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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 Applications20132013:526

https://doi.org/10.1186/1029-242X-2013-526

• Received: 23 April 2013
• Accepted: 8 October 2013
• Published:

## Abstract

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.

## Keywords

• Poincaré-type inequalities
• Orlicz norm
• sharp maximal operator
• Green’s operator

## 1 Introduction

The ${L}^{p}$-theory of solutions of the homogeneous A-harmonic equation ${d}^{\star }A\left(x,du\right)=0$ for differential forms u has been very well developed in recent years. Many ${L}^{p}$-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 ${d}^{\star }A\left(x,du\right)=B\left(x,du\right)$ has just begun . In this paper, we focus on a class of differential forms satisfying the well-known nonhomogeneous A-harmonic equation ${d}^{\star }A\left(x,du\right)=B\left(x,du\right)$.

Let us first introduce some necessary notation and terminology. Ω will refer to a bounded, convex domain in ${\mathbb{R}}^{n}$ unless otherwise stated and B is a ball in ${\mathbb{R}}^{n}$, $n\ge 2$. We use σB to denote the ball with the same center as B and with $diam\left(\sigma B\right)=\sigma diam\left(B\right)$, $\sigma >0$. We do not distinguish the balls from cubes in this paper. We use $|E|$ to denote the n-dimensional Lebesgue measure of the set $E\subseteq {\mathbb{R}}^{n}$. We say w is a weight if $w\in {L}_{\mathrm{loc}}^{1}\left({\mathbb{R}}^{n}\right)$ and $w>0$ a.e. For a function u, we denote the average of u over B by
${u}_{B}=\frac{1}{|B|}{\int }_{B}u\phantom{\rule{0.2em}{0ex}}dx,$
where $|B|$ is the volume of B and the μ-average of u over B by
${u}_{B,\mu }=\frac{1}{\mu \left(B\right)}{\int }_{B}u\phantom{\rule{0.2em}{0ex}}d\mu .$
Let ${\wedge }^{l}={\wedge }^{l}\left({\mathbb{R}}^{n}\right)$ be the set of all l-forms in ${\mathbb{R}}^{n}$, let ${D}^{\prime }\left(\mathrm{\Omega },{\wedge }^{l}\right)$ be the space of all differential l-forms on Ω, and let ${L}^{p}\left(\mathrm{\Omega },{\wedge }^{l}\right)$ be the l-forms $u\left(x\right)={\sum }_{I}{u}_{I}\left(x\right)\phantom{\rule{0.2em}{0ex}}d{x}_{I}$ on Ω satisfying ${\int }_{\mathrm{\Omega }}{|{u}_{I}|}^{p}\phantom{\rule{0.2em}{0ex}}dx<\mathrm{\infty }$ for all ordered l-tuples I, $l=1,2,\dots ,n$. We denote the exterior derivative by $d:{D}^{\prime }\left(\mathrm{\Omega },{\wedge }^{l}\right)\to {D}^{\prime }\left(\mathrm{\Omega },{\wedge }^{l+1}\right)$ for $l=0,1,\dots ,n-1$, and define the Hodge star operator $\star :{\wedge }^{k}\to {\wedge }^{n-k}$ as follows. If $u={u}_{I}\phantom{\rule{0.2em}{0ex}}d{x}_{I}$, ${i}_{1}<{i}_{2}<\cdots <{i}_{k}$, is a differential k-form, then $\star u={\left(-1\right)}^{\sum \left(I\right)}{u}_{I}\phantom{\rule{0.2em}{0ex}}d{x}_{J}$, where $I=\left({i}_{1},{i}_{2},\dots ,{i}_{k}\right)$, $J=\left\{1,2,\dots ,n\right\}-I$, and $\sum \left(I\right)=\frac{k\left(k+1\right)}{2}+{\sum }_{j=1}^{k}{i}_{j}$. The Hodge codifferential operator
${d}^{\star }:{D}^{\prime }\left(\mathrm{\Omega },{\wedge }^{l+1}\right)\to {D}^{\prime }\left(\mathrm{\Omega },{\wedge }^{l}\right)$
is given by ${d}^{\star }={\left(-1\right)}^{nl+1}\star d\star$ on ${D}^{\prime }\left(\mathrm{\Omega },{\wedge }^{l+1}\right)$, $l=0,1,\dots ,n-1$. We write
${\parallel u\parallel }_{s,\mathrm{\Omega }}={\left({\int }_{\mathrm{\Omega }}{|u|}^{s}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/s}.$
The well-known nonhomogeneous A-harmonic equation is
${d}^{\star }A\left(x,du\right)=B\left(x,du\right),$
(1)
where $A:\mathrm{\Omega }×{\wedge }^{l}\left({\mathbb{R}}^{n}\right)\to {\wedge }^{l}\left({\mathbb{R}}^{n}\right)$ and $B:\mathrm{\Omega }×{\wedge }^{l}\left({\mathbb{R}}^{n}\right)\to {\wedge }^{l-1}\left({\mathbb{R}}^{n}\right)$ satisfy the conditions:
$|A\left(x,\xi \right)|\le a{|\xi |}^{p-1},\phantom{\rule{2em}{0ex}}A\left(x,\xi \right)\cdot \xi \ge \phantom{\rule{0.25em}{0ex}}{|\xi |}^{p},\phantom{\rule{2em}{0ex}}|B\left(x,\xi \right)|\le b{|\xi |}^{p-1}$
(2)
for almost every $x\in \mathrm{\Omega }$ and all $\xi \in {\wedge }^{l}\left({\mathbb{R}}^{n}\right)$. Here, $a,b>0$ are constants and $1 is a fixed exponent associated with (1). If the operator $B=0$, equation (1) becomes ${d}^{\star }A\left(x,du\right)=0$, which is called the (homogeneous) A-harmonic equation. A solution to (1) is an element of the Sobolev space ${W}_{\mathrm{loc}}^{1,p}\left(\mathrm{\Omega },{\wedge }^{l-1}\right)$ such that ${\int }_{\mathrm{\Omega }}A\left(x,du\right)\cdot d\phi +B\left(x,du\right)\cdot \phi =0$ for all $\phi \in {W}_{\mathrm{loc}}^{1,p}\left(\mathrm{\Omega },{\wedge }^{l-1}\right)$ with compact support. Let $A:\mathrm{\Omega }×{\wedge }^{l}\left({\mathbb{R}}^{n}\right)\to {\wedge }^{l}\left({\mathbb{R}}^{n}\right)$ be defined by $A\left(x,\xi \right)=\xi {|\xi |}^{p-2}$ with $p>1$. Then A satisfies the required conditions and ${d}^{\star }A\left(x,du\right)=0$ becomes the p-harmonic equation
${d}^{\star }\left(du{|du|}^{p-2}\right)=0$
(3)

