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Abstract and Applied Analysis
Volume 2014, Article ID 596704, 21 pages
http://dx.doi.org/10.1155/2014/596704
Review Article

Recent Advances in -Theory of Homotopy Operator on Differential Forms

1Department of Mathematics, Seattle University, Seattle, WA 98122, USA
2Department of Epidemiology, Harvard University, Boston, MA 02115, USA
3Department of Mathematics, Harbin Institute of Technology, Harbin, China

Received 23 January 2014; Accepted 20 March 2014; Published 4 May 2014

Academic Editor: Yuming Xing

Copyright © 2014 Shusen Ding 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.

Abstract

The purpose of this survey paper is to present an up-to-date account of the recent advances made in the study of -theory of the homotopy operator applied to differential forms. Specifically, we will discuss various local and global norm estimates for the homotopy operator T and its compositions with other operators, such as Green’s operator and potential operator.

1. Introduction

The homotopy operator has been playing an important role in the study of -theory of differential forms. We all know that any differential form can be decomposed as , where is the differential operator and is the homotopy operator. Hence, the homotopy operator provides an effective tool to study various properties of different norms and the related operators. As extensions of functions, differential forms have become invaluable tools for many fields of sciences and engineering, including theoretical physics, general relativity, potential theory, and electromagnetism. They can be used to describe various systems of PDEs and to express different geometrical structures on manifolds. In recent years, much progress has been made in the investigation of differential forms and the related operators; see [17]. The purpose of this survey paper is to present an up-to-date account of the recent advances made in the study of -theory of the homotopy operator and its compositions applied to differential forms. We will first discuss the -norm and -norm inequalities in Sections 2 and 3, respectively. Then, we present Lipschitz and BMO norm inequalities in Sections 4 and 5. We also give some global -inequalities in Section 6. Finally, we discuss the compositions of homotopy operator with the projection operator, potential operator, and Green's operator in Sections 7, 8, and 9. We will keep using the traditional symbols and notations in this survey paper. Specifically, we always assume that is a bounded domain in , , and are the balls with the same center and throughout this paper. We use to denote the -dimensional Lebesgue measure of a set . For a function , the average of over is defined by . All integrals involved in this paper are the Lebesgue integrals. We call a weight if and a.e.. Differential forms are extensions of differentiable functions in . For instance, the function is called a -form. A differential -form in can be written as , where the coefficient functions , , are differentiable. Similarly, a differential -form can be expressed as where , . Let be the set of all -forms in , let be the space of all differential -forms in , and let be the -forms in satisfying for all ordered -tuples , . We denote the exterior derivative by and the Hodge star operator by . The Hodge codifferential operator is given by , .

Let be a bounded, convex domain. The following operator with the case was first introduced by Cartan in [8]. Then, it was extended to the following general version in [9]. For each , there corresponds a linear operator defined by ; and the decomposition . A homotopy operator is defined by averaging over all points in where is normalized by . For simplicity purpose, we write . Then, . By substituting and , we have where the vector function is given by . The integral (3) defines a bounded operator , and the decomposition holds for any differential form . The -form is defined by for all , . Also, for any differential form , we have From [10, Page 16], we know that 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 . Specifically, (i) , (ii) if , and (iii) there exist two constants (we can take and ), so that . Thus, the definition of the homotopy operator can be generalized 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 .

The nonlinear partial differential equation is called nonhomogeneous -harmonic equation, where and satisfy the conditions: for almost every and all . Here are constants and is a fixed exponent associated with (7). A solution to (7) is an element of the Sobolev space such that for all with compact support. If is a function (-form) in , (7) reduces to If the operator , (7) becomes which is called the (homogeneous) -harmonic equation. Let be defined by with . Then, satisfies the required conditions and (11) becomes the -harmonic equation for differential forms. See [1, 1118] for recent results on the -harmonic equations and related topics.

Lemma 1 (see [12]). Let be a solution of the nonhomogeneous -harmonic (7) in a domain and . Then, there exists a constant , independent of , such that for all balls with for some .

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 or Orlicz norm.

Definition 2 (see [19]). 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 [19], each of , and in the above definition is doubling in the sense that its values at and are uniformly comparable for all , and the consequent fact 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 , .

Lemma 3 (see [1]). Let be a solution to the nonhomogeneous -harmonic (7) on and be a constant. Then there exists a constant , independent of , such that for all balls or cubes with and all closed forms . Here .

Lemma 4 (see [1]). Suppose that is a solution to the nonhomogeneous -harmonic (7) on , and . There exists a constant , depending only on , , , , , and , such that for all with .

The following Hölder inequality will be used in this paper.

Lemma 5. Let , , and . If and are measurable functions on , then for any .

2. -Norm Inequalities

The following -norm Poincaré-type inequality for was proved in [13].

Theorem 6. Let , , , be any differential form in a bounded, convex domain and let be the homotopy operator defined in (2). Then, there exists a constant , independent of , such that for all balls with .

Proof. Using (4), (5), and (6), we have We have completed the proof of Theorem 6.

