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.