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Global Poincaré inequalities for the composition of the sharp maximal operator and Green’s operator with Orlicz norms
Journal of Inequalities and Applications volume 2013, Article number: 526 (2013)
In this paper, we establish the global Poincaré-type inequalities for the composition of the sharp maximal operator and Green’s operator with Orlicz norm.
The -theory of solutions of the homogeneous A-harmonic equation for differential forms u has been very well developed in recent years. Many -norm estimates and inequalities, including the Poincaré inequalities, for solution of the homogeneous A-harmonic equation have been established; see [1, 2]. The Poincaré inequalities for differential forms is an important tool in analysis and related fields, including partial differential equations and potential theory. However, the study of the nonhomogeneous A-harmonic equations has just begun [2–4]. In this paper, we focus on a class of differential forms satisfying the well-known nonhomogeneous A-harmonic equation .
Let us first introduce some necessary notation and terminology. Ω will refer to a bounded, convex domain in unless otherwise stated and B is a ball in , . We use σB to denote 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 n-dimensional Lebesgue measure of the set . We say w is a weight if and a.e. For a function u, we denote the average of u over B by
where is the volume of B and the μ-average of u over B by
Let be the set of all l-forms in , let be the space of all differential l-forms on Ω, and let be the l-forms on Ω satisfying for all ordered l-tuples I, . We denote the exterior derivative by for , and define the Hodge star operator as follows. If , , is a differential k-form, then , where , , and . The Hodge codifferential operator
is given by on , . We write
The well-known nonhomogeneous A-harmonic equation is
where and satisfy the conditions:
for almost every and all . Here, are constants and is a fixed exponent associated with (1). If the operator , equation (1) becomes , which is called the (homogeneous) A-harmonic equation. A solution to (1) is an element of the Sobolev space such that for all with compact support. Let be defined by with . Then A satisfies the required conditions and becomes the p-harmonic equation
for differential forms. If u is a function (0-form), equation (3) reduces to the usual p-harmonic equation for functions. A remarkable progress has been made recently in the study of different versions of the harmonic equations, see  for more details.
Let be the space of smooth l-forms on Ω and
The harmonic l-fields are defined by
The orthogonal complement of ℋ in is defined by
Then 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.
In harmonic analysis, a fundamental operator is the Hardy-Littlewood maximal operator. The maximal function is a classical tool in harmonic analysis but recently it has been successfully used in studying Sobolev functions and partial differential equations. For any locally -integrable form , we define the Hardy-Littlewood maximal operator by
where is the ball of radius r, centered at x, . We write if . Similarly, for a locally -integrable form , we define the sharp maximal operator by
The purpose of this paper is to estimate the global Poincaré-type inequalities for the composition of the sharp maximal operator and Green’s operator with Orlicz norm.
2 Definitions and lemmas
We now introduce the following definition and lemmas that will be used in this paper.
Definition 1 We say the weight satisfies the condition, , write if a.e., and
for any ball .
Definition 2 A proper subdomain is called a δ-John domain, , if there exists a point which can be joined with any other point by a continuous curve so that
for each . Here is the Euclidean distance between ξ and ∂ Ω.
Lemma 1 
Each Ω has a modified Whitney cover of cubes such that
and some , and if , then there exists a cube R (this cube need not be a member of ) in such that . Moreover, if Ω is δ-John, then there is a distinguished cube which can be connected with every cube by a chain of cubes from and such that , , for some .
3 Poincaré inequalities
In this section, we prove the global Poincaré inequalities for the composition of the sharp maximal operator and Green’s operator with norm.
To get our result, we rewrite our Theorem 2 in  as follows.
Lemma 2 Let u be a smooth differential form satisfying A-harmonic equation (1) in a bounded domain Ω, let G be Green’s operator, and let be the sharp maximal operator defined in (4) with . Then there exists a constant C, independent of u, such that
for all balls B with , and a constant , where the measure μ is defined by and with for some r >1 and a constant δ.
Theorem 1 Let , , be a smooth differential form satisfying A-harmonic equation (1), let G be Green’s operator, and let be the sharp maximal operator defined in (4) with . Then there exists a constant , independent of u, such that
for any bounded and convex δ-John domain , where the fixed cube , the constant appeared in Lemma 1, and the measure μ is defined by and with for some r >1 and a constant .
Proof First, we use Lemma 1 for the bounded and convex δ-John domain Ω. There is a modified Whitney cover of cubes for Ω such that , and for some . Moreover, there is a distinguished cube which can be connected with every cube by a chain of cubes from and such that , , for some . Then, by the elementary inequality , , we have
The first sum in (8) can be estimated by using Lemma 2.
where the measure μ is defined by and with for some r >1 and a constant .
To estimate the second sum in (8), we need to use the property of δ-John domain. Fix a cube and let be the chain in Lemma 1. Then we have
The chain also has the property that for each i, , . Thus, there exists a cube such that and , . So,
By (11), (12) and Lemma 2, we have
Then, by (10), (13) and the elementary inequality , we finally obtain
Substituting (9) and (14) in (8), we have completed the proof of Theorem 1. □
4 Poincaré inequality with Orlicz norm
In this section, we give a global Poincaré inequality with Orlicz norm for the composition of the sharp maximal operator and Green’s operator.
Definition 3 Let φ be a continuously increasing convex function on with , and let Λ be a domain with . If u is a measurable function in Λ, then we define the Orlicz norm of u by
A continuously increasing function with is called an Orlicz function. A convex Orlicz function φ is often called a Young function.
In , Buckley and Koskela gave the following class of functions.
Definition 4 We say a Young function φ lies in the class , , , if (i) and (ii) for all , where g is a convex increasing function and h is a concave increasing function on .
From  and , we know that the class contains some very interesting functions, such as and , , , and each of φ, g and h is doubling in the sense that its values at t and 2t are uniformly comparable for all , and the consequent fact that
where and are constants.
Now, we are ready to give our another global Poincaré inequality with Orlicz norm.
Theorem 2 Let φ be a Young function in the class , , , let , , be a smooth differential form satisfying A-harmonic equation (1) in Ω, let G be Green’s operator, and let be the sharp maximal operator defined in (4) with . Then there exists a constant C, independent of u, such that
for any bounded and convex δ-John domain with , where the fixed cube appeared in Lemma 1, and the measure μ is defined by and with for some r >1 and a constant .
Proof Let g, h be the functions in the condition. Note that φ is an increasing function. Using Theorem 1, (i) in Definition 4, and Jensen’s inequality, we obtain
Again, from (i) in Definition 4, we have
Thus, we obtain
Combining (17) and (18) yields
Now, using Jensen’s inequality for , (16) and (ii) in Definition 4, and noticing that φ is doubling, we see
Substituting (19) into (20) and using the fact that φ is doubling, we get
Therefore, from Definition 3, we have
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The first author was supported by NSF of P.R. China (No. 11071048).
The authors declare that they have no competing interests.
All authors completed the paper together. All authors read and approved the final manuscript.
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Gejun, B., Ling, Y. Global Poincaré inequalities for the composition of the sharp maximal operator and Green’s operator with Orlicz norms. J Inequal Appl 2013, 526 (2013). https://doi.org/10.1186/1029-242X-2013-526
- Poincaré-type inequalities
- Orlicz norm
- sharp maximal operator
- Green’s operator