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.