain, , if there exists a point , which can be joined with any other point by a continuous curve , so that
for each . Here is the Euclidean distance between and .

Lemma 28 (see [16] (Covering Lemma)). *Each has a modified Whitney cover of cubes such that and some , and if , then there exists a cube (this cube need not be a member of ) in such that . Moreover, if is -John, then there is a distinguished cube which can be connected with every cube by a chain of cubes from and such that , , for some .*

Theorem 29. *Let be any bounded and convex -John domain and let be a solution of the nonhomogeneous -harmonic equation (4), , and . Then, there exists a constant , independent of , such that
**
where the Radon measures and are induced by and , respectively, with , , and , is the center of with , and . Here are some constants with , , and the fixed cube , the constant , and the cubes appeared in Lemma 28.*

*Proof. *We may assume a.e. Otherwise, let and ; then . We define the new function by
Also, we choose the constant ; then for any . Therefore, , and satisfies all conditions required for , particularly, and
with . Hence, we may suppose that a.e. and have
We use the notation appearing in Lemma 28. There is a modified Whitney cover of cubes for such that , and for some . Since , for any , it follows that for some . It is easy to check that all conditions in Theorem 24 are satisfied. Applying Theorem 24 to , we obtain