for differential forms. If u is a function (0-form), equation (3) reduces to the usual p-harmonic equation $div\left(\mathrm{\nabla }u{|\mathrm{\nabla }u|}^{p-2}\right)=0$ for functions. A remarkable progress has been made recently in the study of different versions of the harmonic equations, see  for more details.

Let ${C}^{\mathrm{\infty }}\left(\mathrm{\Omega },{\wedge }^{l}\right)$ be the space of smooth l-forms on Ω and
The harmonic l-fields are defined by
The orthogonal complement of in ${L}^{1}$ is defined by
Then Green’s operator G is defined as
$G:{C}^{\mathrm{\infty }}\left(\mathrm{\Omega },{\wedge }^{l}\right)\to {\mathcal{H}}^{\perp }\cap {C}^{\mathrm{\infty }}\left(\mathrm{\Omega },{\wedge }^{l}\right)$

by assigning $G\left(u\right)$ to be the unique element of ${\mathcal{H}}^{\perp }\cap {C}^{\mathrm{\infty }}\left(\mathrm{\Omega },{\wedge }^{l}\right)$ satisfying Poisson’s equation $\mathrm{\Delta }G\left(u\right)=u-H\left(u\right)$, where H is the harmonic projection operator that maps ${C}^{\mathrm{\infty }}\left(\mathrm{\Omega },{\wedge }^{l}\right)$ onto so that $H\left(u\right)$ 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 ${L}^{s}$-integrable form $u\left(y\right)$, we define the Hardy-Littlewood maximal operator ${\mathcal{M}}_{s}$ by
${\mathcal{M}}_{s}\left(u\right)={\mathcal{M}}_{s}\left(u\right)\left(x\right)=\underset{r>0}{sup}{\left(\frac{1}{|B\left(x,r\right)|}{\int }_{B\left(x,r\right)}{|u\left(y\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dy\right)}^{\frac{1}{s}},$
(4)
where $B\left(x,r\right)$ is the ball of radius r, centered at x, $1\le s<\mathrm{\infty }$. We write $\mathcal{M}\left(u\right)={\mathcal{M}}_{1}\left(u\right)$ if $s=1$. Similarly, for a locally ${L}^{s}$-integrable form $u\left(y\right)$, we define the sharp maximal operator ${\mathcal{M}}_{s}^{\mathrm{#}}$ by
${\mathcal{M}}_{s}^{\mathrm{#}}\left(u\right)={\mathcal{M}}_{s}^{\mathrm{#}}\left(u\right)\left(x\right)=\underset{r>0}{sup}{\left(\frac{1}{|B\left(x,r\right)|}{\int }_{B\left(x,r\right)}{|u\left(y\right)-{u}_{B\left(x,r\right)}|}^{s}\phantom{\rule{0.2em}{0ex}}dy\right)}^{\frac{1}{s}}.$
(5)

