Weighted Decomposition Estimates for Differential Forms

Abstract

After introducing the definition of -weights, we establish the -weighted decomposition estimates and -weighted Caccioppoli-type estimates for -harmonic tensors. Furthermore, by Whitney covering lemma, we obtain the global results in domain . These results can be used to study the integrability of differential forms and to estimate the integrals for differential forms.

1. Introduction

Let denote the standard orthogonal basis of . Suppose that is the linear space of all -vectors, spanned by the exterior product corresponding to all ordered -tuples , . Throughout this paper, we always assume that is an open subset of . We use to denote the space of all differential -forms and to denote the -forms on with the coefficient for all ordered -tuples , where is a manifold. Thus is a Banach space with the norm

(1.1)

For a differential -form , its vector-valued differential form is composed of the differential -form , , here the partial derivatives are with respect to the coefficients of . Usually, suppose that is the space consisting of all , the th one of . We denote the exterior differential operator of -forms by and define the Hodge differential operator with , where , and is the Hodge star operator. We denote a ball or cube by and the ball or cube with the same center as and by . The following nonlinear elliptic equation:

(1.2)

is called the nonhomogeneous -harmonic equation of differential forms, where

(1.3)

are operators satisfying the following conditions for almost all and :

(1.4)

where are two constants, and is a fixed exponent dependent on (1.2).

If the operator in (1.2), it degenerates into the homogeneous -harmonic equation

(1.5)

The solutions to (1.5) are called -harmonic tensors. See [1] for the recent research on -harmonic equation.

2. -Weighted Caccioppoli-Type Inequalities

Caccioppoli-type estimates have been widely studied and frequently used in analysis and related fields, including partial differential equations and the theory of elasticity. These inequalities provide upper bounds for the -norm of if is a function or if is a form with the -norm of the differential form . Different versions of the Caccioppoli-type inequality have been established in [2, 3].

We first introduce the following definition of -weights (or the two-weight), and then establish the local -weighted Caccioppoli-type inequality for solutions to the homogeneous -harmonic equation.

Definition 2.1.

Assume that and . One says a pair of weights satisfies the -condition, writes , if for all balls , one has that

(2.1)

where is a constant.

In [4], Nolder obtains the following local Caccioppoli-type estimate.

Lemma 2.2.

Let be an -harmonic tensor in and let . Then there exists a constant , independent of and , such that

(2.2)

for all balls or cubes with and all closed forms . Here .

The next lemma is the generalized Hölder inequality which will be widely used in this paper.

Lemma 2.3.

Let , , and . If and are measurable functions on , then

(2.3)

for any measurable set .

The following weak reverse Hölder inequality plays an important role in founding the integral estimate of the nonhomogeneous and homogeneous -harmonic tensor; see [4].

Lemma 2.4.

Let be a solution to (1.2) in , and , . Then there exists a constant , depending only on , and , such that

(2.4)

for all balls or cubes with .

The following Whitney covering lemma appears in [4].

Lemma 2.5.

Each has a modified Whitney cover of cubes such that

(2.5)

for all and some , where is the characteristic function for a set . Moreover, if , then there exists a cube (this cube does not need to be a member of ) in such that .

Now we are ready to prove the local weighted Caccioppoli-type inequality for homogeneous -harmonic tensors.

Theorem 2.6.

Let , , be a solution to the homogeneous -harmonic equation (1.5) and for some and . Then there exists a constant , independent of and , such that

(2.6)

for all balls or cubes with and all closed forms .

Specially, if , one has

(2.7)

Here is any constant with .

Proof.

Choosing and by Lemma 2.3, we have

(2.8)

Choosing and by Lemma 2.2, we know that

(2.9)

where is a closed form. Putting (2.9) into (2.8), we have

(2.10)

Since is a closed form and is a solution to (1.5), is still a solution to (1.5).

When , taking and by Lemma 2.4, we obtain

(2.11)

By Lemma 2.3, we have

(2.12)

By the condition , we obtain

(2.13)

where . By (2.10), (2.11), (2.12), and (2.13), we conclude that

(2.14)

Since , inequality (2.14) can be rewritten as

(2.15)

By simple computation, we know that

(2.16)

Then we can change inequality (2.15) into

(2.17)

When , taking and by Lemma 2.4, we obtain

(2.18)

By Lemma 2.3, it follows that

(2.19)

By (2.10), (2.18), and (2.19), we conclude that

(2.20)

Since , inequality (2.20) can be rewritten as

(2.21)

Because of , we have that

(2.22)

The proof of Theorem 2.6 is completed.

