- Haiyu Wen
^{1}Email author

**2009**:851236

https://doi.org/10.1155/2009/851236

© HaiyuWen 2009

**Received: **10 March 2009

**Accepted: **18 May 2009

**Published: **16 June 2009

## Abstract

## Keywords

## 1. Introduction

In the recent years, the -harmonic equations for differential forms have been widely investigated, see [1], and many interesting and important results have been found, such as some weighted integral inequalities for solutions to the -harmonic equations; see [2–7]. Those results are important for studying the theory of differential forms and both qualitative and quantitative properties of the solutions to the different versions of -harmonic equation. In the different versions of -harmonic equation, the nonhomogeneous -harmonic equation has received increasing attentions, in [8] Ding has presented some estimates to such equation. In this paper, we extend some estimates that Ding has presented in [8] into the two-weight case. Our results are more general, so they can be used broadly.

It is well-known that the Lipschitz norm , where the supremum is over all local cubes , as is the BMO norm , so the natural limit of the space locLipk( ) as is the space BMO( ). In Section 3, we establish a relation between these two norms and -norm. We first present the local two-weight Poincaré inequality for -harmonic tensors. Then, as the application of this inequality and the result in [8], we prove some weighted Lipschitz norm inequalities and BMO norm inequalities for differential forms satisfying the different nonhomogeneous -harmonic equations. These results can be used to study the basic properties of the solutions to the nonhomogeneous -harmonic equations.

Now, we first introduce related concepts and notations.

Throughout this paper we assume that is a bounded connected open subset of . We assume that is a ball in with diameter and is the ball with the same center as with . We use to denote the Lebesgue measure of . We denote a weight if and a.e.. Also in general . For , we write if the weighted -norm of over satisfies , where is a real number. A differential -form on is a schwartz distribution on with value in , we denote the space of differential -forms by . We write for the -forms with for all ordered -tuples , , . Thus is a Banach space with norm . We denote the exterior derivative by for . Its formal adjoint operator is given by on , . A differential -form is called a closed form if in . Similarly, a differential -form is called a coclosed form if . The -form is defined by , and , , for all , , here is a homotopy operator, for its definition, see [8].

Then, we introduce some -harmonic equations.

for almost every and all . Here is a constant and is a fixed exponent associated with (1.1) and . Note that if we choose in (1.1), then (1.1) will reduce to the conjugate -harmonic equation .

Definition 1.1.

in , and exists in , we call and conjugate -harmonic tensors in .

Definition 1.2.

We call an -harmonic tensor in if satisfies the -harmonic equation (1.4) in .

## 2. The Local and Global -Weighted Estimates

In this section, we will extend Lemma 2.3, see in [8], to new version with weight both locally and globally.

Definition 2.1.

See [9] for properties of -weights. We will need the following generalized Hölder's inequality.

Lemma 2.2.

We also need the following lemma; see [8].

Lemma 2.3.

Theorem 2.4.

Proof.

The proof of Theorem 2.4 has been completed.

Using the same method, we have the following two-weighted -estimate for .

Theorem 2.5.

for all balls with . Here is any positive constant with , , and .

As applications of the local results, we prove the following global norm comparison theorem.

Lemma 2.6.

for all and some and if , then there exists a cube (this cube does not need be a member of ) in such that .

Theorem 2.7.

Proof.

Since is bounded. The proof of inequality (2.24) has been completed. Similarly, using Theorem 2.5 and Lemma 2.6, inequality (2.25) can be proved immediately. This ends the proof of Theorem 2.7.

Definition 2.8.

We see that -weight reduce to the usual -weight if and ; see [10].

And, if and in Theorem 2.7, it is easy to obtain Theorems and in [8].

## 3. Estimates for Lipschitz Norms and BMO Norms

In [11] Ding has presented some estimates for the Lipchitz norms and BMO norms. In this section, we will prove another estimates for the Lipchitz norms and BMO norms.

Definition 3.1.

for some . When is a -form, (3.2) reduces to the classical definition of BMO .

We also discuss the weighted Lipschitz and BMO norms.