Some interesting results about these operators have been established, see [3, 4] and  for more details.

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 $w\left(x\right)$ satisfies the ${A}_{r}\left(\mathrm{\Omega }\right)$ condition, $r>1$, write $w\in {A}_{r}\left(\mathrm{\Omega }\right)$ if $w\left(x\right)>1$ a.e., and
$\underset{B}{sup}\left(\frac{1}{|B|}{\int }_{B}w\phantom{\rule{0.2em}{0ex}}dx\right){\left(\frac{1}{|B|}{\int }_{B}{\left(\frac{1}{w}\right)}^{\frac{1}{r-1}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{r-1}<\mathrm{\infty }$
(6)

for any ball $B\subset \mathrm{\Omega }$.

Definition 2 A proper subdomain $\mathrm{\Omega }\subset {\mathbb{R}}^{n}$ is called a δ-John domain, $\delta >0$, if there exists a point ${x}_{0}\in \mathrm{\Omega }$ which can be joined with any other point $x\in \mathrm{\Omega }$ by a continuous curve $\gamma \subset \mathrm{\Omega }$ so that
$d\left(\xi ,\partial \mathrm{\Omega }\right)\ge \delta |x-\xi |$

for each $\xi \in \gamma$. Here $d\left(\xi ,\partial \mathrm{\Omega }\right)$ is the Euclidean distance between ξ and Ω.

Lemma 1 

Each Ω has a modified Whitney cover of cubes $\mathcal{V}=\left\{{Q}_{i}\right\}$ such that
$\bigcup _{i}{Q}_{i}=\mathrm{\Omega },\phantom{\rule{2em}{0ex}}\sum _{{Q}_{i}\in \mathcal{V}}{\chi }_{\sqrt{\frac{5}{4}}{Q}_{i}}\le N{\chi }_{\mathrm{\Omega }}$

and some $N>1$, and if ${Q}_{i}\cap {Q}_{j}\ne \mathrm{\varnothing }$, then there exists a cube R (this cube need not be a member of ) in ${Q}_{i}\cap {Q}_{j}$ such that ${Q}_{i}\cup {Q}_{j}\subset NR$. Moreover, if Ω is δ-John, then there is a distinguished cube ${Q}_{0}\in \mathcal{V}$ which can be connected with every cube $Q\in \mathcal{V}$ by a chain of cubes ${Q}_{0}={Q}_{{j}_{0}},{Q}_{{j}_{1}},\dots ,{Q}_{{j}_{k}}=Q$ from and such that $Q\subset \rho {Q}_{{j}_{i}}$, $i=0,1,2,\dots ,k$, for some $\rho =\rho \left(n,\delta \right)$.

## 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 ${L}^{p}$ 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 ${\mathcal{M}}_{s}^{\mathrm{♯}}$ be the sharp maximal operator defined in (4) with $1. Then there exists a constant C, independent of u, such that
$\begin{array}{r}{\left({\int }_{B}{|{\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\mathcal{M}}_{s}^{\mathrm{♯}}{\left(G\left(u\right)\right)}_{B}|}^{q}\phantom{\rule{0.2em}{0ex}}d\mu \right)}^{1/q}\\ \phantom{\rule{1em}{0ex}}\le C\left(\delta ,\mathrm{\Omega }\right){|B|}^{1+\frac{1}{n}-\frac{1}{p}+\frac{1}{q}}{\left({\int }_{\sigma B}{|u|}^{p}\phantom{\rule{0.2em}{0ex}}d\mu \right)}^{1/p}\end{array}$

for all balls B with $\sigma B\subset \mathrm{\Omega }$, and a constant $\sigma >1$, where the measure μ is defined by $d\mu =w\left(x\right)dx$ and $w\left(x\right)\in {A}_{r}\left(\mathrm{\Omega }\right)$ with $w\ge \delta >0$ for some r >1 and a constant δ.

