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Norm Comparison Inequalities for the Composite Operator
Journal of Inequalities and Applications volume 2009, Article number: 212915 (2009)
Abstract
We establish norm comparison inequalities with the Lipschitz norm and the BMO norm for the composition of the homotopy operator and the projection operator applied to differential forms satisfying the A-harmonic equation. Based on these results, we obtain the two-weight estimates for Lipschitz and BMO norms of the composite operator in terms of the -norm.
1. Introduction
The purpose of this paper is to establish the Lipschitz norm and BMO norm inequalities for the composition of the homotopy operator and the projection operator
applied to differential forms in
,
. The harmonic projection operator
, one of the key operators in the harmonic analysis, plays an important role in the Hodge decomposition theory of differential forms. In the meanwhile, the homotopy operator
is also widely used in the decomposition and the
-theory of differential forms. In many situations, we need to estimate the various norms of the operators and their compositions.
We always assume that is a bounded, convex domain and
is a ball in
,
, throughout this paper. Let
be the ball with the same center as
and with
,
. We do not distinguish the balls from cubes in this paper. For any subset
, we use
to denote the Lebesgue measure of
. We call
a weight if
and
a.e. Differential forms are extensions of functions in
. For example, the function
is called a 0-form. Moreover, if
is differentiable, then it is called a differential 0-form. A differential
-form
is generated by
,
, that is,
where
,
, and
are differentiable functions. Let
be the set of all
-forms in
,
be the space of all differential
-forms on
and
be the
-forms
on
satisfying
for all ordered
-tuples
,
. We denote the exterior derivative by
for
. The Hodge codifferential operator
is given by
on
,
. We write
and
, where
is a weight. Let
be the
th exterior power of the cotangent bundle and
be the space of smooth
-forms on
. We set
has generalized gradient
. The harmonic
-fields are defined by
for some
The orthogonal complement of
in
is defined by
for all
The harmonic projection operator
is the operator involved in the Poisson's equation
where
is the Green's operator. See [1–4] for more propeties of the projection operator and Green's operator.
The differential equation is called the
-harmonic equation and the nonlinear elliptic partial differential equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ1_HTML.gif)
is called the nonhomogeneous -harmonic equation for differential forms, where
and
satisfy the conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ2_HTML.gif)
for almost every and all
. Here
are constants and
is a fixed exponent associated with (1.1). A solution to (1.1) is an element of the Sobolev space
such that
for all
with compact support. Let
be defined by
with
. Then
satisfies required conditions and
becomes the
-harmonic equation
for differential forms. If
is a function (a 0-form), the above equation reduces to the usual
-harmonic equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ3_HTML.gif)
for functions. Some results have been obtained in recent years about different versons of the -harmonic equation, see [2–9].
Let ,
. We write
,
, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ4_HTML.gif)
for some . The factor
here is for convenience and in fact the norm
is independent of this expansion factor, see [8]. Further, we write
for those forms whose coefficients are in the usual Lipschitz space with exponent
and write
for this norm. Similarly, for
,
, we write
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ5_HTML.gif)
for some . Again, the factor
here is for convenience and the norm
is independent of the expansion factor
, see [8]. When
is a
-form, (1.5) reduces to the classical definition of BMO
.
The following operator with the case
was first introduced by Cartan in [10]. Then, it was extended to the following version in [6]. For each point
, there is a linear operator
defined by
and the decomposition
A homotopy operator
is defined by
, averaging
over all points
in
, where
is normalized by
and the decomposition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ6_HTML.gif)
holds for any differential form . The
-form
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ7_HTML.gif)
for all ,
. From [6], we know that for any differential form
,
,
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ8_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ9_HTML.gif)
2. Lipschitz Norm Estimates
The following Hölder inequality will be used in the proofs of main theorems.
Lemma 2.1.
Let,
and
. If
and
are measurable functions on
, then
for any
.
Lemma 2.2 (see [1]).
Let and
,
. Then, there exists a positive constant
, independent of
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ10_HTML.gif)
We first prove the following Poincaré-type inequality for the composition of the homotopy operator and the projection operator.
Theorem 2.3.
Let,
,
, be a smooth differential form in a bounded, convex domain
,
be the projection operator and
be the homotopy operator. Then, there exists a constant
, independent of
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ11_HTML.gif)
for all balls with
.
Proof.
Let be the projection operator and
be the homotopy operator. For any differential form
, we know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ12_HTML.gif)
Replacing by
in (2.3) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ13_HTML.gif)
Since and
, by Lemma 2.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ14_HTML.gif)
Using (1.9), (2.4), and (2.5), we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ15_HTML.gif)
The proof of Theorem 2.3 has been completed.
Using Theorem 2.3, we estimate the following Lipschitz norm of the composite operator .
Theorem 2.4.
Let,
,
, be a smooth differential form in a bounded, convex domain
,
be the projection operator and
be the homotopy operator. Then, there exists a constant
, independent of
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ16_HTML.gif)
where is a constant with
.
Proof.
From Theorem 2.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ17_HTML.gif)
for all balls with
. Using the Hölder inequality with
, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ18_HTML.gif)
where we have used . Now, from the definition of Lipschitz norm, (2.9) and
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ19_HTML.gif)
The proof of Theorem 2.4 has been completed.
In order to prove Theorem 2.6, we extend [11, Lemma 8.2.2] into the following version for differential forms.
Lemma 2.5.
Let be a strictly increasing convex function on
with
, and
be a bounded domain in
. Assume that
is a smooth differential form in
such that
for any real number
and
, where
is a Radon measure defined by
for a weight
. Then, for any positive constant
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ20_HTML.gif)
where is a positive constant.
Proof.
