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Weighted Inequalities for Potential Operators on Differential Forms
Journal of Inequalities and Applications volume 2010, Article number: 713625 (2010)
Abstract
We develop the weak-type and strong-type inequalities for potential operators under two-weight conditions to the versions of differential forms. We also obtain some estimates for potential operators applied to the solutions of the nonhomogeneous A-harmonic equation.
1. Introduction
In recent years, differential forms as the extensions of functions have been rapidly developed. Many important results have been obtained and been widely used in PDEs, potential theory, nonlinear elasticity theory, and so forth; see [1–3]. In many cases, the process to solve a partial differential equation involves various norm estimates for operators. In this paper, we are devoted to develop some two-weight norm inequalities for potential operator to the versions of differential forms.
We first introduce some notations. Throughout this paper we always use to denote an open subset of
,
. Assume that
is a ball and
is the ball with the same center as
and with
. Let
,
, be the linear space of all
-forms
with summation over all ordered
-tuples
,
. The Grassman algebra
is a graded algebra with respect to the exterior products
. Moreover, if the coefficient
of
-form
is differential on
, then we call
a differential
-form on
and use
to denote the space of all differential
-forms on
. In fact, a differential
-form
is a Schwarz distribution on
with value in
. For any
and
, the inner product in
is defined by
with summation over all
-tuples
and all
. As usual, we still use
to denote the Hodge star operator. Moreover, the norm of
is given by
. Also, we use
to denote the differential operator and use
to denote the Hodge codifferential operator defined by
on
,
.
A weight is a nonnegative locally integrable function on
. The Lebesgue measure of a set
is denoted by
.
is a Banach space with norm

Similarly, for a weight , we use
to denote the weighted
space with norm
.
From [1], if is a differential form defined in a bounded, convex domain
, then there is a decomposition

where is called a homotopy operator. Furthermore, we can define the
-form
by

for all ,
.
For any differential -form
, we define the potential operator
by

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,

where is a function defined on
. Associated with
, the functional
is defined as

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

and there exists such that

On the potential operator and the functional
, see [4] for details.
For any locally -integrable form
, the Hardy-Littlewood maximal operator
is defined by

where is the ball of radius
, centered at
,
.
Consider the nonhomogeneous -harmonic equation for differential forms as follows:

where and
are two operators satisfying the conditions

for almost every and all
. Here
are some constants and
is a fixed exponent associated with (1.10). A solution to (1.10) is an element of the Sobolev space
such that

for all with compact support. Here
are those differential
-forms on
whose coefficients are in
. The notation
is self-explanatory.
2. Weak Type
Inequalities for Potential Operators
In this section, we establish the weighted weaks type inequalities for potential operators applied to differential forms. To state our results, we need the following definitions and lemmas.
We first need the following generalized Hölder inequality.
Lemma 2.1.
Let ,
, and
. If
and
are two measurable functions on
, then

for any .
Definition 2.2.
A pair of weights
satisfies the
-condition in a set
; write
for some
and
with
if

Proposition 2.3.
If for some
and
with
, then
satisfies the following condition:

Proof.
Choose and
. From the Hölder inequality, we have the estimate

Since

we obtain that satisfies (2.3) as required.
In [4], Martell proved the following two-weight weak type norm inequality applied to functions.
Lemma 2.4.
Let and
. Assume that
is the potential operator defined in (1.5) and that
is a functional satisfying (1.7) and (1.8). Let
be a pair of weights for which there exists
such that

Then the potential operator verifies the following weak type
inequality:

where for any set
and
.
The following definition is introduced in [5].
Definition 2.5.
A kernel
on
satisfies the standard estimates if there exist
,
, and constant
such that for all distinct points
and
in
, and all
with
, the kernel
satisfies
;
;
.
Theorem 2.6.
Let be the potential operator defined in (1.4) with the kernel
satisfying the condition
of the standard estimates and let
be a differential form in a domain
. Assume that
satisfies (2.3) for some
and
. Then, there exists a constant
, independent of
, such that the potential operator
satisfies the following weak type
inequality:

where for any set
and
.
Proof.
Since satisfies condition
of the standard estimates, for any ball
of radius
, we have

Here and
are two constants independent of
. Therefore,
and
are some constants independent of
. Thus, from
satisfying (2.3) for some
and
, it follows that

Set and
, where
corresponds to all ordered
-tuples and
. It is easy to find that there must exist some
such that
whenever
. Since the reverse is obvious, we immediately get
. Thus, using Lemma 2.4 and the elementary inequality
, where
is any constant, we have

Combining the above inequality (2.11), the elementary inequality and Lemma 2.4 yield