Since there are fore real parameters, , in Theorem 2.6, we can obtain some desired versions of weighted Caccioppoli-type estimates by different choices of them. Let in Theorem 2.6, we obtain the following corollaries.

Corollary 2.7.

Let , be a solution to the homogeneous -harmonic equation (1.5) and for some and . Then there exists a constant , independent of and , such that

(2.23)

for all closed forms .

Furthermore, choosing in Corollary 2.7, we get the following result.

Corollary 2.8.

Let , be a solution to the homogeneous -harmonic equation (1.5) and for some . Then there exists a constant , independent of and , such that

(2.24)

for all closed forms .

3. The Local Weighted Estimates for the Decomposition

The Hodge decomposition theorem has been playing an important part in partial differential equation, the operator theory, and so on. In the recent years, there are some interesting conclusions on the Hodge decomposition of differential forms; see [5, 6]. In [7], there is the following decomposition theorem for differential form .

Lemma 3.1.

Let , , . Then, there exist differential forms and , such that

(3.1)

(3.2)

here is a positive constant.

Definition 3.2.

The weight satisfies the -condition on the set , write , if , a.e., and for all balls , one has that

(3.3)

where .

The next lemma states the reverse Hölder inequality for -weight; see [8].

Lemma 3.3.

Assume that . Then, there exists a constant , independent of , for all balls , such that

(3.4)

where is a real number.

In this section, we will extend (3.2) to the weighted form.

Theorem 3.4.

Let be the solution to the nonhomogeneous -harmonic equation (1.2), , . Then, there exist differential forms and , such that

(3.5)

(3.6)

here , , , , and is a constant.

Proof.

From (3.2), it follows that

(3.7)

(3.8)

for any .

When , take . By Lemma 2.3 with and (3.7), we have

(3.9)

Similarly, using Lemma 2.3 and (3.8), we have

(3.10)

Combining (3.9) with (3.10), we have

(3.11)

Take , then and . By Lemma 2.4, it follows that

(3.12)

here is a constant. Putting (3.11) into (3.12), we get

(3.13)

By Lemma 2.3, the following inequality holds:

(3.14)

Combining (3.13) with (3.14), we obtain the following estimate:

(3.15)

Since , we have

(3.16)

Putting (3.16) into (3.15), we get

(3.17)

Recall that , , then we have

(3.18)

So (3.17) can be rewritten as

(3.19)

When , by Lemma 3.3, we know that

(3.20)

where .

Let , then and . By Lemma 2.3 with and (3.20), we have that

(3.21)

Similarly, by Lemma 2.3 and (3.20), we have that

(3.22)

Putting (3.7) into (3.21), we have

(3.23)

Putting (3.8) into (3.22), we conclude that

(3.24)

Combining (3.23) with (3.24), we obtain

(3.25)

Let , then . By Lemma 2.4, we have

(3.26)

where . Putting (3.26) into (3.25), we get

(3.27)

By Lemma 2.3, we get

(3.28)

Putting (3.28) into (3.27), it follows that

(3.29)

Since , we have

(3.30)

Putting (3.30) into (3.29), we get the following inequality:

(3.31)

Since , (3.31) can be rewritten as

(3.32)

So inequality (3.4) is true for .

4. The Global Weighted Estimates

Based on the local weighted estimate for the decomposition and the Whitney covering lemma, we get the global weighted estimate on domain .

Theorem 4.1.

Let be a differential form satisfying the nonhomogeneous -harmonic equation (1.2) in a bounded domain , , , and , . Then, there exist differential forms and , such that

(4.1)

where and is a constant.

Proof.

By Lemma 2.5, has a modified Whitney cover of cubes . Let .

First, we assume that all cubes satisfy . By Lemma 2.5 and (3.6), we know that

(4.2)

In the above proof, if there exists one cube , such that can not be contained in completely, the proof is as follows. Set , and define on as follows:

(4.3)

Then, we know that the following formulas are true:

(4.4)

Note that (4.4) hold, and then the following formula is true:

(4.5)

The proof of Theorem 4.1 is completed.

In Theorems 3.4 and 4.1, there is a real parameter , which makes the results more flexible. By choosing different value of the parameter , we get the estimates in different forms. For example, if we take , we have the following corollary.

Corollary 4.2.

Let be the solution to the nonhomogeneous -harmonic equation (1.2), , , and , . Then, there exist differential forms and , such that

(4.6)

where is a constant.