Open Access

Weighted Norm Inequalities for Solutions to the Nonhomogeneous -Harmonic Equation

Journal of Inequalities and Applications20092009:851236

DOI: 10.1155/2009/851236

Received: 10 March 2009

Accepted: 18 May 2009

Published: 16 June 2009

Abstract

We first prove the local and global two-weight norm inequalities for solutions to the nonhomogeneous -harmonic equation for differential forms. Then, we obtain some weighed Lipschitz norm and BMO norm inequalities for differential forms satisfying the different nonhomogeneous -harmonic equations.

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 [27]. 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.

In this paper we consider solutions to the nonhomogeneous -harmonic equation
(1.1)
for differential forms, where and satisfies the following conditions:
(1.2)

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.

We call and a pair of conjugate -harmonic tensor in if and satisfy the conjugate -harmonic equation
(1.3)

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

We also consider solutions to the equation of the form
(1.4)
here and satisfy the conditions:
(1.5)
for almost every and all . Here are constants and is a fixed exponent associated with (1.4). A solution to (1.4) is an element of the Sobolev space such that
(1.6)

for all with compact support.

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.

We say a pair of weights satisfies the -condition in a domain and write for some and with , if
(2.1)

for any ball .

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

Lemma 2.2.

Let , , and , if and are measurable functions on , then
(2.2)

for any .

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

Lemma 2.3.

Let and be a pair of solutions to the nonhomogeneous -harmonic equation (1.1) in a domain . If and , then if and only if . Moreover, there exist constants and , independent of and , such that
(2.3)

for all balls with .

Theorem 2.4.

Let and be a pair of solutions to the nonhomogeneous -harmonic equation (1.1) in a domain . Assume that for some and with . Then, there exists a constants , independent of and , such that
(2.4)
for all balls with . Here is any positive constant with , , and . Note that (2.4) can be written as the following symmetric form:
(2.5)

Proof.

Choose , since , using Hölder inequality, we find that
(2.6)
Applying the elementary inequality and Lemma 2.3, we obtain
(2.7)
Choose , using Hölder inequality with again yields
(2.8)
Then, choosing , using Hölder inequality once again, we have
(2.9)
We know that
(2.10)
and hence
(2.11)
Note that
(2.12)
Since
(2.13)
then,
(2.14)
Combining (2.11) and (2.14), we obtain
(2.15)
Using the similar method, we can easily get that
(2.16)
Combining (2.6) and (2.7) gives
(2.17)
Substituting (2.8), (2.15), and (2.16) into (2.17), we have
(2.18)
Since , then
(2.19)
Putting (2.19) into (2.18), we obtain the desired result
(2.20)

The proof of Theorem 2.4 has been completed.

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

Theorem 2.5.

Let and be a pair of solutions to the nonhomogeneous -harmonic equation (1.1) in a domain . Assume that for some and with . Then, there exists a constants , independent of and , such that
(2.21)

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

It is easy to see that the inequality (2.21) is equivalent to
(2.22)

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

Lemma 2.6.

Each has a modified Whitney cover of cubes such that
(2.23)

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.

Let and be a pair of solutions to the nonhomogeneous -harmonic equation (1.1) in a bounded domain . Assume that for some and with . Then, there exist constants and , independent of and , such that
(2.24)
Here is any positive constant with , , , and
(2.25)

for and .

Proof.

Applying Theorem 2.4 and Lemma 2.6, we have
(2.26)

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 say the weight satisfies the -condition in a domain write for some with , if
(2.27)

for any ball .

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.

Let , . We write , , if
(3.1)

for some .

Similarly, we write BMO if
(3.2)

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.

Let , . We write , , if
(3.3)
Similarly, for , . We write BMO , if
(3.4)

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.

Let and , then is in with and
(3.5)

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

Lemma 3.4.

Suppose that is a solution to (1.4), and . There exists a constant , depending only on , , , , , and , such that
(3.6)

for all balls with .

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

Theorem 3.5.

Let be an -harmonic tensor in a domain and , . Assume that , , and for some and with . Then, there exists a constant , independent of , such that
(3.7)

for all balls with . Here is any constant with .

Proof.

Choose , since , using Hölder inequality, we find that
(3.8)
Taking , then , using Lemmas 3.4 and 3.3 and the same method as [2, Proof of Theorem ], we obtain
(3.9)
where . Using Hölder inequality with again yields
(3.10)
Substituting (3.10) in (3.9), we have
(3.11)
Since , then
(3.12)
Combining (3.11) and (3.12) gives
(3.13)
Note that
(3.14)
Finally, we obtain the desired result
(3.15)

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]).

Let and be a pair of solution to the conjugate -harmonic tensor in . Assume for some . Then, there exists a constant , independent of , such that
(3.16)

Here is any positive constant with , and .

Theorem 3.7.

Let be an -harmonic tensor in a domain , and all with , and , . Assume that and for some and with for any . Then, there exist constants and , independent of , such that
(3.17)
(3.18)

where and are constants with and

Proof.

We note that implies that
(3.19)
for any ball . Using (3.7) and the Hölder inequality with , we have
(3.20)
From the definition of the Lipschitz norm (3.3), (3.19), and (3.20), we obtain
(3.21)

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

Then, we prove the theorem for BMO norm
(3.22)
From (3.21) we find
(3.23)
Using (3.17) we have
(3.24)

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.

Let be an -harmonic tensor in a domain , and all with , and , . Assume that and for with for any . Then, there exist constants and , independent of , such that
(3.25)
(3.26)

where and are constants with and

If , we have
(3.27)

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

Theorem 3.9.

Let and be a pair of conjugate -harmonic tensor in a domain , then if and only if where the measure is defined by , and all with . Assume that for with for any . Then, there exist constants and , independent of and , such that
(3.28)

where and are positive constants with and , for ,

Proof.

From (3.25), we have
(3.29)
Choose , , using Lemma 3.6, it is easy to obtain the desire result
(3.30)
Using the similar method for BMO norm, we have
(3.31)
If , we have
(3.32)

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

(1)
Department of Mathematics, Harbin Institute of Technology

References

  1. 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
  2. Ding S, Nolder CA: Weighted Poincaré inequalities for solutions to -harmonic equations. Illinois Journal of Mathematics 2002,46(1):199–205.MathSciNetMATHGoogle Scholar
  3. 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
  4. Xing Y: Weighted Poincaré-type estimates for conjugate -harmonic tensors. Journal of Inequalities and Applications 2005, (1):1–6.Google Scholar
  5. 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
  6. 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
  7. 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
  8. 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
  9. Neugebauer CJ: Inserting -weights. Proceedings of the American Mathematical Society 1983,87(4):644–648.MathSciNetMATHGoogle Scholar
  10. 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
  11. Ding S: Lipschitz and BOM norm inequalities for operators. Proceedings of the 5th World Congress of Nonliner Analysis, July 2008, Orlando Fla, USAGoogle Scholar

Copyright

© HaiyuWen 2009

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.