Weighted Inequalities for Potential Operators with Lipschitz and BMO Norms
© Y. Tong and J. Gu. 2011
Received: 1 January 2011
Accepted: 7 March 2011
Published: 15 March 2011
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.
On the potential operator and the functional , see  for details.
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 .
Lemma 2.1 (see ).
We will establish the following estimate for potential operators.
We have completed the proof of Theorem 2.2.
In this section, we introduce the weight appeared in .
The desired result is obtained.
which is called the strong doubling property of weights; see .
4. The Weighted Inequality for Potential Operators
Lemma 4.1 (see ).
Lemma 4.3 (see ).
The authors are supported by NSF of Hebei Province (A2010000910) and Scientific Research Fund of Zhejiang Provincial Education Department (Y201016044).
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