- Research Article
- Open access
- Published:
Weighted Norm Inequalities for Solutions to the Nonhomogeneous
-Harmonic Equation
Journal of Inequalities and Applications volume 2009, Article number: 851236 (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 [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.
In this paper we consider solutions to the nonhomogeneous -harmonic equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ1_HTML.gif)
for differential forms, where and
satisfies the following conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ3_HTML.gif)
in , and
exists in
, we call
and
conjugate
-harmonic tensors in
.
We also consider solutions to the equation of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ4_HTML.gif)
here and
satisfy the conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ10_HTML.gif)
for all balls with
. Here
is any positive constant with
,
, and
. Note that (2.4) can be written as the following symmetric form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ11_HTML.gif)
Proof.
Choose , since
, using Hölder inequality, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ12_HTML.gif)
Applying the elementary inequality and Lemma 2.3, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ13_HTML.gif)
Choose , using Hölder inequality with
again yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ14_HTML.gif)
Then, choosing , using Hölder inequality once again, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ15_HTML.gif)
We know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ16_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ17_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ18_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ19_HTML.gif)
then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ20_HTML.gif)
Combining (2.11) and (2.14), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ21_HTML.gif)
Using the similar method, we can easily get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ22_HTML.gif)
Combining (2.6) and (2.7) gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ23_HTML.gif)
Substituting (2.8), (2.15), and (2.16) into (2.17), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ24_HTML.gif)
Since , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ25_HTML.gif)
Putting (2.19) into (2.18), we obtain the desired result
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ26_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ27_HTML.gif)
for all balls with
. Here
is any positive constant with
,
, and
.
It is easy to see that the inequality (2.21) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ28_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ29_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ30_HTML.gif)
Here is any positive constant with
,
,
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ31_HTML.gif)
for and
.
Proof.
Applying Theorem 2.4 and Lemma 2.6, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ32_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ33_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ34_HTML.gif)
for some .
Similarly, we write BMO
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ35_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ36_HTML.gif)
Similarly, for ,
. We write
BMO
, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ37_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ38_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ39_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ40_HTML.gif)
for all balls with
. Here
is any constant with
.
Proof.
Choose , since
, using Hölder inequality, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ41_HTML.gif)
Taking , then
, using Lemmas 3.4 and 3.3 and the same method as [2, Proof of Theorem
], we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ42_HTML.gif)
where . Using Hölder inequality with
again yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ43_HTML.gif)
Substituting (3.10) in (3.9), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ44_HTML.gif)
Since , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ45_HTML.gif)
Combining (3.11) and (3.12) gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ46_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ47_HTML.gif)
Finally, we obtain the desired result
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ48_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ49_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ50_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ51_HTML.gif)
where and
are constants with
and
Proof.
We note that implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ52_HTML.gif)
for any ball . Using (3.7) and the Hölder inequality with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ53_HTML.gif)
From the definition of the Lipschitz norm (3.3), (3.19), and (3.20), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ54_HTML.gif)
Since and
. The desired result for Lipschitz norm has been completed.
Then, we prove the theorem for BMO norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ55_HTML.gif)
From (3.21) we find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ56_HTML.gif)
Using (3.17) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ57_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ58_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ59_HTML.gif)
where and
are constants with
and
If , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ60_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ61_HTML.gif)
where and
are positive constants with
and
, for
,
Proof.
From (3.25), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ62_HTML.gif)
Choose ,
, using Lemma 3.6, it is easy to obtain the desire result
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ63_HTML.gif)
Using the similar method for BMO norm, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ64_HTML.gif)
If , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F851236/MediaObjects/13660_2009_Article_2016_Equ65_HTML.gif)
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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).
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Wen, H. Weighted Norm Inequalities for Solutions to the Nonhomogeneous -Harmonic Equation.
J Inequal Appl 2009, 851236 (2009). https://doi.org/10.1155/2009/851236
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DOI: https://doi.org/10.1155/2009/851236