Global Poincaré inequalities for the composition of the sharp maximal operator and Green’s operator with Orlicz norms
© Gejun and Ling; licensee Springer. 2013
Received: 23 April 2013
Accepted: 8 October 2013
Published: 11 November 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 .
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.
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.
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.
for any ball .
for each . Here is the Euclidean distance between ξ and ∂ Ω.
Lemma 1 
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.
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.
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 .
where the measure μ is defined by and with for some r >1 and a constant .
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.
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 .
where and are constants.
Now, we are ready to give our another global Poincaré inequality with Orlicz norm.
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 .
The first author was supported by NSF of P.R. China (No. 11071048).
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