Recent Advances in 𝐿^𝑝-Theory of Homotopy Operator on Differential Forms

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 [1–7]. 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, 11–18] 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 Combination of (77) and (78) yields . The proof of Theorem 33 has been completed.

As in the proof of Theorem 33, using inequality (75) and Theorem 31, we obtain the following result immediately.

Theorem 34. 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

5. Weighted Lipschitz and BMO Norm Inequalities

In this section, we present the weighted Lipschitz and norms inequalities. For , , we write , , if for some , where is a bounded domain, the measure is defined by , is a weight, and is a real number. For convenience, we will write the following simple notation for . Similarly, for , , we will write if for some , where the measure is defined by , is a weight, and is a real number. Again, we will write to replace when it is clear that the integral is weighted.

Theorem 35. Let , , , be a solution of the nonhomogeneous -harmonic (7) in a bounded, convex domain and let be the homotopy operator defined in (2), where the measure is defined by and for some with for any . Then, there exists a constant , independent of , such that where and are constants with and .

Proof. First, we note that , which implies that for any ball . Using (30) and the Hölder inequality with , we find that Next, from the definition of the weighted Lipschitz norm, (80), and (84), we obtain since and . We have completed the proof of Theorem 35.

Next, we present the norm estimate. Let , , , in a bounded domain . From the definitions of the weighted Lipschitz and the weighted norms, we have where is a positive constant. Hence, we have obtained the following theorem.

Theorem 36. Let , , , be any differential form in a bounded domain , where is a weight for some . Then, and where and are constants with .

Theorem 37. Let , , , be a solution of the nonhomogeneous -harmonic (7) in a bounded, convex domain and let be the homotopy operator defined in (2), where the measure is defined by and for some with for any . Then, there exists a constant , independent of , such that where is a constant with .

Proof. Replacing by in Theorem 36, we have where is a constant with . Now, from Theorem 35, we find that
Substituting (90) into (89), we obtain . The proof of Theorem 37 has been completed.

6. Global -Inequalities

In this section, we discuss the global inequalities in the following -averaging domains. See [13] for detailed proofs.

Definition 38 (see [20]). Let be an increasing convex function on with . We call a proper subdomain an -averaging domain, if and there exists a constant such that for some ball and all such that , where are constants with , and the supremum is over all balls .

From the above definition, we see that -averaging domains and -averaging domains are special -averaging domains when in Definition 38. Also, uniform domains and John domains are very special -averaging domains; see [20, 21] for more results about domains.

Theorem 39. Let be a Young function in the class , , , and let be any bounded -averaging 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 where is some fixed ball.

Proof. From Definition 38, (48), and noticing that is doubling, we have We have completed the proof of Theorem 39.

Similar to the local case, the following global inequality with the Orlicz norm holds if all conditions in Theorem 39 are satisfied. Also, by the same way, we can extend Theorem 28 into the following global result in -averaging domains.

Theorem 40. Let be a Young function in the class , , , be a bounded -averaging domain and , and , , be the homotopy operator defined in (2). Assume that and . Then, there exists a constant , independent of , such that where is some fixed ball.

Note that (95) can be written as It has been proved that any John domain is a special -averaging domain. Hence, we have the following results.

Corollary 41. Let be a Young function in the class , , , and let be a bounded John 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 where is some fixed ball.

Choosing in Theorems 39 and 40, respectively, we obtain the following Poincaré inequalities with the -norms.

Corollary 42. Let , , , and let , , be the homotopy operator defined in (2). Assume that and is a solution of the nonhomogeneous -harmonic (7). Then, there exists a constant , independent of , such that for any bounded -averaging domain and is some fixed ball.

Note that (98) can be written as the following version with the Luxemburg norm provided the conditions in Corollary 42 are satisfied.

Corollary 43. Let , , , be a bounded -averaging domain and , and , , be the homotopy operator defined in (2). Assume that , . Then, there exists a constant , independent of , such that where is some fixed ball.

