Advances in Study of Poincaré Inequalities and Related Operators

Xing, Yuming; Ding, Shusen

doi:https://doi.org/10.1155/2014/971595

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Recent Advances in Inequalities and p-Harmonic Equations

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Volume 2014 | Article ID 971595 | https://doi.org/10.1155/2014/971595

Advances in Study of Poincaré Inequalities and Related Operators

Yuming Xing¹and Shusen Ding²

Academic Editor: Peilin Shi

Received11 Jan 2014

Accepted06 Mar 2014

Published06 May 2014

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 where is a constant. Using the elementary inequality , , we have for a fixed . The first sum in (51) can be estimated by using Theorem 24 and the Covering Lemma We use the properties of -John domain to estimate the second sum in (51). Fix a cube and let be the chain in Lemma 28. Consider The chain also has property that, for each , , with , there exists a cube such that and , . Consider For such , , Let ; then By (49), (53), (55), and (50), we have Since for , , , and when , using (56), we find that Taking the th root both sides in (57) and using , (53), and (55), we obtain for every . Raising both sides of inequality (58) to powers and then integrating over both sides, we have Notice that Using elementary inequality for , we finally have Substituting (52) and (61) into (51), we obtain (46). The proof of Theorem 29 has been completed.

Similarly, choose and in Theorem 29, where is the center of with , , and . We have the following version of the Poincaré inequality with the Radon measures.

Corollary 30. Let be a solution of the nonhomogeneous -harmonic equation (4), , . Then, there exists a constant , independent of , such that for any bounded and convex -John domain and , where the Radon measures and are induced by and , respectively, where is the center of with . Here are constants with , , and the fixed cube , the constant , and the cubes appeared in Lemma 28.

Using (2) and (46), and noticing the fact that since is bounded, we have Thus, we have the following global Sobolev-Poincaré imbedding inequality with the Radon measure.

Corollary 31. Assume that all conditions in Theorem 29 are satisfied. Then, there exists a constant , independent of , such that for any bounded and convex -John domain .

4. Poincaré Inequalities with Luxemburg Norms

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 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 32 (see [28]). We say 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 [28], 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 is that 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 , .
The following results were proved in [29].

Theorem 33. Let be a Young function in the class , , and let be a bounded domain. Assume that and is a solution of the nonhomogeneous -harmonic equation (3) in , . Then, for any ball with , there exists a constant , independent of , such that

Proof. From (3.5) in [30], we have for any . Note that if is a solution of the nonhomogeneous -harmonic equation (3), then is also a solution of (3). Since is a closed form, from Lemma 3, it follows that for any positive numbers and . From (69), (i) in Definition 32, and 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 32, we find that . Thus, Combining (70) and (71) yields Using Jensen’s inequality for , (66), and noticing that and are doubling, we obtain Substituting (72) into (73) and noticing that is doubling, we have We have completed the proof of Theorem 33.

Since each of , and in Definition 32 is doubling, from the proof of Theorem 33 or directly from (67), we have for all balls with and any constant . From (65) and (75), the following Poincaré inequality with the Luxemburg norm holds under the conditions described in Theorem 33.

Theorem 34. Let be a Young function in the class , , , a bounded domain, and . Assume that is any differential -form, , , and . Then, for any ball , there exists a constant , independent of , such that

Proof. From (73), it follows that If , by assumption, we have . Using the Poincaré-type inequality for differential forms we find that Note that the -norm of increases with and as ; it follows that (80) still holds when . Since is increasing, from (78) and (80), we obtain Applying (81), (i) in Definition 32, Jensen’s inequality, and noticing that and are doubling, we have Using (i) in Definition 32 again yields Combining (82) and (83), we obtain The proof of Theorem 34 has been completed.

Similar to (76), from (65) and (77), the following Luxemburg norm Poincaré inequality holds if all conditions of Theorem 34 are satisfied.

Based on the above discussion, we extend the local Poincaré inequalities into the global cases in the following -averaging domains.

Definition 35 (see [31]). 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 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 35. Also, uniform domains and John domains are very special -averaging domains; see [2, 26, 32, 33] for more results about domains.

Theorem 36. Let be a Young function in the class , , , and any bounded -averaging domain. Assume that and are a solution of the nonhomogeneous -harmonic equation (4) in , . Then, there exists a constant , independent of , such that where is some fixed ball.

Similar to the local case, the following global Poincaré inequality with the Luxemburg norm holds if all conditions in Theorem 36 are satisfied. Also, by the same way, we can extend Theorem 34 into the following global result in -averaging domains.

Theorem 37. Let be a Young function in the class , , , a bounded -averaging domain, and . Assume that and and . Then, there exists a constant , independent of , such that where is some fixed ball.

Note that (89) can be written as

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

Corollary 38. Let be a Young function in the class , , , and a bounded John domain. Assume that and is a solution of the nonhomogeneous -harmonic equation (4) in , . Then, there exists a constant , independent of , such that where is some fixed ball.

