Global Poincaré inequalities for the composition of the sharp maximal operator and Green’s operator with Orlicz norms

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.

1 Introduction

The $L^p$-theory of solutions of the homogeneous A-harmonic equation $d^{\star }A(x,du)=0$ 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 $d^{\star }A(x,du)=B(x,du)$ has just begun [2–4]. In this paper, we focus on a class of differential forms satisfying the well-known nonhomogeneous A-harmonic equation $d^{\star }A(x,du)=B(x,du)$.

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 $diam(\sigma B)=\sigma diam(B)$, $\sigma >0$. 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 $w\in L_{\mathrm{loc}}^1({\mathbb{R}}^n)$ and $w>0$ a.e. For a function u, we denote the average of u over B by

$$u_B=\frac{1}{|B|}{\int }_{B}udx,$$

where $|B|$ is the volume of B and the μ-average of u over B by

$$u_{B,\mu }=\frac{1}{\mu (B)}{\int }_{B}ud\mu .$$

Let ${\wedge }^l={\wedge }^l({\mathbb{R}}^n)$ be the set of all l-forms in ${\mathbb{R}}^n$, let $D^{\prime }(\mathrm{\Omega },{\wedge }^l)$ be the space of all differential l-forms on Ω, and let $L^p(\mathrm{\Omega },{\wedge }^l)$ be the l-forms $u(x)={\sum }_I{u}_I(x)d{x}_I$ on Ω satisfying ${\int }_{\mathrm{\Omega }}{|u_I|}^{p}dx<\mathrm{\infty }$ for all ordered l-tuples I, $l=1,2,\dots ,n$. We denote the exterior derivative by $d:D^{\prime }(\mathrm{\Omega },{\wedge }^l)\to D^{\prime }(\mathrm{\Omega },{\wedge }^{l+1})$ for $l=0,1,\dots ,n-1$, and define the Hodge star operator $\star :{\wedge }^k\to {\wedge }^{n-k}$ as follows. If $u=u_{I}d{x}_I$, $i_1<i_2<\cdots <i_k$, is a differential k-form, then $\star u={(-1)}^{\sum (I)}{u}_{I}d{x}_J$, where $I=(i_1,i_2,\dots ,i_k)$, $J=\{1,2,\dots ,n\}-I$, and $\sum (I)=\frac{k(k+1)}{2}+{\sum }_{j=1}^k{i}_j$. The Hodge codifferential operator

$$d^{\star }:D^{\prime }(\mathrm{\Omega },{\wedge }^{l+1})\to D^{\prime }(\mathrm{\Omega },{\wedge }^l)$$

is given by $d^{\star }={(-1)}^{nl+1}\star d\star$ on $D^{\prime }(\mathrm{\Omega },{\wedge }^{l+1})$, $l=0,1,\dots ,n-1$. We write

$${\parallel u\parallel }_{s,\mathrm{\Omega }}={\left({\int }_{\mathrm{\Omega }}{|u|}^{s}dx\right)}^{1/s}.$$

The well-known nonhomogeneous A-harmonic equation is

$$d^{\star }A(x,du)=B(x,du),$$

(1)

where $A:\mathrm{\Omega }\times {\wedge }^l({\mathbb{R}}^n)\to {\wedge }^l({\mathbb{R}}^n)$ and $B:\mathrm{\Omega }\times {\wedge }^l({\mathbb{R}}^n)\to {\wedge }^{l-1}({\mathbb{R}}^n)$ satisfy the conditions:

$$|A(x,\xi )|\le a{|\xi |}^{p-1},A(x,\xi )\cdot \xi \ge {|\xi |}^p,|B(x,\xi )|\le b{|\xi |}^{p-1}$$

