- Yuxia Tong
^{1}Email author, - Juan Li
^{2}and - Jiantao Gu
^{1}

**2010**:589040

https://doi.org/10.1155/2010/589040

© Yuxia Tong et al. 2010

**Received: **29 December 2009

**Accepted: **31 March 2010

**Published: **27 June 2010

## Absract

## 1. Introduction

Green's operator is often applied to study the solutions of various differential equations and to define Poisson's equation for differential forms. Green's operator has been playing an important role in the study of PDEs. In many situations, the process to study solutions of PDEs involves estimating the various norms of the operators. Hence, we are motivated to establish some Lipschitz norm inequalities and BMO norm inequalities for Green's operator in this paper.

In the meanwhile, there have been generally studied about -weighted [1, 2] and -weighted [3, 4] different inequalities and their properties. Results for more applications of the weight are given in [5, 6]. The purpose of this paper is to derive the new weighted inequalities with the Lipschitz norm and BMO norm for Green's operator applied to differential forms. We will introduce -weight, which can be considered as a further extension of the -weight.

We keep using the traditional notation.

Let be a connected open subset of , let be the standard unit basis of , and let be the linear space of -covectors, spanned by the exterior products , corresponding to all ordered -tuples , , . We let . The Grassman algebra is a graded algebra with respect to the exterior products. For and , the inner product in is given by with summation over all -tuples and all integers .

We define the Hodge star operator by the rule and for all . The norm of is given by the formula . The Hodge star is an isometric isomorphism on with and .

Balls are denoted by and is the ball with the same center as and with . We do not distinguish balls from cubes throughout this paper.

where the partial differentiations are applied to the coefficients of .

We denote the exterior derivative by for . Its formal adjoint operator is given by on , . Let be the th exterior power of the cotangent bundle and let be the space of smooth -forms on . We set . The harmonic -fields are defined by = . The orthogonal complement of in is defined by = . Then, Green's operator is defined as by assigning to 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 [8] for more properties of Green's operator.

for differential forms. If is a function (a 0-form), (1.10) reduces to the usual -harmonic equation for functions. We should notice that if the operator equals in (1.7), then (1.7) reduces to the homogeneous -harmonic equation. Some results have been obtained in recent years about different versions of the -harmonic equation; see [9–11].

for some . When is a 0-form, (1.12) reduces to the classical definition of .

for some , where the Radon measure is defined by , is a weight and is a real number. Again, we use to replace whenever it is clear that the integral is weighted.

## 2. Preliminary Knowledge and Lemmas

Definition 2.1.

If we choose and in Definition 2.1, we will obtain the usual -weight. If we choose , and in Definition 2.1, we will obtain the -weight [3]. If we choose , and in Definition 2.1, we will obtain the -weight [12].

Lemma 2.2 (see [1]).

We need the following generalized Hölder inequality.

Lemma 2.3.

The following version of weak reverse Hölder inequality appeared in [13].

Lemma 2.4.

Lemma 2.5 (see [14]).

We need the following Lemma 2.6 (Caccioppoli inequality) that was proved in [8].

Lemma 2.6 (see [8]).

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

Lemma 2.7 (see [14]).

Lemma 2.8 (see [14]).

## 3. Main Results and Proofs

Theorem 3.1.

where is a constant with , and is a constant with .

Proof.

We have completed the proof of Theorem 3.1.

Remark 3.

Next, we will establish the following weighted norm comparison theorem between the Lipschitz and the BMO norms.

Theorem 3.2.

where is a constant with , and is a constant with .

Proof.

where is a constant and . We have completed the proof of Theorem 3.2.

Now, we will prove the following weighted inequality between the BMO norm and the Lipschitz norm for Green's operator.

Theorem 3.3.

Proof.

We have completed the proof of Theorem 3.3.

Using the same methods, and by Lemmas 2.7 and 2.8, we can estimate Lipschitz norm and BMO norm of Green's operator in terms of norm.

Theorem 3.4.

where is a constant with , and is a constant with .

Theorem 3.5.

Remark 3.

Note that the differentiable functions are special differential forms (0-forms). Hence, the usual -harmonic equation for functions is the special case of the -harmonic equation for differential forms. Therefore, all results that we have proved for solutions of the -harmonic equation in this paper are still true for -harmonic functions.

## Declarations

### Acknowledgments

The first author is supported by NSFC (No:10701013), NSF of Hebei Province (A2010000910) and Tangshan Science and Technology projects (09130206c). The second author is supported by NSFC (10771110 and 60872095) and NSF of Nongbo (2008A610018).

## Authors’ Affiliations

## References

- Garnett JB:
*Bounded Analytic Functions, Pure and Applied Mathematics*.*Volume 96*. Academic Press, New York, NY, USA; 1981:xvi+467.Google Scholar - Heinonen J, Kilpeläinen T, Martio O:
*Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs*. Oxford University Press, Oxford, UK; 1993:vi+363.Google 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.1996.0380MATHMathSciNetView ArticleGoogle Scholar - Ding S: Estimates of weighted integrals for differential forms.
*Journal of Mathematical Analysis and Applications*2001, 256(1):312–323. 10.1006/jmaa.2000.7329MATHMathSciNetView ArticleGoogle Scholar - Cruz-Uribe D, Pérez C: Two-weight, weak-type norm inequalities for fractional integrals, Calderón-Zygmund operators and commutators.
*Indiana University Mathematics Journal*2000, 49(2):697–721.MATHMathSciNetView ArticleGoogle Scholar - García-Cuerva J, Martell JM: Two-weight norm inequalities for maximal operators and fractional integrals on non-homogeneous spaces.
*Indiana University Mathematics Journal*2001, 50(3):1241–1280.MATHMathSciNetView ArticleGoogle Scholar - Iwaniec T, Lutoborski A: Integral estimates for null Lagrangians.
*Archive for Rational Mechanics and Analysis*1993, 125(1):25–79. 10.1007/BF00411477MATHMathSciNetView ArticleGoogle 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/S0002-9939-04-07347-2MATHMathSciNetView ArticleGoogle Scholar - Wang Y, Wu C: Sobolev imbedding theorems and Poincaré inequalities for Green's operator on solutions of the nonhomogeneous -harmonic equation.
*Computers & Mathematics with Applications*2004, 47: 1545–1554. 10.1016/j.camwa.2004.06.006MATHMathSciNetView ArticleGoogle Scholar - Xing Y: 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)00036-2MATHMathSciNetView ArticleGoogle Scholar - Xing Y: Two-weight imbedding inequalities for solutions to the -harmonic equation.
*Journal of Mathematical Analysis and Applications*2005, 307(2):555–564. 10.1016/j.jmaa.2005.03.019MATHMathSciNetView ArticleGoogle 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.1998.6096MATHMathSciNetView ArticleGoogle Scholar - Ding S, Nolder CA: Weighted Poincaré inequalities for solutions to -harmonic equations.
*Illinois Journal of Mathematics*2002, 46(1):199–205.MATHMathSciNetGoogle Scholar - Xing Y, Ding S: Inequalities for Green's operator with Lipschitz and BMO norms.
*Computers & Mathematics with Applications*2009, 58(2):273–280. 10.1016/j.camwa.2009.03.096MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.