We complete the proof of Theorem 2.6.
3. The Strong Type
Inequalities for Potential Operators
In this section, we give the strong type inequalities for potential operators applied to differential forms. The result in last section shows that
-weights are stronger than those of condition (2.3), which is sufficient for the weak
inequalities, while the following conclusions show that
-condition is sufficient for strong
inequalities.
The following weak reverse Hölder inequality appears in [6].
Lemma 3.1.
Let ,
be a solution of the nonhomogeneous A-harmonic equation in
,
and
. Then there exists a constant
, independent of
, such that

for all balls with
.
The following two-weight inequality appears in [7].
Lemma 3.2.
Let and
. Assume that
is the potential operator defined in (1.5) and
is a functional satisfying (1.7) and (1.8). Let
be a pair of weights for which there exists
such that

Then, there exists a constant , independent of
, such that

Lemma 3.3.
Let ,
,
, be a differential form defined in a domain
and
be the potential operator defined in (1.4) with the kernel
satisfying condition
of standard estimates. Assume that
for some
and
. Then, there exists a constant
, independent of
, such that

Proof.
By the proof of Theorem 2.6, note that (3.2) still holds whenever satisfies the
-condition. Therefore, using Lemma 3.2, we have

Also, Lemma 3.2 yields that

for all ordered -tuples
. From (3.5) and (3.6), it follows that

We complete the proof of Lemma 3.3.
Lemma 3.3 shows that the two-weight strong inequality still holds for differential forms. Next, we develop the inequality to the parametric version.
Theorem 3.4.
Let ,
,
, be the solution of the nonhomogeneous A-harmonic equation in a domain
and let
be the potential operator defined in (1.4) with the kernel
satisfying condition
of standard estimates. Assume that
for some
and
. Then, there exists a constant
, independent of
, such that

for all balls with
. Here
and
are constants with
.
Proof.
Take . By
, where
and the Hölder inequality, we have

for all balls with
. Choosing
to be a ball and
in Lemma 3.3, then there exists a constant
, independent of
, such that

Choosing and using Lemma 3.1, we obtain

where . Combining (3.9), (3.10), and (3.11), it follows that

Since , using the Hölder inequality with
, we obtain

From the condition , we have

Combining (3.12), (3.13), and (3.14) yields

for all balls with
. Thus, we complete the proof of Theorem 3.4.
Next, we extend the weighted inequality to the global version, which needs the following lemma about Whitney cover that appears in [6].
Lemma 3.5.
Each open set has a modified Whitney cover of cubes
such that


for all and some
, where
is the characteristic function for a set
.
Theorem 3.6.
Let ,
,
, be the solution of the nonhomogeneous A-harmonic equation in a domain
and let
be the potential operator defined in (1.4) with the kernel
satisfying condition
of standard estimates. Assume that
for some
and
. Then, there exists a constant
, independent of
, such that

where is some constant with
.
Proof.
From Lemma 3.5, we note that has a modified Whitney cover
. Hence, by Theorem 3.4, we have that

This completes the proof of Theorem 3.6.
Remark 3.7.
Note that if we choose the kernel to satisfy the standard estimates, then the potential operators
reduce to the Calderón-Zygmund singular integral operators. Hence, Theorems 3.4 and 3.6 as well as Theorem 2.6 in last section still hold for the Calderón-Zygmund singular integral operators applied to differential forms.
4. Applications
In this section, we apply our results to some special operators. We first give the estimate for composite operators. The following lemma appears in [8].
Lemma 4.1.
Let be the Hardy-Littlewood maximal operator defined in (1.9) and let
,
,
, be a differential form in a domain
. Then,
and

for some constant independent of
.
Observing Lemmas 4.1 and 3.3, we immediately have the following estimate for the composition of the Hardy-Littlewood maximal operator and the potential operator
.
Theorem 4.2.
Let ,
,
, be a differential form defined in a domain
,
be the Hardy-Littlewood maximal operator defined in (1.9),
, and let
be the potential operator with the kernel
satisfying condition
of standard estimates. Then, there exists a constant
, independent of
, such that

Next, applying our results to some special kernels, we have the following estimates.
Consider that the function is defined by

where . For any
, we write
. It is easy to see that
and
. Such functions are called mollifiers. Choosing the kernel
and setting each coefficient of
satisfing
, we have the following estimate.
Theorem 4.3.
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

Proof.
By the decomposition for differential forms, we have

where is the homotopy operator. Also, from [1], we have

for any differential form defined in
. Therefore,

Note that

where the notation denotes convolution. Hence, we have

Since , it is easy to find that
. Therefore, we have

From (4.7) and (4.10), we obtain

This ends the proof of Theorem 4.3.
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Bi, H. Weighted Inequalities for Potential Operators on Differential Forms. J Inequal Appl 2010, 713625 (2010). https://doi.org/10.1155/2010/713625
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DOI: https://doi.org/10.1155/2010/713625
Keywords
- Differential Form
- Potential Operator
- Standard Estimate
- Weak Type
- Norm Inequality