Some Estimates of Integrals with a Composition Operator
© Bing Liu. 2010
Received: 27 December 2009
Accepted: 16 March 2010
Published: 30 March 2010
The purpose of this paper is to establish the Poincaré-type inequalities for the composition of the homotopy operator , differential operator , and Green's operator under Lipschitz and norms. One of the reasons that we consider this composition operator is due to the Hodge theorem. It is well known that Hodge decomposition theorem plays important role in studying harmonic analysis and differential forms; see [1–3]. It gives a relationship of the three key operators in harmonic analysis, namely, Green's operator , the Laplacian operator , and the harmonic projection operator . This relationship offers us a tool to apply the composition of the three operators under the consideration to certain harmonic forms and to obtain some estimates for certain integrals which are useful in studying the properties of the solutions of PDEs. We also consider the integrals of this composition operator with a singular factor because of their broad applications in solving differential and integral equations; see .
We first give some notations and definitions which are commonly used in many books and papers; for example, see [1, 4–12]. We use to denote a Riemannian, compact, oriented, and smooth manifold without boundary on . Let be the th exterior power of the cotangent bundle, and let be the space of smooth -forms on and has generalized gradient . The harmonic -fields are defined by , for some The orthogonal complement of in is defined by for all Then, Green's operator is defined as by assigning as the unique element of satisfying Poisson's equation , where is the harmonic projection operator that maps onto so that is the harmonic part of . In this paper, we also assume that is a bounded and convex domain in . The -dimensional Lebesgue measure of a set is denoted by . The operator with the case was first introduced by Cartan in . Then, it was extended to the following version in . To each there corresponds a linear operator defined by and the decomposition A homotopy operator is defined by averaging over all points :
where satisfy the conditions: , and for almost every and all . Here and are constants, and is a fixed exponent associated with the equation. A significant progress has been made recently in the study of different versions of the harmonic equations; see [1, 4–12].
We will use the following results.
Lemma 1.1 (see ).
Lemma 1.2 (see ).
Lemma 1.3 (see ).
2. The Estimates for Lipschitz and BMO Norms
Theorem 2.1 is proved.
We learned from  that the norm and the Lipschitz norm are related in the following inequality.
Lemma 2.2 (see ).
3. The Lipschitz and BMO Norms with a Singular Factor
We considered the integrals with singular factors in . Here, we will give estimates to Poincaré type inequalities with singular factors in the Lipschitz and norms. If we use the formula (1.7) in Lemma 1.1 and follow the same proof of Lemma in , we obtain the following theorem.
We extend Theorem 3.1 to the Lipschitz norm with a singular factor and have the following result.
We have completed the proof of Theorem 3.2.
We omit the proof since it is the same as the proof of Theorem 3.2.
4. The Weighted Inequalities
Lemma 4.3 (see ).
We extend the Lemma 4.3 to the version with the Lipschitz norm as the following.
Example 5.2 (see ).
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