- Open Access
Norm inequalities for composition of the Dirac and Green’s operators
© Ding and Liu; licensee Springer. 2013
Received: 27 June 2013
Accepted: 27 August 2013
Published: 13 September 2013
We first prove a norm inequality for the composition of the Dirac operator and Green’s operator. Then, we estimate for the Lipschitz and BMO norms of the composite operator in terms of the norm of a differential form.
MSC:26B10, 30C65, 31B10, 46E35.
The purpose of this paper is to derive the norm inequalities for the composite operator of the Hodge-Dirac operator D and Green’s operator G on differential forms. Specifically, we will develop the upper bounds for norms of the composite operator applied to differential form u in terms of the norm of u. We all know that there are different versions of Dirac operators, such as the Hodge-Dirac operator associated to a Riemannian manifold and the euclidean Dirac operator arising in Clifford analysis. The Dirac operator studied in this paper is the Hodge-Dirac operator defined by , where d is the exterior derivative, and is the Hodge codifferential, which is the formal adjoint to d. Both the Dirac operator D and Green’s operator G are widely studied and used in mathematics and physics. Since it was initiated by Paul Dirac in order to get a form of quantum theory compatible with special relativity, Dirac operators have been playing an important role in many fields of mathematics and physics, such as quantum mechanics, Clifford analysis and PDEs. Green’s operator is a key operator, which has been very well used in several areas of mathematics. In many situations, the process of studying solutions to PDEs involves estimating the various norms of the operators and their compositions. Hence, we are motivated to establish the upper bounds for the composite operators in this paper. See [1–8] for recent work on the Dirac operator, Green’s operator and their applications.
Let M be a bounded domain and B be a ball in , , throughout this paper. We use σB to express the ball with the same center as B and with , . We do not distinguish the balls from cubes in this paper. We use to denote the Lebesgue measure of a set . We call w a weight if and a.e. Let be the standard unit basis of , and let be the linear space of l-vectors, which is spanned by the exterior products , corresponding to all ordered l-tuples , , . The Grassman algebra is a graded algebra with respect to the exterior products. For any and , the inner product in ∧ is defined by , with summation over all l-tuples and all integers . The Hodge star operator ⋆: ∧→∧ is defined by the rule and for all α, . The norm of is given by the formula . The Hodge star is an isometric isomorphism on Λ with and .
A differential l-form ω on M is a de Rham current (see [, Chapter III]) on M with values in . Differential forms are extensions of functions. For example, in , the function is called a 0-form. Moreover, if is differentiable, then it is called a differential 0-form. The 1-form in can be written as . If the coefficient functions , , are differentiable, then is called a differential 1-form. Similarly, a differential k-form is generated by , , that is, , where , . Let be the space of all differential l-forms on M, and let be the l-forms on M satisfying for all ordered l-tuples I, . We denote the exterior derivative by for . The Hodge codifferential operator is given by on , . The Dirac operator D involved in this paper is defined by . It is easy to check that , where is the Laplace-Beltrami operator. Let be the l th exterior power of the cotangent bundle, be the space of smooth l-forms on M and . The harmonic l-fields are defined by . The orthogonal complement of ℋ in is defined by . Then the Green’s operator G is defined as by assigning to be the unique element of satisfying Poisson’s equation , where H is the harmonic projection operator that maps onto ℋ so that is the harmonic part of u. See  for more properties of these operators. We write and , where is a weight.
for some . When ω is a 0-form, (1.2) reduces to the classical definition of . The definitions of Lipschitz and BMO norms above appeared in .
2 norm inequalities
Hence, we have the following lemma.
for any ball.
The following results about the homotopy operator T can be found in .
for all , .
We will use the following generalized Hölder’s inequality repeatedly in this paper.
We now prove the following norm inequality for the composite operator of the Dirac operator D and Green’s operator G applied to differential forms.
for all balls.
We have completed the proof of Lemma 2.4.
Next, we prove the Poincaré-type inequality for the composition of the Dirac operator and Green’s operator, which forms the foundation of this paper. □
for all balls.
We have completed the proof of Theorem 2.5. □
3 Upper bounds for Lipschitz and BMO norms
In this section, we establish the upper bounds for Lipschitz norms and BMO norms in terms of norms. Using Theorem 2.5, we now obtain the upper bounds for Lipschitz norm of the composite operator .
wherekis a constant with.
The proof of Theorem 3.1 has been completed.
where is a positive constant. Hence, we have the following inequality between the Lipschitz norm and the BMO norm. □
whereCis a constant.
Combining Theorems 3.1 and Lemma 3.2, we obtain the following inequality between the BMO norm and the norm.
Combining (3.7) and (3.8) gives . The proof of Theorem 3.3 has been completed. □
We will need the following lemma that appeared in .
whereCis a positive constant.
The following -class of differential forms was introduced in .
holds for any ball B with , where and are constants.
wherekis a constant with.
The proof of Theorem 3.6 has been completed. □
Replacing u by in Lemma 3.2, we obtain the following comparison inequality between the Lipschitz norm and the BMO norm.
4 Weighted inequalities
for some , where the measure μ is defined by , w is a weight. Again, we write to replace when it is clear that the integral is weighted.
for any ball .
for all balls B with , where is a constant.
wherekis a constant with.
since and . We have completed the proof of Theorem 4.2.
where is a positive constant. Thus, we have obtained the following result. □
whereCis a constant.
holds for any bounded domainM.
This ends the proof of Theorem 4.4. □
As applications of the results proved in this paper, we consider the following examples.
In fact, it would be very hard to estimate directly from calculation of the operator norm.
Remark (i) The Poincaré-type inequalities for the composition of the Dirac operator and Green’s operator presented in (2.3) and (4.3) can be extended into the global case. (ii) It should be noticed that the domains involved in this paper are general bounded domains, which largely increases the flexibility and applicability of our results.
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