7. Composition of Homotopy and Projection Operators

In this section, we present the norm estimates for the composition of the homotopy operator and projection operator. The results presented in this section can be found in [15, 16]. We assume that is a domain in an oriented, compact, smooth Riemannian manifold of dimension . Let be the th exterior power of the cotangent bundle, and let be the space of smooth -forms on and has generalized gradient}.The harmonic -fields are defined by for some . The orthogonal complement of in is defined by for all . Then, Green's operator is defined as by assigning be 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 [1, 22, 23] for more properties of these operators.

Lemma 44 (see [20]). Let be a strictly increasing convex function on with , and let be a domain in . Assume that is a function in such that and for any constant , where is a Radon measure defined by for a weight . Then, we have for any positive constant , where .

Lemma 45 (see [24]). Let and , . Then, there exists a positive constant , independent of , such that

Lemma 46 (see [12]). 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 .

Lemma 47. Let , , , be the projection operator, and be the homotopy operator. Then, there exists a constant , independent of , such that for all balls .

Proof. Let be the homotopy operator and let be locally integrable form. Then, there exists a constant , independent of , such that By using Lemma 45, we have Thus, by (104) and (105), we have which ends the proof of Lemma 47.

Lemma 48. Let , , , be a solution of the nonhomogeneous -harmonic (7) 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 , , and any real number and with and . Here is the center of the ball.

Theorem 49. Let , , , be a solution of the nonhomogeneous -harmonic (7) in a bounded 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 50. Let , , , be a smooth differential form in a bounded domain , let be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that for all balls .

In applications, such as in calculating electric or magnetic fields, we often face the fact that the integrand contains a singular factor. So, the above result was extended into the following singular weighted case.

Theorem 51. Let , , , be a solution of the nonhomogeneous -harmonic equation in a bounded domain , let H be the projection operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that for all balls with and any real numbers and with , where and is the center of ball and is a constant.

Proof. Let be small enough such that and be any ball with center and radius . Choose ; then, . Write , and using the Hölder inequality and Theorem 49, we have where is a constant. We may assume that . Otherwise, we can move the center to the origin by a simple transformation. Then, for any , . By using the polar coordinate substitution, we have Choose , then . By the reverse Hölder inequality, we find that where is a constant. By the Hölder inequality again, we obtain Note that Substituting (112), (113), and (114) in (111) and using (115), we have We have completed the proof of Theorem 51.

Remark 52. (1) Replacing by and by in Theorem 51, we have (2) If , inequality (110) reduces to which does not contain a singular factor in the integral on the right side of the inequality.
The following definition of -averaging domains can be found in [1]. We call a proper subdomain an -averaging domain, , if , and there exists a constant such that for some ball and all . Here the supremum is over all balls with and is a measure defined by for a weight and .

Theorem 53. Let be a solution of the nonhomogeneous -harmonic equation, let be the projection operator, and let be the homotopy operator. Assume that is a fixed exponent associated with the nonhomogeneous -harmonic equation. Then, there exists a constant , independent of , such that for any bounded and convex -averaging domain . Here is a fixed ball and and are constants with .

Proof. Let be the radius of a ball . We may assume the center of is . Then, for any . Therefore, for any . Similar to the proof of Theorem 51, we have for all balls with , , and any real numbers and with , where . Write . Then, , and hence . Since is an -averaging domain, using (121) and noticing that , we have which is equivalent to We have completed the proof of Theorem 53.

We recall the following definition of -John domains with .

Definition 54. 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 .

Theorem 55. Let be a solution of the nonhomogeneous -harmonic (7), let be the projection operator, and let be the homotopy operator. Assume that is a fixed exponent associated with the nonhomogeneous -harmonic equation. Then, there exists a constant , independent of , such that for any bounded and convex -John domain , where . Here and are constants with , and the fixed cube , the cubes , and the constant appeared in Lemma 46.

Proof. We use the notation appearing in Lemma 46. There is a modified Whitney cover of cubes for such that , and for some . Since , for any , it follows that for some . Applying Lemma 48 to , we have where is a constant. Let and be the Radon measures defined by and , respectively. Then, where is a positive constant. Then, by the elementary inequality , , we have for a fixed . The first sum in (128) can be estimated by using Lemma 44 with , , and Lemma 48: To estimate the second sum in (128), we need to use the property of -John domain. Fix a cube and let be the chain in Lemma 46. The chain also has property that, for each , , with ; there exists a cube such that and , : For such , , let ; then By (127), (132), and Lemma 48, we have Since for , , , from (133) We know that since is bounded and when . Thus, from , (130), and (134), for every . Then, Notice that Using elementary inequality , we finally have Substituting (129) and (138) in (128), we have proved Theorem 55.