Theorem 1 Let $u\in {L}_{\mathrm{loc}}^{t}\left(\mathrm{\Omega },{\wedge }^{l}\right)$, $l=1,2,\dots ,n$, be a smooth differential form satisfying A-harmonic equation (1), let G be Green’s operator, and let ${\mathcal{M}}_{s}^{\mathrm{♯}}$ be the sharp maximal operator defined in (4) with $1. Then there exists a constant $C\left(n,t,{\delta }_{0},N,\mathrm{\Omega }\right)$, independent of u, such that
$\begin{array}{r}{\left({\int }_{\mathrm{\Omega }}{|{\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{0}}|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \right)}^{1/t}\\ \phantom{\rule{1em}{0ex}}\le C\left(n,t,{\delta }_{0},N,\mathrm{\Omega }\right){\left({\int }_{\mathrm{\Omega }}{|u|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \right)}^{1/t}\end{array}$
(7)

for any bounded and convex δ-John domain $\mathrm{\Omega }\subset {\mathbb{R}}^{n}$, where the fixed cube ${Q}_{0}\subset \mathrm{\Omega }$, the constant $N>1$ appeared in Lemma 1, and the measure μ is defined by $d\mu =w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$ and $w\left(x\right)\in {A}_{r}\left(\mathrm{\Omega }\right)$ with $w\ge {\delta }_{0}>0$ for some r >1 and a constant ${\delta }_{0}$.

Proof First, we use Lemma 1 for the bounded and convex δ-John domain Ω. There is a modified Whitney cover of cubes $\mathcal{V}=\left\{{Q}_{i}\right\}$ for Ω such that $\mathrm{\Omega }=\bigcup {Q}_{i}$, and ${\sum }_{{Q}_{i}\in \mathcal{V}}{\chi }_{\sqrt{\frac{5}{4}}{Q}_{i}}\le N{\chi }_{\mathrm{\Omega }}$ for some $N>1$. Moreover, there is a distinguished cube ${Q}_{0}\in \mathcal{V}$ which can be connected with every cube $Q\in \mathcal{V}$ by a chain of cubes ${Q}_{0}={Q}_{{j}_{0}},{Q}_{{j}_{1}},\dots ,{Q}_{{j}_{k}}=Q$ from and such that $Q\subset \rho {Q}_{{j}_{i}}$, $i=0,1,2,\dots ,k$, for some $\rho =\rho \left(n,\delta \right)$. Then, by the elementary inequality ${\left(a+b\right)}^{t}\le {2}^{t}\left({|a|}^{t}+{|b|}^{t}\right)$, $t\ge 0$, we have
$\begin{array}{r}{\left({\int }_{\mathrm{\Omega }}{|{\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{0}}|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \right)}^{1/t}\\ \phantom{\rule{1em}{0ex}}\le \left(\sum _{{Q}_{i}\in \mathcal{V}}\left({2}^{t}{\int }_{{Q}_{i}}{|{\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{i}}|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \\ {\phantom{\rule{2em}{0ex}}+{2}^{t}{\int }_{{Q}_{i}}{|{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{i}}-{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{0}}|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \right)\right)}^{1/t}\\ \phantom{\rule{1em}{0ex}}\le {C}_{1}\left(t\right)\left({\left(\sum _{{Q}_{i}\in \mathcal{V}}{\int }_{{Q}_{i}}{|{\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{i}}|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \right)}^{1/t}\\ \phantom{\rule{2em}{0ex}}+{\left(\sum _{{Q}_{i}\in \mathcal{V}}{\int }_{{Q}_{i}}{|{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{i}}-{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{0}}|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \right)}^{1/t}\right).\end{array}$
(8)
The first sum in (8) can be estimated by using Lemma 2.
$\begin{array}{r}\sum _{{Q}_{i}\in \mathcal{V}}{\int }_{{Q}_{i}}{|{\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{i}}|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}\le {C}_{2}\left(n,t,{\delta }_{0},\mathrm{\Omega }\right)\sum _{{Q}_{i}\in \mathcal{V}}{\int }_{{\rho }_{i}{Q}_{i}}{|u|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}\le {C}_{3}\left(n,t,{\delta }_{0},\mathrm{\Omega }\right)\sum _{{Q}_{i}\in \mathcal{V}}{\int }_{\mathrm{\Omega }}{|u|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}\le {C}_{4}\left(n,t,N,{\delta }_{0},\mathrm{\Omega }\right){\int }_{\mathrm{\Omega }}{|u|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu ,\end{array}$
(9)

where the measure μ is defined by $d\mu =w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$ and $w\left(x\right)\in {A}_{r}\left(\mathrm{\Omega }\right)$ with $w\ge {\delta }_{0}>0$ for some r >1 and a constant ${\delta }_{0}$.