The basic -norm inequality (18) can be extended into different weighted cases. Let us recall some weight classes as follows. We first introduce the Muckenhoupt weights.

Definition 7. We say the weight satisfies the condition, , and write , if a.e., and for any ball .

Definition 8. A weight is called a doubling weight and write if there exists a constant such that for all balls with . Here the measure is defined by . If this condition holds only for all balls with , then is weak doubling and we write .

Definition 9. Let . It is said that satisfies a weak reverse Hölder inequality and write when there exist constants and such that for all balls with . We say that satisfies a reverse Hölder inequality when (22) holds with , and write . In fact the space is independent of .

If satisfies the -condition for all balls with , we write . Also we write and .

It is well known that if and only if . This is also true for and . Moreover, .

Definition 10. Let be a locally integrable nonnegative function in and assume that a.e.. We say that belongs to the class, , and or that is an -weight, and write or when it will not cause any confusion, if for all balls .
It is clear that is the usual -class; see [1] for more properties of -weights. We prove some properties of the -weights. The following theorem says that is an increasing class with respect to .
The following result shows that -weights have the property similar to the strong doubling property of -weights: if , , and the measure is defined by , then where is a ball in and is a measurable subset of .
If we put (24), then we have which is called the strong doubling property of -weights. It is well known that an -weight satisfies the following reverse Hölder inequality.
The definitions of the following several weight classes can be found in [1] and these weight classes have been widely used recently in the study of the integral properties of differential forms.

Definition 11. We say that the weight satisfies the -condition, and , and write , if for any ball . Here is a subset of .

Definition 12. A pair of weights satisfies the -condition in a set , and write , for some and with , if

Definition 13. A pair of weights satisfies the -condition in a set , and write for some and , if for any ball .

Definition 14. A pair of weights satisfies the -condition in a set , and write for some and , if for any ball .

Using the basic Poincaré-type estimate for the homotopy operator established in Theorem 6, we have the following -weighted inequality.

Theorem 15. Let , , , be a solution of the nonhomogeneous -harmonic (7) in a bounded domain and let be the homotopy operator defined in (2). Assume that and for some . Then, there exists a constant , independent of , such that for all balls with and any real number with .

The above -norm inequality can also be written in the integral form as

Also, using the procedure developed to extend the local inequalities into the John domains, we have the following global Poincaré-type inequality.

Theorem 16. Let be a solution of the nonhomogeneous -harmonic (7) and , , be the homotopy operator defined in (2). Assume that for some and is a fixed exponent associated with the -harmonic (7). Then, there exists a constant , independent of , such that for any bounded -John domain . Here is a fixed cube.

By the same method used to prove the imbedding inequalities, we can prove the following local and global imbedding inequalities, Theorems 17 and 18, respectively.

Theorem 17. Let , , , be a smooth differential form in a bounded domain and let be the homotopy operator defined in (2). Assume that and for some . Then, there exists a constant , independent of , such that for all balls with and any real number with .

Theorem 18. Let be a solution of the nonhomogeneous -harmonic (7) and let , , be the homotopy operator defined in (2). Assume that for some and is a fixed exponent associated with the -harmonic (7). Then, there exists a constant , independent of , such that for any bounded -John domain . Here is a fixed cube.

So far, we have presented the -weighted Poincaré-type estimates for the homotopy operator . Now, we state other estimates with different weights, such as -weights and -weights.

Theorem 19. Let , , , be a differential form satisfying the nonhomogeneous -harmonic (7) in a bounded domain and let be the homotopy operator defined in (2). Assume that for some and . Then, there exists a constant , independent of , such that for all balls with and any real number with . Here is some constant.

Note that inequality (35) can be written as

Theorem 20. Let , , , be a differential form satisfying (7) in a bounded domain and let be the homotopy operator defined in (2). Assume that and for some and . Then, there exists a constant , independent of , such that for all balls with and any real number with .

The above inequalities have integral representations; for example, inequality (38) can be written as The above estimates can be extended into the following two-weight case.

Theorem 21. Let , , , be a solution of the nonhomogeneous -harmonic (7) in a bounded domain and let be the homotopy operator defined in (2). Suppose that and for some and . Then, there exists a constant , independent of , such that for all balls with and any real number with .

In Theorem 21, we have assumed that . If the weights and satisfy some other condition, say , we have the following version of Poincaré-type inequality.

Theorem 22. Let , , , be a differential form satisfying (7) in a bounded domain and let be the homotopy operator defined in (2). Suppose that and for some and . Then, there exists a constant , independent of , such that for all balls with and any real number with .

Note that inequality can be written as Similarly, if , we have the following version of two-weight Poincaré inequality for differential forms.

Theorem 23. Let , , , be a differential form satisfying (7) in a bounded domain and let be the homotopy operator defined in (2). Suppose that for some and . If and , then there exists a constant , independent of , such that for all balls with .

If we choose in Theorem 23, we have the following version of the Poincaré inequality with .