The following -imbedding inequality with a singular factor in the John domain was also proved in [12].

Theorem 56. Let be a solution of the nonhomogeneous -harmonic (7), let be the projection operator, and let be the homotopy operator. Assume that is a fixed exponent associated with the nonhomogeneous -harmonic equation. Then, there exists a constant , independent of , such that for any bounded and convex -John domain . Here the weights are defined by and , respectively. and are constants with .

Theorem 57. Let be a solution of the nonhomogeneous -harmonic (7), let be the projection operator, and let be the homotopy operator. Assume that is a fixed exponent associated with the nonhomogeneous -harmonic equation. Then, there exists a constant , independent of , such that for any bounded, convex -John domain . Here the weights are defined by and , and are constants with , and the fixed cube and the constant appeared in Lemma 46.

Proof. Since is a closed form, . Thus, by using Theorem 55 and (139), we have Thus, (141) holds. We have completed the proof of Theorem 57.

Remark 58. Since the usual -harmonic equation for functions is the special case of the nonhomogeneous -harmonic equation for differential forms, all results proved in Theorems 55, 56, and 57 are still true for -harmonic functions.

8. Composition of Homotopy and Potential Operators

Recently, Bi extended the definition of the potential operator to the case of differential forms; see [3]. For any differential -form , the potential operator is defined by where the kernel is a nonnegative measurable function defined for and the summation is over all ordered -tuples . The case reduces to the usual potential operator: where is a function defined on . See [3, 25] for more results about the potential operator. We say a kernel on satisfies the standard estimates if there exist , , and constant such that for all distinct points and in , and all with , the kernel satisfies (i) ; (ii) ; and (iii) .

In this paper, we always assume that is the potential operator defined in (143) with the kernel satisfying condition (i) of the standard estimates. Recently, Bi in [3] proved the following inequality for the potential operator: where , is a differential form defined in a bounded and convex domain and is a constant.

In this section, we prove the local imbedding inequalities for applied to solutions of the nonhomogeneous -harmonic equation in a bounded domain. For any subset , we use to denote the Orlicz-Sobolev space of -forms which equals with norm If we choose , in , we obtain the usual norm for In 2013, the following Theorems 59 to 61 were recently proved in [18].

Theorem 59. Let be a Young function in the class , , , be a bounded domain, , , be the homotopy operator defined in (2), and let be the potential operator defined in (143) with the kernel satisfying condition (i) of the standard estimates. 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 for some .

Theorem 60. Let be a Young function in the class , , , be a bounded domain, be the homotopy operator defined in (2), and let be the potential operator defined in (143) with the kernel satisfying condition (i) of the standard estimates. 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 for some .

Theorem 61. Let be a Young function in the class , , , be a bounded domain, be the homotopy operator defined in (2), and let be the potential operator defined in (143) with the kernel satisfying condition (i) of the standard estimates. 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 for some .

The following local -imbedding theorem was also obtained in [18].

Theorem 62. Let be a Young function in the class , , , be a bounded domain, be the homotopy operator defined in (2), and let be the potential operator defined in (143) with the kernel satisfying condition (i) of the standard estimates. 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 for some .

Proof. From , (147), and (148), we have
for all balls with , where . The proof of Theorem 62 has been completed.

The following version of local imbedding will be used to establish a global imbedding theorem which indicates that the operator is bounded.

Theorem 63. Let be a Young function in the class , , , be a bounded domain, be the homotopy operator defined in (2), and let be the potential operator defined in (143) with the kernel satisfying condition (i) of the standard estimates. 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 for some .

Proof. Applying (6) to , then using (145), we find that for any differential form and all balls with , where is a constant. Starting with (152) and using the similar method developed in the proof of Theorem 61, we obtain respectively, where and are constants. From , (153), we have where . The proof of Theorem 63 has been completed.