To estimate the second sum in (8), we need to use the property of δ-John domain. Fix a cube ${Q}_{i}\in \mathcal{V}$ and let ${Q}_{0}={Q}_{{j}_{0}},{Q}_{{j}_{1}},\dots ,{Q}_{{j}_{k}}={Q}_{i}$ be the chain in Lemma 1. Then we have
$|{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{i}}-{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{0}}|\le \sum _{i=0}^{k-1}|{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{{j}_{i}}}-{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{{j}_{i+1}}}|.$
(10)
The chain $\left\{{Q}_{{j}_{i}}\right\}$ also has the property that for each i, $i=0,1,\dots ,k-1$, ${Q}_{{j}_{i}}\cap {Q}_{{j}_{i+1}}\ne \mathrm{\varnothing }$. Thus, there exists a cube ${D}_{i}$ such that ${D}_{i}\subset {Q}_{{j}_{i}}\cap {Q}_{{j}_{i+1}}$ and ${Q}_{{j}_{i}}\cup {Q}_{{j}_{i+1}}\subset N{D}_{i}$, $N>1$. So,
$\frac{max\left\{|{Q}_{{j}_{i}}|,|{Q}_{{j}_{i+1}}|\right\}}{|{Q}_{{j}_{i}}\cap {Q}_{{j}_{i+1}}|}\le \frac{max\left\{|{Q}_{{j}_{i}}|,|{Q}_{{j}_{i+1}}|\right\}}{|{D}_{i}|}\le N.$
(11)
Note that
$\mu \left(Q\right)={\int }_{Q}\phantom{\rule{0.2em}{0ex}}d\mu ={\int }_{Q}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\ge {\int }_{Q}{\delta }_{0}\phantom{\rule{0.2em}{0ex}}dx={\delta }_{0}|Q|.$
(12)
By (11), (12) and Lemma 2, we have
$\begin{array}{r}{|{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{{j}_{i}}}-{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{{j}_{i+1}}}|}^{t}\\ \phantom{\rule{1em}{0ex}}=\frac{1}{\mu \left({Q}_{{j}_{i}}\cap {Q}_{{j}_{i+1}}\right)}{\int }_{{Q}_{{j}_{i}}\cap {Q}_{{j}_{i+1}}}{|{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{{j}_{i}}}-{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{{j}_{i+1}}}|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{{\delta }_{0}|{Q}_{{j}_{i}}\cap {Q}_{{j}_{i+1}}|}{\int }_{{Q}_{{j}_{i}}\cap {Q}_{{j}_{i+1}}}{|{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{{j}_{i}}}-{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{{j}_{i+1}}}|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}\le \frac{N}{{\delta }_{0}max\left\{|{Q}_{{j}_{i}}|,|{Q}_{{j}_{i+1}}|\right\}}{\int }_{{Q}_{{j}_{i}}\cap {Q}_{{j}_{i+1}}}{|{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{{j}_{i}}}-{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{{j}_{i+1}}}|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}\le {C}_{5}\left(n,t,{\delta }_{0},N,\mathrm{\Omega }\right)\sum _{k=i}^{i+1}\frac{1}{|{Q}_{{j}_{k}}|}{\int }_{{Q}_{{j}_{k}}}{|{\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{{j}_{k}}}|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}\le {C}_{6}\left(n,t,{\delta }_{0},N,\mathrm{\Omega }\right)\sum _{k=i}^{i+1}\frac{{|{Q}_{{j}_{k}}|}^{1+\frac{1}{n}}}{|{Q}_{{j}_{k}}|}{\int }_{{\sigma }_{{j}_{k}}{Q}_{{j}_{k}}}{|u|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}={C}_{6}\left(n,t,{\delta }_{0},N,\mathrm{\Omega }\right)\sum _{k=i}^{i+1}{|{Q}_{{j}_{k}}|}^{\frac{1}{n}}{\int }_{{\sigma }_{{j}_{k}}{Q}_{{j}_{k}}}{|u|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}\le {C}_{7}\left(n,t,{\delta }_{0},N,\mathrm{\Omega }\right)\sum _{k=i}^{i+1}{|\mathrm{\Omega }|}^{\frac{1}{n}}{\int }_{\mathrm{\Omega }}{|u|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}\le {C}_{8}\left(n,t,{\delta }_{0},N,\mathrm{\Omega }\right)\sum _{{Q}_{i}\in \mathcal{V}}{\int }_{\mathrm{\Omega }}{|u|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}\le {C}_{9}\left(n,t,{\delta }_{0},N,\mathrm{\Omega }\right){\int }_{\mathrm{\Omega }}{|u|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu .\end{array}$
(13)
Then, by (10), (13) and the elementary inequality ${|{\sum }_{i=1}^{M}{t}_{i}|}^{s}\le {M}^{s-1}{\sum }_{i=1}^{M}{|{t}_{i}|}^{s}$, we finally obtain
$\begin{array}{r}\sum _{{Q}_{i}\in \mathcal{V}}{\int }_{{Q}_{i}}{|{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{i}}-{\left({\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)\right)}_{{Q}_{0}}|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}\le {C}_{10}\left(n,t,{\delta }_{0},N,\mathrm{\Omega }\right)\sum _{{Q}_{i}\in \mathcal{V}}{\int }_{{Q}_{i}}\left({\int }_{\mathrm{\Omega }}{|u|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \right)\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}={C}_{10}\left(n,t,{\delta }_{0},N,\mathrm{\Omega }\right)\left(\sum _{{Q}_{i}\in \mathcal{V}}{\int }_{{Q}_{i}}\phantom{\rule{0.2em}{0ex}}d\mu \right){\int }_{\mathrm{\Omega }}{|u|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}\le {C}_{11}\left(n,t,{\delta }_{0},N,\mathrm{\Omega }\right)\left({\int }_{\mathrm{\Omega }}\phantom{\rule{0.2em}{0ex}}d\mu \right){\int }_{\mathrm{\Omega }}{|u|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}={C}_{11}\left(n,t,{\delta }_{0},N,\mathrm{\Omega }\right)\mu \left(\mathrm{\Omega }\right){\int }_{\mathrm{\Omega }}{|u|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}={C}_{12}\left(n,t,{\delta }_{0},N,\mathrm{\Omega }\right){\int }_{\mathrm{\Omega }}{|u|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu .\end{array}$
(14)

