© Yuxia Tong et al. 2010
Received: 29 December 2009
Accepted: 31 March 2010
Published: 27 June 2010
Green's operator is often applied to study the solutions of various differential equations and to define Poisson's equation for differential forms. Green's operator has been playing an important role in the study of PDEs. 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 Green's operator in this paper.
In the meanwhile, there have been generally studied about -weighted [1, 2] and -weighted [3, 4] different inequalities and their properties. Results for more applications of the weight are given in [5, 6]. The purpose of this paper is to derive the new weighted inequalities with the Lipschitz norm and BMO norm for Green's operator applied to differential forms. We will introduce -weight, which can be considered as a further extension of the -weight.
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 denote the exterior derivative by for . Its formal adjoint operator is given by on , . Let be the th exterior power of the cotangent bundle and let be the space of smooth -forms on . We set . The harmonic -fields are defined by = . The orthogonal complement of in is defined by = . Then, Green's operator is defined as by assigning to be the unique element of satisfying Poisson's equation = , where is the harmonic projection operator that maps onto , so that is the harmonic part of . See  for more properties of Green's operator.
for differential forms. If is a function (a 0-form), (1.10) reduces to the usual -harmonic equation for functions. We should notice that if the operator equals in (1.7), then (1.7) reduces to the homogeneous -harmonic equation. Some results have been obtained in recent years about different versions of the -harmonic equation; see [9–11].
2. Preliminary Knowledge and Lemmas
If we choose and in Definition 2.1, we will obtain the usual -weight. If we choose , and in Definition 2.1, we will obtain the -weight . If we choose , and in Definition 2.1, we will obtain the -weight .
Lemma 2.2 (see ).
We need the following generalized Hölder inequality.
The following version of weak reverse Hölder inequality appeared in .
Lemma 2.5 (see ).
We need the following Lemma 2.6 (Caccioppoli inequality) that was proved in .
Lemma 2.6 (see ).
Lemma 2.7 (see ).
Lemma 2.8 (see ).
3. Main Results and Proofs
We have completed the proof of Theorem 3.1.
Next, we will establish the following weighted norm comparison theorem between the Lipschitz and the BMO norms.
Now, we will prove the following weighted inequality between the BMO norm and the Lipschitz norm for Green's operator.
We have completed the proof of Theorem 3.3.
Note that the differentiable functions are special differential forms (0-forms). Hence, the usual -harmonic equation for functions is the special case of the -harmonic equation for differential forms. Therefore, all results that we have proved for solutions of the -harmonic equation in this paper are still true for -harmonic functions.
The first author is supported by NSFC (No:10701013), NSF of Hebei Province (A2010000910) and Tangshan Science and Technology projects (09130206c). The second author is supported by NSFC (10771110 and 60872095) and NSF of Nongbo (2008A610018).
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