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Weighted Inequalities for Potential Operators with Lipschitz and BMO Norms
Advances in Difference Equations volume 2011, Article number: 659597 (2011)
Abstract
Some Lipschitz norm and BMO norm inequalities for potential operator to the versions of differential forms are obtained, and some properties of a new kind of weight are derived.
1. Introduction
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 potential operator to the versions of differential forms.
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. The
-dimensional Lebesgue measure of a set
is denoted by
. We call
a weight if
and that is,
. For
and a weight
, we denote the weighted
-norm of a measurable function
over
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ1_HTML.gif)
where is a real number.
Differential forms are important generalizations of real functions and distributions; note that a 0-form is the usual function in . A differential
-form
on
is a Schwartz distribution on
with values in
. We use
to denote the space of all differential
-forms
. We write
for the
-forms with
for all ordered
-tuples
. Thus,
is a Banach space with norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ2_HTML.gif)
For , the vector-valued differential form
consists of differential forms
, where the partial differentiations are applied to the coefficients of
. As usual,
is used to denote the Sobolev space of
-forms, which equals
with norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ3_HTML.gif)
The notations and
are self-explanatory. For
and a weight
, the weighted norm of
over
is denoted by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ4_HTML.gif)
where is a real number. We denote the exterior derivative by
 : 
for
. Its formal adjoint operator
 : 
is given by
on
.
Let . We write
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ5_HTML.gif)
for some . 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%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ6_HTML.gif)
for some . When
is a 0-form, (1.6) reduces to the classical definition of
.
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%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ7_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 write
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ8_HTML.gif)
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.
From [1], if is a differential form defined in a bounded, convex domain
, then there is a decomposition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ9_HTML.gif)
where is called a homotopy operator. Furthermore, we can define the
-form
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ10_HTML.gif)
for all .
For any differential -form
, we define the potential operator
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ11_HTML.gif)
where the kernel is a nonnegative measurable function defined for
, and the summation is over all ordered
-tuples
. It is easy to find that the case
reduces to the usual potential operator. That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ12_HTML.gif)
where is a function defined on
. Associated with
, the functional
is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ13_HTML.gif)
where is some sufficiently small constant and
is a ball with radius
. Throughout this paper, we always suppose that
satisfies the following conditions: there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ14_HTML.gif)
and there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ15_HTML.gif)
On the potential operator and the functional
, see [2] for details.
The nonlinear elliptic partial differential equation is called the homogeneous
-harmonic equation or the
-harmonic equation, and the differential equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ16_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%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ17_HTML.gif)
for almost every and all
. Here
are constants and
is a fixed exponent associated with (1.16). A solution to (1.16) is an element of the Sobolev space
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ18_HTML.gif)
for all with compact support. When
is a 0-form, that is,
is a function, (1.16) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ19_HTML.gif)
Lots of results have been obtained in recent years about different versions of the -harmonic equation, see [3–5].
2. The Estimate for Potential Operators with Lipschitz Norm and BMO Norm
In this section, we give the estimate for potential operators with Lipschitz norm and BMO norm applied to differential forms. The following strong type inequality for potential operators appears in [6].
Lemma 2.1 (see [6]).
Let , be a differential form defined in a bounded, convex domain
, and let
be coefficient of
with
for all ordered
-tuples
. Assume that
and
is the potential operator with
for any
, then there exists a constant
, independent of
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ20_HTML.gif)
We will establish the following estimate for potential operators.
Theorem 2.2.
Let , be a differential form defined in a bounded, convex domain
, and let
be coefficient of
with
for all ordered
-tuples
. Assume that
and
is the potential operator with
for any
, then there exists a constant
, independent of
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ21_HTML.gif)
Proof.
By the definition of the Lipschitz norm, (2.1), and hölder's inequality with , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ22_HTML.gif)
since and
, where
is a constant and
.
By the definition of the BMO norm, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ23_HTML.gif)
We have completed the proof of Theorem 2.2.
3. The
Weight
In this section, we introduce the weight appeared in [7].
Definition 3.1.
Let be two locally integrable nonnegative functions in
and assume that
almost everywhere. We say that
belongs to the
class,
and
, or that
is an
weight, write
or
when it will not cause any confusion, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ24_HTML.gif)
for all balls .
The following results show that the weights have the properties similar to those of the
weights.
Theorem 3.2.
If , then
.
Proof.
Let . Since
, by Hölder's inequality,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ25_HTML.gif)
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ26_HTML.gif)
Thus, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ27_HTML.gif)
for all balls since
. Therefore,
, and hence
.
Theorem 3.3.
If and the measures
are defined by
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ28_HTML.gif)
where is a ball in
and
is a measurable subset of
.
Proof.
By Hölder's inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ29_HTML.gif)
This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ30_HTML.gif)
Note that , by Hölder's inequality again, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ31_HTML.gif)
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ32_HTML.gif)
Hence, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ33_HTML.gif)
Since , there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ34_HTML.gif)
Combining (3.7), (3.10), and (3.11), we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ35_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ36_HTML.gif)
The desired result is obtained.
If we choose and
in Theorem 3.3, we will obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ37_HTML.gif)
which is called the strong doubling property of weights; see [8].
4. The Weighted Inequality for Potential Operators
In this section, we are devoted to develop some two-weight norm inequalities for potential operator to the versions of differential forms. We need the following lemmas.
Lemma 4.1 (see [9]).
If , then there exist constants
and
, independent of
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ38_HTML.gif)
for all balls .
Lemma 4.2.
Let ,
, and
. If
and
are measurable functions on
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ39_HTML.gif)
for any .
Lemma 4.3 (see [10]).
Let be a solution of the nonhomogeneous A-harmonic equation in
and
, then there exists a constant
, independent of
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ40_HTML.gif)
for all with
.
Theorem 4.4.
Let , be a solution of the nonhomogeneous
-harmonic equation (1.16) in a bounded domain
and
is the potential operator with
for any
, where the Radon measures
and
are defined by
. Assume that
and
for some
with
for any
, then there exists a constant
, independent of
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ41_HTML.gif)
where is a constant with
.
Proof.
Since , using Lemma 4.1, there exist constants
and
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ42_HTML.gif)
for any ball .
Since , by Lemma 4.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ43_HTML.gif)
Choose where
,
, then
and
. Since
, by Lemma 4.2 and (4.5), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ44_HTML.gif)
From Lemma 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ45_HTML.gif)
Applying Lemma 4.3 (the weak reverse Hölder inequality for the solutions of the nonhomogeneous -harmonic equation), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ46_HTML.gif)
where is a constant and
. Choosing
, then
. Using Hölder's inequality with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ47_HTML.gif)
Since , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ48_HTML.gif)
Since , combining with (4.6)–(4.11), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ49_HTML.gif)
From the definition of the BMO norm, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F659597/MediaObjects/13662_2011_Article_64_Equ50_HTML.gif)
for all balls with
and
. We have completed the proof of Theorem 4.4.
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Acknowledgments
The authors are supported by NSF of Hebei Province (A2010000910) and Scientific Research Fund of Zhejiang Provincial Education Department (Y201016044).
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Tong, Y., Gu, J. Weighted Inequalities for Potential Operators with Lipschitz and BMO Norms. Adv Differ Equ 2011, 659597 (2011). https://doi.org/10.1155/2011/659597
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DOI: https://doi.org/10.1155/2011/659597