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 $\left[0,\mathrm{\infty }\right)$ with $\phi \left(0\right)=0$, and let Λ be a domain with $\mu \left(\mathrm{\Lambda }\right)<\mathrm{\infty }$. If u is a measurable function in Λ, then we define the Orlicz norm of u by
${\parallel u\parallel }_{L\left(\phi ,\mathrm{\Lambda },\mu \right)}=inf\left\{k>0:\frac{1}{\mu \left(\mathrm{\Lambda }\right)}{\int }_{\mathrm{\Lambda }}\phi \left(\frac{|u\left(x\right)|}{k}\right)\phantom{\rule{0.2em}{0ex}}d\mu \le 1\right\}.$
(15)

A continuously increasing function $\psi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ with $\phi \left(0\right)=0$ 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 $G\left(p,q,C\right)$, $1\le p, $C\ge 1$, if (i) $1/C\le \phi \left({t}^{1/p}\right)/g\left(t\right)\le C$ and (ii) $1/C\le \phi \left({t}^{1/q}\right)/h\left(t\right)\le C$ for all $t>0$, where g is a convex increasing function and h is a concave increasing function on $\left[0,\mathrm{\infty }\right)$.

From  and , we know that the class $G\left(p,q,C\right)$ contains some very interesting functions, such as $\phi \left(t\right)={t}^{p}$ and $\phi \left(t\right)={t}^{p}{log}_{+}^{\alpha }\left(t\right)$, $p\ge 1$, $\alpha \in \mathbb{R}$, and each of φ, g and h is doubling in the sense that its values at t and 2t are uniformly comparable for all $t>0$, and the consequent fact that
${C}_{1}{t}^{q}\le {h}^{-1}\left(\phi \left(t\right)\right)\le {C}_{2}{t}^{q},\phantom{\rule{2em}{0ex}}{C}_{1}{t}^{p}\le {g}^{-1}\left(\phi \left(t\right)\right)\le {C}_{2}{t}^{p},$
(16)