(2)

for almost every $x\in \mathrm{\Omega }$ and all $\xi \in {\wedge }^l({\mathbb{R}}^n)$. Here, $a,b>0$ are constants and $1<p<\mathrm{\infty }$ is a fixed exponent associated with (1). If the operator $B=0$, equation (1) becomes $d^{\star }A(x,du)=0$, which is called the (homogeneous) A-harmonic equation. A solution to (1) is an element of the Sobolev space $W_{\mathrm{loc}}^{1,p}(\mathrm{\Omega },{\wedge }^{l-1})$ such that ${\int }_{\mathrm{\Omega }}A(x,du)\cdot d\phi +B(x,du)\cdot \phi =0$ for all $\phi \in W_{\mathrm{loc}}^{1,p}(\mathrm{\Omega },{\wedge }^{l-1})$ with compact support. Let $A:\mathrm{\Omega }\times {\wedge }^l({\mathbb{R}}^n)\to {\wedge }^l({\mathbb{R}}^n)$ be defined by $A(x,\xi )=\xi {|\xi |}^{p-2}$ with $p>1$. Then A satisfies the required conditions and $d^{\star }A(x,du)=0$ 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 $div(\mathrm{\nabla }u{|\mathrm{\nabla }u|}^{p-2})=0$ for functions. A remarkable progress has been made recently in the study of different versions of the harmonic equations, see [1] for more details.

Let $C^{\mathrm{\infty }}(\mathrm{\Omega },{\wedge }^l)$ be the space of smooth l-forms on Ω and

$$\mathcal{W}(\mathrm{\Omega },{\wedge }^l)=\{u\in L_{\mathrm{loc}}^1(\mathrm{\Omega },{\wedge }^l):u\text{ has generalized gradient}\}.$$

The harmonic l-fields are defined by

$$\mathcal{H}(\mathrm{\Omega },{\wedge }^l)=\{u\in \mathcal{W}(\mathrm{\Omega },{\wedge }^l):du=d^{\star }u=0,u\in L^p\text{ for some }1<p<\mathrm{\infty }\}.$$

The orthogonal complement of ℋ in $L^1$ is defined by

$${\mathcal{H}}^{\perp }=\{u\in L^1:〈u,h〉=0\text{ for all }h\in \mathcal{H}\}.$$

Then Green’s operator G is defined as

$$G:C^{\mathrm{\infty }}(\mathrm{\Omega },{\wedge }^l)\to {\mathcal{H}}^{\perp }\cap C^{\mathrm{\infty }}(\mathrm{\Omega },{\wedge }^l)$$

by assigning $G(u)$ to be the unique element of ${\mathcal{H}}^{\perp }\cap C^{\mathrm{\infty }}(\mathrm{\Omega },{\wedge }^l)$ satisfying Poisson’s equation $\mathrm{\Delta }G(u)=u-H(u)$, where H is the harmonic projection operator that maps $C^{\mathrm{\infty }}(\mathrm{\Omega },{\wedge }^l)$ onto ℋ so that $H(u)$ is the harmonic part of u. See [5] 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 $u(y)$, we define the Hardy-Littlewood maximal operator ${\mathcal{M}}_s$ by

$${\mathcal{M}}_s(u)={\mathcal{M}}_s(u)(x)=\underset{r>0}{sup}{\left(\frac{1}{|B(x,r)|}{\int }_{B(x,r)}{|u(y)|}^{s}dy\right)}^{\frac{1}{s}},$$

(4)

where $B(x,r)$ is the ball of radius r, centered at x, $1\le s<\mathrm{\infty }$. We write $\mathcal{M}(u)={\mathcal{M}}_1(u)$ if $s=1$. Similarly, for a locally $L^s$-integrable form $u(y)$, we define the sharp maximal operator ${\mathcal{M}}_s^{\mathrm{\#}}$ by

$${\mathcal{M}}_s^{\mathrm{\#}}(u)={\mathcal{M}}_s^{\mathrm{\#}}(u)(x)=\underset{r>0}{sup}{\left(\frac{1}{|B(x,r)|}{\int }_{B(x,r)}{|u(y)-u_{B(x,r)}|}^{s}dy\right)}^{\frac{1}{s}}.$$

(5)

Some interesting results about these operators have been established, see [3, 4] and [6] 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 $w(x)$ satisfies the $A_r(\mathrm{\Omega })$ condition, $r>1$, write $w\in A_r(\mathrm{\Omega })$ if $w(x)>1$ a.e., and

$$\underset{B}{sup}\left(\frac{1}{|B|}{\int }_{B}wdx\right){\left(\frac{1}{|B|}{\int }_B{\left(\frac{1}{w}\right)}^{\frac{1}{r-1}}dx\right)}^{r-1}<\mathrm{\infty }$$

(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

$$d(\xi ,\partial \mathrm{\Omega })\ge \delta |x-\xi |$$

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

Lemma 1 [7]

Each Ω has a modified Whitney cover of cubes $\mathcal{V}=\{Q_i\}$ such that

$$\bigcup _i{Q}_i=\mathrm{\Omega },\sum _{Q_i\in \mathcal{V}}{\chi }_{\sqrt{\frac{5}{4}}{Q}_i}\le N{\chi }_{\mathrm{\Omega }}$$

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 $\rho =\rho (n,\delta )$.