Choosing in Theorems 36 and 37, respectively, we obtain the following Poincaré inequalities with the -norms.

Corollary 39. Let , , and . Assume that and is a solution of the nonhomogeneous -harmonic equation (4), . Then, there exists a constant , independent of , such that for any bounded -averaging domain and is some fixed ball.

Note that (92) can be written as the following version with the Luxemburg norm provided the conditions in Corollary 39 are satisfied.

Corollary 40. Let , , and and let be a bounded -averaging domain and . Assume that , , and . Then, there exists a constant , independent of , such that where is some fixed ball.

5. Inequalities for Green’s Operator

In this section, we say that has a generalized gradient, if, for each coordinate system, the pullbacks of the coordinate function of have generalized gradient in the familiar sense; see [21]. We write As usual, the harmonic -fields are defined by The orthogonal complement of in is defined by We define Green’s operator by setting equal to the unique element of satisfying Poisson’s equation where is either the harmonic projection or the harmonic part of . It has been proved in [34] that for and , .

We will need the following lemma about -estimates for Green’s operator which appeared in [34].

Lemma 41. Let , . For , there exists constant , independent of , such that

The following results about the composition of the Laplace-Beltrami operator and Green’s operator were proved in [35].

Theorem 42. Let , , and . Then, there exists a constant , independent of , such that

Theorem 43. Let , . Assume that . Then, there exists a constant , independent of , such that

Using Minkowski’s inequality and combining Theorems 42 and 43, we obtain the following corollary immediately.

Corollary 44. Let , . For , there exists a constant , independent of , such that

Theorem 45. Let , . If , then there exists a constant , independent of , such that for any convex and bounded with .

Corollary 46. Let , . Assume that . Then, for any convex and bounded with , there exists a constant , independent of , such that for any closed form , and

For , the vector-valued differential form consists of differential forms , where the partial differentiation is applied to the coefficients of . The notations and are self-explanatory. For and a weight , the weighted norm of over is denoted by where is a real number.

We have made necessary preparation in the previous section to prove the following Poincaré-type inequality for Green’s operator.

Theorem 47. Let , . Assume that . Then, there exists a constant , independent of , such that for all balls with .

As an application of Theorem 47, now we will prove the following Sobolev-Poincaré imbedding theorem for Green’s operator applied to a differential form .

Theorem 48. Let , . Assume that . Then, there exists a constant , independent of , such that for all balls with .

Proof. Since is a closed form for any form , it follows that is a closed form and Note that . Using (108) and (111), we obtain From (109) and (112), we have

Remark 49. Since Green’s operators can commute with and , Theorem 48 can be proved by applying Corollary 4.1 in [20] to and using (100).
We notice that all the results developed so far in this section are about differential forms. We do not require that differential form must satisfy any differential equation. However, if satisfies some version of harmonic equation, we can extend above inequalities into the weighted cases. In fact, now we will prove the following -weighted Sobolev-Poincaré imbedding theorem for Green’s operator .

Theorem 50. Let , , be an -harmonic tensor on a manifold . Assume that , , and for some . Then, there exists a constant , independent of , such that for all balls with .

Similarly, we can extend inequalities (104) and (106) to the following -weighted version.

Theorem 51. Let , , be an -harmonic tensor on a manifold . Assume that , , and for some . Then, there exists a constant , independent of , such that for all balls with and any real number with .

In [12], Wang and Wu proved the following global weighted Poincaré-type inequality for Green’s operator applied to the solutions of the nonhomogeneous -harmonic equation (3).

Theorem 52. Let be a solution of the nonhomogeneous -harmonic equation (3) and for some . Assume that is a fixed exponent associated with the -harmonic equation (3), . Then, there exists a constant , independent of , such that for any bounded -John domain . Here is a fixed cube appearing in Lemma 28.

Proof. From Theorem 51, we have where the measure is defined by . We use the notation and the covering described in Lemma 28 (Covering Lemma) and the properties of the measure . If , then for each cube with (see [36]) and for the sequence of cubes described in the Covering Lemma. From the elementary inequality , , we find that
The first sum can be estimated by (117) and the Covering Lemma Now we will estimate the second sum in (120). Fix a cube and let be the chain in the Covering Lemma. We have Using (117) and (118), we get Since for , , we have According to (122) and , we have (note ) for every . Hence, Now from (126), it follows that Notice that Finally, using the elementary inequality , we conclude that by the Covering Lemma. Combining (120), (121), and (129), we obtain the required inequality.

Using Theorem 52 and the proof of Theorem 3.2.10 in [2], we obtain the following global Sobolev imbedding inequality for Green’s operator applied to the solutions of the nonhomogeneous -harmonic equation in the John domain.

Theorem 53. Let be a solution of the nonhomogeneous -harmonic equation (3) and for some . Assume that is a fixed exponent associated with the -harmonic equation (3), . Then, there exists a constant , independent of , such that for any -John domain . Here is a fixed cube, which appears in Lemma 28.

Conflict of Interests

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

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Copyright © 2014 Yuming Xing and Shusen Ding. 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.

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