where ${C}_{1}$ and ${C}_{2}$ 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 $G\left(p,q,{C}_{0}\right)$, $1\le p, ${C}_{0}\ge 1$, let $u\in {L}_{\mathrm{loc}}^{t}\left(\mathrm{\Omega },{\wedge }^{l}\right)$, $l=1,2,\dots ,n$, be a smooth differential form satisfying A-harmonic equation (1) in Ω, let G be Green’s operator, and let ${\mathcal{M}}_{s}^{\mathrm{♯}}$ be the sharp maximal operator defined in (4) with $1. Then there exists a constant C, independent of u, such that
${\parallel {\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\mathcal{M}}_{s}^{\mathrm{♯}}{\left(G\left(u\right)\right)}_{{Q}_{0}}\parallel }_{L\left(\phi ,\mathrm{\Omega },\mu \right)}\le C{\parallel u\parallel }_{L\left(\phi ,\mathrm{\Omega },\mu \right)}$

for any bounded and convex δ-John domain $\mathrm{\Omega }\subset {\mathbb{R}}^{n}$ with $\mu \left(\mathrm{\Omega }\right)<\mathrm{\infty }$, where the fixed cube ${Q}_{0}\subset \mathrm{\Omega }$ appeared in Lemma 1, and the measure μ is defined by $d\mu =w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$ and $w\left(x\right)\in {A}_{r}\left(\mathrm{\Omega }\right)$ with $w\ge {\delta }_{0}>0$ for some r >1 and a constant ${\delta }_{0}$.