Definition 3.2.

for some , where is a bounded domain, the measure is defined by , is a weight, and is a real number.

We need the following classical Poincaré inequality; see [10].

Lemma 3.3.

We also need the following lemma; see [2].

Lemma 3.4.

We need the following local weighted Poincaré inequality for -harmonic tensors.

Theorem 3.5.

for all balls with . Here is any constant with .

Proof.

This ends the proof of Theorem 3.5.

Similarly, if setting and in Theorem 3.5, we obtain Theorem in [2]. And we choose in Theorem 3.5, we have the classical Poincaré inequality (3.5).

Lemma 3.6 (see [8]).

Here is any positive constant with , and .

Theorem 3.7.

where and are constants with and

Proof.

Since and . The desired result for Lipschitz norm has been completed.

Now, we have completed the proof of Theorem 3.7.

Similarly, if setting and in Theorem 3.7, we obtain the following theorem.

Theorem 3.8.

where and are constants with and

Using Lemma 3.6, we can also obtain the following theorem.

Theorem 3.9.

where and are positive constants with and , for ,

Proof.

## Declarations

### Acknowledgment

This work was supported by Science Research Foundation in Harbin Institute of Technology (HITC200709) and Development Program for Outstanding Young Teachers in HIT (HITQNJS.2006.052).

## Authors’ Affiliations

## References

- Agarwal RP, Ding S:
**Advances in differential forms and the -harmonic equation.***Mathematical and Computer Modelling*2003,**37**(12–13):1393–1426. 10.1016/S0895-7177(03)90049-5MathSciNetView ArticleMATHGoogle Scholar - Ding S, Nolder CA:
**Weighted Poincaré inequalities for solutions to -harmonic equations.***Illinois Journal of Mathematics*2002,**46**(1):199–205.MathSciNetMATHGoogle Scholar - Liu B:
**-weighted Caccioppoli-type and Poincaré-type inequalities for -harmonic tensors.***International Journal of Mathematics and Mathematical Sciences*2002,**31**(2):115–122. 10.1155/S0161171202107046MathSciNetView ArticleMATHGoogle Scholar - Xing Y: Weighted Poincaré-type estimates for conjugate -harmonic tensors. Journal of Inequalities and Applications 2005, (1):1–6.Google Scholar
- Yuming X:
**Weighted integral inequalities for solutions of the -harmonic equation.***Journal of Mathematical Analysis and Applications*2003,**279**(1):350–363. 10.1016/S0022-247X(03)00036-2MathSciNetView ArticleMATHGoogle Scholar - Ding S, Ling Y:
**Weighted norm inequalities for conjugate -harmonic tensors.***Journal of Mathematical Analysis and Applications*1996,**203**(1):278–288. 10.1006/jmaa.1996.0380MathSciNetView ArticleMATHGoogle Scholar - Ding S, Shi P:
**Weighted Poincaré-type inequalities for differential forms in -averaging domains.***Journal of Mathematical Analysis and Applications*1998,**227**(1):200–215. 10.1006/jmaa.1998.6096MathSciNetView ArticleMATHGoogle Scholar - Ding S:
**Local and global norm comparison theorems for solutions to the nonhomogeneous -harmonic equation.***Journal of Mathematical Analysis and Applications*2007,**335**(2):1274–1293. 10.1016/j.jmaa.2007.02.048MathSciNetView ArticleMATHGoogle Scholar - Neugebauer CJ:
**Inserting -weights.***Proceedings of the American Mathematical Society*1983,**87**(4):644–648.MathSciNetMATHGoogle Scholar - Ding S:
**Two-weight Caccioppoli inequalities for solutions of nonhomogeneous -harmonic equations on Riemannian manifolds.***Proceedings of the American Mathematical Society*2004,**132**(8):2367–2375. 10.1090/S0002-9939-04-07347-2MathSciNetView ArticleMATHGoogle Scholar - Ding S: Lipschitz and BOM norm inequalities for operators. Proceedings of the 5th World Congress of Nonliner Analysis, July 2008, Orlando Fla, USAGoogle Scholar

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