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 [4] 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 $1<s\le p,q<\mathrm{\infty }$. Then there exists a constant C, independent of u, such that

$$\begin{array}{r}{\left({\int }_B{|{\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\mathcal{M}}_s^{\mathrm{♯}}{(G(u))}_B|}^{q}d\mu \right)}^{1/q}\\ \le C(\delta ,\mathrm{\Omega }){|B|}^{1+\frac{1}{n}-\frac{1}{p}+\frac{1}{q}}{\left({\int }_{\sigma B}{|u|}^{p}d\mu \right)}^{1/p}\end{array}$$

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

Theorem 1 Let $u\in L_{\mathrm{loc}}^t(\mathrm{\Omega },{\wedge }^l)$, $l=1,2,\dots ,n$, 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 $1<s<t<\mathrm{\infty }$. Then there exists a constant $C(n,t,{\delta }_0,N,\mathrm{\Omega })$, independent of u, such that

$$\begin{array}{r}{\left({\int }_{\mathrm{\Omega }}{|{\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_0}|}^{t}d\mu \right)}^{1/t}\\ \le C(n,t,{\delta }_0,N,\mathrm{\Omega }){\left({\int }_{\mathrm{\Omega }}{|u|}^{t}d\mu \right)}^{1/t}\end{array}$$

(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 $d\mu =w(x)dx$ and $w(x)\in A_r(\mathrm{\Omega })$ 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 $\mathcal{V}=\{Q_i\}$ 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 $\rho =\rho (n,\delta )$. Then, by the elementary inequality ${(a+b)}^t\le 2^t({|a|}^t+{|b|}^t)$, $t\ge 0$, we have

$$\begin{array}{r}{\left({\int }_{\mathrm{\Omega }}{|{\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_0}|}^{t}d\mu \right)}^{1/t}\\ \le (\sum _{Q_i\in \mathcal{V}}(2^t{\int }_{Q_i}{|{\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_i}|}^{t}d\mu \\ {+2^t{\int }_{Q_i}{|{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_i}-{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_0}|}^{t}d\mu ))}^{1/t}\\ \le C_1(t)({\left(\sum _{Q_i\in \mathcal{V}}{\int }_{Q_i}{|{\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_i}|}^{t}d\mu \right)}^{1/t}\\ +{\left(\sum _{Q_i\in \mathcal{V}}{\int }_{Q_i}{|{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_i}-{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_0}|}^{t}d\mu \right)}^{1/t}).\end{array}$$

(8)

The first sum in (8) can be estimated by using Lemma 2.

$$\begin{array}{r}\sum _{Q_i\in \mathcal{V}}{\int }_{Q_i}{|{\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_i}|}^{t}d\mu \\ \le C_2(n,t,{\delta }_0,\mathrm{\Omega })\sum _{Q_i\in \mathcal{V}}{\int }_{{\rho }_i{Q}_i}{|u|}^{t}d\mu \\ \le C_3(n,t,{\delta }_0,\mathrm{\Omega })\sum _{Q_i\in \mathcal{V}}{\int }_{\mathrm{\Omega }}{|u|}^{t}d\mu \\ \le C_4(n,t,N,{\delta }_0,\mathrm{\Omega }){\int }_{\mathrm{\Omega }}{|u|}^{t}d\mu ,\end{array}$$

(9)

where the measure μ is defined by $d\mu =w(x)dx$ and $w(x)\in A_r(\mathrm{\Omega })$ 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

$$|{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_i}-{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_0}|\le \sum _{i=0}^{k-1}|{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_{j_i}}-{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_{j_{i+1}}}|.$$

(10)