Let Note that
. Then, there exists a constant
such that
that is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ21_HTML.gif)
Since is an increasing convex function, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ22_HTML.gif)
The proof of Lemma 2.5 is completed.
Theorem 2.6.
Let,
, be a smooth differential form satisfying the nonhomogeneous
-harmonic equation in a bounded, convex domain
and the Lebesgue
for any ball
. Assume that
is the projection operator and
is the homotopy operator. Then, there exists a constant
, independent of
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ23_HTML.gif)
where is a constant with
.
Proof.
Using Lemma 2.5 with and the weight
over the ball
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ24_HTML.gif)
From Theorem 2.3 and (2.15), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ25_HTML.gif)
From the definition of the Lipschitz norm, the Hölder inequality with and (2.16), for any ball
with
, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ26_HTML.gif)
Next, from the weak reverse Hölder inequality for solutions of the nonhomogeneous -harmonic equation, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ27_HTML.gif)
for some constant . Combination of (2.17) and (2.18) gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ28_HTML.gif)
Hence, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ29_HTML.gif)
Thus, taking the supremum on both sides of (2.20) over all balls with
and using the definitions of the Lipschitz and BMO norms, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ30_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ31_HTML.gif)
The proof of Theorem 2.6 has been completed.
Note that inequality (2.14) implies that the norm of
can be controlled by the norm
when
is a 1-form.
Theorem 2.7.
Let,
,
, be a smooth differential form in a bounded, convex domain
,
be the projection operator and
be the homotopy operator. Then, there exists a constant
, independent of
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ32_HTML.gif)
Proof.
From the definitions of the Lipschitz and BMO norms, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ33_HTML.gif)
that is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ34_HTML.gif)
where and
are constants with
. We have completed the proof of Theorem 2.7.
3. BMO Norm Estimates
We have developed some estimates for the Lipschitz norm in last section. Now, we estimates the BMO norm
. We first prove the following inequality between the BMO norm and the Lipschitz norm for the composite operator.
Theorem 3.1.
Let,
,
, be a smooth differential form in a bounded, convex domain
,
be the projection operator and
be the homotopy operator. Then, there exists a constant
, independent of
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ35_HTML.gif)
Proof.
From Theorems 2.4 and 2.7, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ36_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ37_HTML.gif)
respectively. Combination of (3.2) and (3.3) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ38_HTML.gif)
The proof of Theorem 3.1 has been completed.
Based on the above results, we discuss the weighted Lipschitz and BMO norms. For ,
, we write
,
, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ39_HTML.gif)
for some , where
is a bounded domain, the Radon measure
is defined by
,
is a weight and
is a real number. For convenience, we will write the following simple notation
for
. Similarly, for
,
, we will write
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ40_HTML.gif)
for some , where the Radon measure
is defined by
,
is a weight and
is a real number. Again, the factor
in the definitions of the norms
and
is for convenience and in fact these norms are independent of this expansion factor. We also write
to replace
when it is clear that the integral is weighted.
Definition 3.2.
We say a pair of weights satisfies the
-condition in a set
, write
, for some
and
with
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ41_HTML.gif)
Lemma (see [8]).
Let be a smooth differential form satisfying the nonhomogeneous
-harmonic equation in
,
and
. Then there exists a constant
, independent of
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ42_HTML.gif)
for all balls or cubes with
.
Using the reverse Hölder inequality (Lemma 3.3) and Theorem 2.3, one obtains the following weighted version:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ43_HTML.gif)
for all balls with
, where
, and
,
,
,
and
are constants with
,
,
,
and
.
Theorem 3.4.
Let,
,
, be a solution of the nonhomogeneous
-harmonic equation in a bounded, convex domain
,
be the projection operator and
be the homotopy operator, where the measure
and
are defined by
,
and
for some
and
with
for any
. Then, there exists a constant
, independent of
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ44_HTML.gif)
where and
are constants with
and
.
Proof.
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ45_HTML.gif)
for any ball . Using (3.9) and the Hölder inequality with
, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ46_HTML.gif)
Notice that and
, from (3.5), (3.11), and (3.12), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ47_HTML.gif)
We have completed the proof of Theorem 3.4.
We now estimate the norm in terms of the
norm.
Theorem 3.5.
Let,
,
, be a solution of the nonhomogeneous
-harmonic equation in a bounded domain
be the projection operator and
be the homotopy operator, where the measure
and
are defined by
,
and
for some
and
with
for any
. Then, there exists a constant
, independent of
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ48_HTML.gif)
where is a constant with
.
Proof.
From the definitions of the weighted Lipschitz and the weighted BMO norms, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ49_HTML.gif)
where is a positive constant. Replacing
by
in (3.15), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ50_HTML.gif)
where is a constant with
. Now, from Theorem 3.4, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ51_HTML.gif)
Substituting (3.17) into (3.16), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ52_HTML.gif)
The proof of Theorem 3.5 has been completed.
Theorem 3.6.
Let and
,
,
, be a smooth differential form in a bounded and convex domain
. Then, there exists a constant
, independent of
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ53_HTML.gif)
Proof.
From the decomposition (1.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ54_HTML.gif)
Using (1.5)), (3.20) and the Hölder inequality, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F212915/MediaObjects/13660_2008_Article_1921_Equ55_HTML.gif)
This ends the proof of Theorem 3.6.
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Acknowledgments
The authors thank the referee and the editor, Professor András Rontó, for their precious and thoughtful suggestions on this paper. The first author was supported by Science Research Foundation in Harbin Institute of Technology (HITC200709) and Development Program for Outstanding Young Teachers in HIT.
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Xing, Y., Ding, S. Norm Comparison Inequalities for the Composite Operator. J Inequal Appl 2009, 212915 (2009). https://doi.org/10.1155/2009/212915
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DOI: https://doi.org/10.1155/2009/212915