Proof Let g, h be the functions in the $G\left(p,q,{C}_{0}\right)$ condition. Note that φ is an increasing function. Using Theorem 1, (i) in Definition 4, and Jensen’s inequality, we obtain
$\begin{array}{r}\phi \left(\frac{1}{k}{\left({\int }_{\mathrm{\Omega }}{|{\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\mathcal{M}}_{s}^{\mathrm{♯}}{\left(G\left(u\right)\right)}_{{Q}_{0}}|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \right)}^{1/t}\right)\\ \phantom{\rule{1em}{0ex}}\le \phi \left(\frac{1}{k}{C}_{1}{\left({\int }_{\mathrm{\Omega }}{|u|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \right)}^{1/t}\right)\\ \phantom{\rule{1em}{0ex}}=\phi \left({\left(\frac{1}{{k}^{t}}{C}_{1}^{t}{\int }_{\mathrm{\Omega }}{|u|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \right)}^{1/t}\right)\\ \phantom{\rule{1em}{0ex}}\le {C}_{0}g\left(\frac{1}{{k}^{t}}{C}_{1}^{t}{\int }_{\mathrm{\Omega }}{|u|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \right)\\ \phantom{\rule{1em}{0ex}}={C}_{0}g\left({\int }_{\mathrm{\Omega }}\frac{1}{{k}^{t}}{C}_{1}^{t}{|u|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \right)\\ \phantom{\rule{1em}{0ex}}\le {C}_{0}{\int }_{\mathrm{\Omega }}g\left(\frac{1}{{k}^{t}}{C}_{1}^{t}{|u|}^{t}\right)\phantom{\rule{0.2em}{0ex}}d\mu .\end{array}$
(17)
Again, from (i) in Definition 4, we have
$g\left(x\right)\le {C}_{0}\phi \left({x}^{\frac{1}{t}}\right).$
Thus, we obtain
${\int }_{\mathrm{\Omega }}g\left(\frac{1}{{k}^{t}}{C}_{1}^{t}{|u|}^{t}\right)\phantom{\rule{0.2em}{0ex}}d\mu \le {C}_{0}{\int }_{\mathrm{\Omega }}\phi \left(\frac{1}{k}{C}_{1}|u|\right)\phantom{\rule{0.2em}{0ex}}d\mu .$
(18)
Combining (17) and (18) yields
$\begin{array}{r}\phi \left(\frac{1}{k}{\left({\int }_{\mathrm{\Omega }}{|{\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\mathcal{M}}_{s}^{\mathrm{♯}}{\left(G\left(u\right)\right)}_{{Q}_{0}}|}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \right)}^{1/t}\right)\\ \phantom{\rule{1em}{0ex}}\le {C}_{0}^{2}{\int }_{\mathrm{\Omega }}\phi \left(\frac{1}{k}{C}_{1}|u|\right)\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}={C}_{2}{\int }_{\mathrm{\Omega }}\phi \left(\frac{1}{k}{C}_{1}|u|\right)\phantom{\rule{0.2em}{0ex}}d\mu .\end{array}$
(19)
Now, using Jensen’s inequality for ${h}^{-1}$, (16) and (ii) in Definition 4, and noticing that φ is doubling, we see
$\begin{array}{r}{\int }_{\mathrm{\Omega }}\phi \left(\frac{|{\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\mathcal{M}}_{s}^{\mathrm{♯}}{\left(G\left(u\right)\right)}_{{Q}_{0}}|}{k}\right)\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}=h\left({h}^{-1}\left({\int }_{\mathrm{\Omega }}\phi \left(\frac{|{\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\mathcal{M}}_{s}^{\mathrm{♯}}{\left(G\left(u\right)\right)}_{{Q}_{0}}|}{k}\right)\phantom{\rule{0.2em}{0ex}}d\mu \right)\right)\\ \phantom{\rule{1em}{0ex}}\le h\left({\int }_{\mathrm{\Omega }}{h}^{-1}\left(\phi \left(\frac{|{\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\mathcal{M}}_{s}^{\mathrm{♯}}{\left(G\left(u\right)\right)}_{{Q}_{0}}|}{k}\right)\right)\phantom{\rule{0.2em}{0ex}}d\mu \right)\\ \phantom{\rule{1em}{0ex}}\le h\left({C}_{3}{\int }_{\mathrm{\Omega }}{\left(\frac{|{\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\mathcal{M}}_{s}^{\mathrm{♯}}{\left(G\left(u\right)\right)}_{{Q}_{0}}|}{k}\right)}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \right)\\ \phantom{\rule{1em}{0ex}}\le {C}_{0}\phi \left({\left({C}_{3}{\int }_{\mathrm{\Omega }}{\left(\frac{|{\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\mathcal{M}}_{s}^{\mathrm{♯}}{\left(G\left(u\right)\right)}_{{Q}_{0}}|}{k}\right)}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \right)}^{\frac{1}{t}}\right)\\ \phantom{\rule{1em}{0ex}}={C}_{0}\phi \left(\frac{1}{k}{\left({C}_{3}{\int }_{\mathrm{\Omega }}{\left(|{\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\mathcal{M}}_{s}^{\mathrm{♯}}{\left(G\left(u\right)\right)}_{{Q}_{0}}|\right)}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \right)}^{\frac{1}{t}}\right)\\ \phantom{\rule{1em}{0ex}}\le {C}_{4}\phi \left(\frac{1}{k}{\left({\int }_{\mathrm{\Omega }}{\left(|{\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\mathcal{M}}_{s}^{\mathrm{♯}}{\left(G\left(u\right)\right)}_{{Q}_{0}}|\right)}^{t}\phantom{\rule{0.2em}{0ex}}d\mu \right)}^{\frac{1}{t}}\right).\end{array}$
(20)
Substituting (19) into (20) and using the fact that φ is doubling, we get
${\int }_{\mathrm{\Omega }}\phi \left(\frac{|{\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\mathcal{M}}_{s}^{\mathrm{♯}}{\left(G\left(u\right)\right)}_{{Q}_{0}}|}{k}\right)\phantom{\rule{0.2em}{0ex}}d\mu$
(21)
$\begin{array}{r}\phantom{\rule{1em}{0ex}}\le {C}_{5}{\int }_{\mathrm{\Omega }}\phi \left(\frac{1}{k}{C}_{1}|u|\right)\phantom{\rule{0.2em}{0ex}}d\mu \\ \phantom{\rule{1em}{0ex}}\le {C}_{6}{\int }_{\mathrm{\Omega }}\phi \left(\frac{1}{k}|u|\right)\phantom{\rule{0.2em}{0ex}}d\mu .\end{array}$
(22)
Therefore, from Definition 3, we have
${\parallel {\mathcal{M}}_{s}^{\mathrm{♯}}\left(G\left(u\right)\right)-{\mathcal{M}}_{s}^{\mathrm{♯}}{\left(G\left(u\right)\right)}_{{Q}_{0}}\parallel }_{L\left(\phi ,\mathrm{\Omega },\mu \right)}\le {C}_{6}{\parallel u\parallel }_{L\left(\phi ,\mathrm{\Omega },\mu \right)}.$

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## Declarations

### Acknowledgements

The first author was supported by NSF of P.R. China (No. 11071048).

## Authors’ Affiliations

(1)
Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, P.R. China
(2)
Department of Mathematical Science, Delaware State University, Dover, 19901, USA

## References 