The chain $\{Q_{j_i}\}$ 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,

$$\frac{max\{|Q_{j_i}|,|Q_{j_{i+1}}|\}}{|Q_{j_i}\cap Q_{j_{i+1}}|}\le \frac{max\{|Q_{j_i}|,|Q_{j_{i+1}}|\}}{|D_i|}\le N.$$

(11)

Note that

$$\mu (Q)={\int }_{Q}d\mu ={\int }_{Q}w(x)dx\ge {\int }_Q{\delta }_{0}dx={\delta }_0|Q|.$$

(12)

By (11), (12) and Lemma 2, we have

$$\begin{array}{r}{|{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_{j_i}}-{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_{j_{i+1}}}|}^t\\ =\frac{1}{\mu (Q_{j_i}\cap Q_{j_{i+1}})}{\int }_{Q_{j_i}\cap Q_{j_{i+1}}}{|{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_{j_i}}-{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_{j_{i+1}}}|}^{t}d\mu \\ \le \frac{1}{{\delta }_0|Q_{j_i}\cap Q_{j_{i+1}}|}{\int }_{Q_{j_i}\cap Q_{j_{i+1}}}{|{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_{j_i}}-{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_{j_{i+1}}}|}^{t}d\mu \\ \le \frac{N}{{\delta }_{0}max\{|Q_{j_i}|,|Q_{j_{i+1}}|\}}{\int }_{Q_{j_i}\cap Q_{j_{i+1}}}{|{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_{j_i}}-{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_{j_{i+1}}}|}^{t}d\mu \\ \le C_5(n,t,{\delta }_0,N,\mathrm{\Omega })\sum _{k=i}^{i+1}\frac{1}{|Q_{j_k}|}{\int }_{Q_{j_k}}{|{\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_{j_k}}|}^{t}d\mu \\ \le C_6(n,t,{\delta }_0,N,\mathrm{\Omega })\sum _{k=i}^{i+1}\frac{{|Q_{j_k}|}^{1+\frac{1}{n}}}{|Q_{j_k}|}{\int }_{{\sigma }_{j_k}{Q}_{j_k}}{|u|}^{t}d\mu \\ =C_6(n,t,{\delta }_0,N,\mathrm{\Omega })\sum _{k=i}^{i+1}{|Q_{j_k}|}^{\frac{1}{n}}{\int }_{{\sigma }_{j_k}{Q}_{j_k}}{|u|}^{t}d\mu \\ \le C_7(n,t,{\delta }_0,N,\mathrm{\Omega })\sum _{k=i}^{i+1}{|\mathrm{\Omega }|}^{\frac{1}{n}}{\int }_{\mathrm{\Omega }}{|u|}^{t}d\mu \\ \le C_8(n,t,{\delta }_0,N,\mathrm{\Omega })\sum _{Q_i\in \mathcal{V}}{\int }_{\mathrm{\Omega }}{|u|}^{t}d\mu \\ \le C_9(n,t,{\delta }_0,N,\mathrm{\Omega }){\int }_{\mathrm{\Omega }}{|u|}^{t}d\mu .\end{array}$$

(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

$$\begin{array}{r}\sum _{Q_i\in \mathcal{V}}{\int }_{Q_i}{|{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_i}-{\left({\mathcal{M}}_s^{\mathrm{♯}}(G(u))\right)}_{Q_0}|}^{t}d\mu \\ \le C_{10}(n,t,{\delta }_0,N,\mathrm{\Omega })\sum _{Q_i\in \mathcal{V}}{\int }_{Q_i}\left({\int }_{\mathrm{\Omega }}{|u|}^{t}d\mu \right)d\mu \\ =C_{10}(n,t,{\delta }_0,N,\mathrm{\Omega })\left(\sum _{Q_i\in \mathcal{V}}{\int }_{Q_i}d\mu \right){\int }_{\mathrm{\Omega }}{|u|}^{t}d\mu \\ \le C_{11}(n,t,{\delta }_0,N,\mathrm{\Omega })\left({\int }_{\mathrm{\Omega }}d\mu \right){\int }_{\mathrm{\Omega }}{|u|}^{t}d\mu \\ =C_{11}(n,t,{\delta }_0,N,\mathrm{\Omega })\mu (\mathrm{\Omega }){\int }_{\mathrm{\Omega }}{|u|}^{t}d\mu \\ =C_{12}(n,t,{\delta }_0,N,\mathrm{\Omega }){\int }_{\mathrm{\Omega }}{|u|}^{t}d\mu .\end{array}$$