Corollary 24. Let , , , be a differential form satisfying (7) in a bounded domain and let be the homotopy operator defined in (2). Suppose that for some . If and , then there exists a constant , independent of , such that for all balls with .

Choosing in Theorem 23, we obtain the following two-weighted Poincaré inequality.

Corollary 25. Let , , , be a differential form satisfying (7) in a bounded domain and let be the homotopy operator defined in (2). Suppose that for some , and , then there exists a constant , independent of , such that for all balls with .

Letting in Corollary 25, we find the following symmetric two-weighted inequality.

Corollary 26. Let , , , be a differential form satisfying (7) in a bounded domain and let be the homotopy operator defined in (2). Suppose that for some and , then there exists a constant , independent of , such that for all balls with .

Finally, when in Theorem 23, we have the following two-weighted inequality.

Corollary 27. Let , , , be a differential form satisfying (7) in a bounded domain and let be the homotopy operator defined in (2). Suppose that for some . If and , then there exists a constant , independent of , such that for all balls with .

3. -Norm Inequalities

The following local Poincaré-type inequality with the -norm was proved in [13], which can be used to establish the global inequality.

Theorem 28. Let be a Young function in the class , , , be a bounded and convex domain, and let , , be the homotopy operator defined in (2). Assume that and is a solution of the nonhomogeneous -harmonic (7) in . Then, there exists a constant , independent of , such that for all balls with .

Proof. From (18), we have for all balls with . From Lemma 1, for any positive numbers and , it follows that where is a constant . Using Jensen's inequality for , (14), (49), (50), and (i) in Definition 2, and noticing the fact that and are doubling and is an increasing function, we obtain Since , then . Hence, we have . Note that is doubling, we obtain Combining (51) and (52) yields We have completed the proof of Theorem 28.

Since each of , and in Definition 2 is doubling, from the proof of Theorem 28 or directly from (48), we have for all balls with and any constant . From (13) and (54), the following Poincaré inequality with the Luxemburg norm holds under the conditions described in Theorem 28.

Theorem 29. Let be a Young function in the class , , , , be a bounded domain, and , , be the homotopy operator defined in (2). Assume that is any differential -form, . Then, there exists a constant , independent of , such that for all balls with .

Proof. From (53), we have If , by assumption, we have . Using the Poincaré-type inequality for differential forms we find that We all know that for any differential form , , and . Hence, Combining (57), (59), and (60), we obtain for . Note that the -norm of increases with and as , it follows that (59) still holds when . Since is increasing, from (57) and (59), we obtain Applying (62), (i) in Definition 2, Jensen's inequality, and noticing that and are doubling, we have Using (i) in Definition 2 again yields Combining (63) and (64), we obtain The proof of Theorem 29 has been completed.

Similar to (55), from (18) and (56), the following Orlicz norm inequality holds if all conditions of Theorem 29 are satisfied.

4. Lipschitz and BMO Norm Inequalities

In this section, we will present Lipschitz and BMO norm inequalities for the homotopy operator. All results presented in this section and next section can be found in [14]. Let us recall the definitions of Lipschitz and BMO norms first.

Let , . We write , , if for some . Further, we write for those forms whose coefficients are in the usual Lipschitz space with exponent and write for this norm. Similarly, for , , we write if for some . When is a -form, (68) reduces to the classical definition of BMO. The definitions of the above Lipschitz and norms can be found in [1].

The following Theorem 30 indicates that we can use the -norm of to estimate the Lipschitz norm of .

Theorem 30. Let , , , be a solution of the -harmonic (1) in a bounded, convex domain and let be the homotopy operator defined in (7). Then, there exists a constant , independent of , such that where is a constant with .

Proof. From Theorem 6, we have for all balls with , where is a constant. Using the Hölder inequality with , we find that Using the definition of the Lipschitz norm, (71), and , we obtain The proof of Theorem 30 has been completed.

Using the similar method involved in the proof of Theorem 30, we have the following Lipschitz norm inequalities for Green's operator and the projection operator ; see [1] for more properties about Green's operator and the projection operator .

Theorem 31. Let , , , be a solution of the -harmonic (7) in a bounded domain , and let be Green's operator and let be the projection operator. Then, there exists a constant , independent of , such that where is a constant with .

We have discussed some estimates for the Lipschitz norm above. Next, we will focus on the estimates for the norm . For this, let , , , and let be a bounded domain. Then, from the definitions of the Lipschitz and norms, we have where is a positive constant. Hence, we have proved the following inequality between the Lipschitz norm and the norm.

Theorem 32. If a differential form , , , in a bounded domain , then and where is a constant.

Using Theorems 32 and 30, we obtain the following inequality between the norm and the norm.

Theorem 33. Let , , , be a solution of the -harmonic (7) in a bounded, convex domain and let be the homotopy operator defined in (2). Then, there exists a constant , independent of , such that

Proof. Since inequality (75) holds for any differential form, we may replace by in inequality (75). Thus, it follows that where is a constant with . On the other hand, from Theorem 30 we have