Note that if we choose or in Theorems 59–63, we will obtain some -norm or -norm inequalities, respectively. For example, let in Theorem 62; we have the following imbedding inequalities for with the -norms.

Corollary 64. Let , , and , and be a bounded domain. Assume that and is a solution of the nonhomogeneous -harmonic (7). Then, there exists a constant , independent of , such that for all balls with , where is a constant.

Selecting in Theorem 62, we obtain the usual imbedding inequalities with the -norms.

for all balls with , where is a constant. Now, we present the global imbedding theorem with the -norm as follows.

Theorem 65. Let be a Young function in the class , , , be any bounded -averaging domain, be the homotopy operator defined in (2), and let be the potential operator defined in (143) with the kernel satisfying condition (i) of the standard estimates. Assume that and is a solution of the nonhomogeneous -harmonic (7) in . Then, there exists a constant , independent of , such that where is some fixed ball.

It is well known that any John domain is a special -averaging domain; see [1]. Hence, we have the following global -imbedding theorem for John domains.

Theorem 66. Let be a Young function in the class , , , be any bounded John domain, be the homotopy operator defined in (2), and let be the potential operator defined in (143) with the kernel satisfying condition (i) of the standard estimates. Assume that and is a solution of the nonhomogeneous -harmonic (7) in . Then, there exists a constant , independent of , such that where is some fixed ball.

Next, let be the set of all solutions of the nonhomogeneous -harmonic equation in . We have the following version of imbedding theorem with norm for any bounded domain, which says that the composite operator maps continuously into . See [18] for the proof of Theorem 67.

Theorem 67. Let be a Young function in the class , , , be the homotopy operator defined in (2), and let be the potential operator defined in (143) with the kernel satisfying condition (i) of the standard estimates. Assume that and in . Then, the composite operator maps continuously into . Furthermore, there exists a constant , independent of , such that holds for any bounded domain .

Selecting in Theorems 65, we have the following version of the imbedding inequality with -norms.

Corollary 68. Let , , be the homotopy operator defined in (2), and let be the potential operator defined in (143). Assume that and is a solution of the nonhomogeneous -harmonic (7) in . Then, there exists a constant , independent of , such that holds for any bounded domain .

Remark 69. (i) We know that the -averaging domains are the special -averaging domains. Thus, Theorem 65 also holds for the -averaging domain; (ii) Theorem 67 holds for any bounded domain in .

9. Composition of Homotopy and Green's Operators

In this section, we estimate the Lipschitz norm or norm of composition in terms of the norm. First, we present the following norm inequality for the composition of the homotopy operator and Green's operator .

Theorem 70. Let , , , be a smooth 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 .

Using Theorem 70, we obtain the following inequality with Lipschitz norm.

Theorem 71. Let , , , be a smooth differential form in a bounded, convex domain , let be Green's operator, and let be the homotopy operator defined in (2). Then, there exists a constant , independent of , such that where is a constant with .

The following Theorem 72 tells us the relationship between the Lipschitz norm and norm of composition .

Theorem 72. Let , , be a solution of the nonhomogeneous -harmonic (7) in a bounded, convex domain . Let be Green's operator and let be the homotopy operator defined in (2). Then, there exists a constant , independent of , such that where is a constant with .

The following theorem gives an estimate for norm of composition in terms of norm.

Theorem 73. Let , , be a solution of the nonhomogeneous -harmonic (7) in a bounded, convex domain . Let be Green's operator and let be the homotopy operator defined in (2). Then, there exists a constant , independent of , such that

Theorem 74. Let , , , be a solution of the nonhomogeneous -harmonic equation in a bounded, convex domain . Let be Green's operator and let be the homotopy operator defined in (2). The measures and are defined by , , and for some and with for any . Then, there exists a constant , independent of , such that where and are constants with and .

Finally, we can estimate the weighted norm in terms of the norm.

Theorem 75. Let , , , be a solution of the nonhomogeneous -harmonic equation in a bounded, convex domain . Let be Green's operator and let be the homotopy operator defined in (2). The measures and are defined by , , and for some and with for any . Then, there exists a constant , independent of , such that where is a constant with .

Conflict of Interests

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