(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 $[0,\mathrm{\infty })$ with $\phi (0)=0$, and let Λ be a domain with $\mu (\mathrm{\Lambda })<\mathrm{\infty }$. If u is a measurable function in Λ, then we define the Orlicz norm of u by

$${\parallel u\parallel }_{L(\phi ,\mathrm{\Lambda },\mu )}=inf\{k>0:\frac{1}{\mu (\mathrm{\Lambda })}{\int }_{\mathrm{\Lambda }}\phi \left(\frac{|u(x)|}{k}\right)d\mu \le 1\}.$$

(15)

A continuously increasing function $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $\phi (0)=0$ is called an Orlicz function. A convex Orlicz function φ is often called a Young function.

In [8], Buckley and Koskela gave the following class of functions.

Definition 4 We say a Young function φ lies in the class $G(p,q,C)$, $1\le p<q<\mathrm{\infty }$, $C\ge 1$, if (i) $1/C\le \phi (t^{1/p})/g(t)\le C$ and (ii) $1/C\le \phi (t^{1/q})/h(t)\le C$ for all $t>0$, where g is a convex increasing function and h is a concave increasing function on $[0,\mathrm{\infty })$.

From [8] and [9], we know that the class $G(p,q,C)$ contains some very interesting functions, such as $\phi (t)=t^p$ and $\phi (t)=t^p{log}_{+}^{\alpha }(t)$, $p\ge 1$, $\alpha \in \mathbb{R}$, 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

$$C_1{t}^q\le h^{-1}(\phi (t))\le C_2{t}^q,C_1{t}^p\le g^{-1}(\phi (t))\le C_2{t}^p,$$

(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 $G(p,q,C_0)$, $1\le p<q<\mathrm{\infty }$, $C_0\ge 1$, let $u\in L_{\mathrm{loc}}^t(\mathrm{\Omega },{\wedge }^l)$, $l=1,2,\dots ,n$, 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 $1<s\le t<\mathrm{\infty }$. Then there exists a constant C, independent of u, such that

$${\parallel {\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\mathcal{M}}_s^{\mathrm{♯}}{(G(u))}_{Q_0}\parallel }_{L(\phi ,\mathrm{\Omega },\mu )}\le C{\parallel u\parallel }_{L(\phi ,\mathrm{\Omega },\mu )}$$

for any bounded and convex δ-John domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with $\mu (\mathrm{\Omega })<\mathrm{\infty }$, where the fixed cube $Q_0\subset \mathrm{\Omega }$ appeared in Lemma 1, and the measure μ is defined by $d\mu =w(x)dx$ and $w(x)\in A_r(\mathrm{\Omega })$ with $w\ge {\delta }_0>0$ for some r >1 and a constant ${\delta }_0$.

Proof Let g, h be the functions in the $G(p,q,C_0)$ condition. Note that φ is an increasing function. Using Theorem 1, (i) in Definition 4, and Jensen’s inequality, we obtain

$$\begin{array}{r}\phi \left(\frac{1}{k}{\left({\int }_{\mathrm{\Omega }}{|{\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\mathcal{M}}_s^{\mathrm{♯}}{(G(u))}_{Q_0}|}^{t}d\mu \right)}^{1/t}\right)\\ \le \phi \left(\frac{1}{k}{C}_1{\left({\int }_{\mathrm{\Omega }}{|u|}^{t}d\mu \right)}^{1/t}\right)\\ =\phi \left({\left(\frac{1}{k^t}{C}_1^t{\int }_{\mathrm{\Omega }}{|u|}^{t}d\mu \right)}^{1/t}\right)\\ \le C_{0}g\left(\frac{1}{k^t}{C}_1^t{\int }_{\mathrm{\Omega }}{|u|}^{t}d\mu \right)\\ =C_{0}g\left({\int }_{\mathrm{\Omega }}\frac{1}{k^t}{C}_1^t{|u|}^{t}d\mu \right)\\ \le C_0{\int }_{\mathrm{\Omega }}g\left(\frac{1}{k^t}{C}_1^t{|u|}^t\right)d\mu .\end{array}$$

(17)

Again, from (i) in Definition 4, we have

$$g(x)\le C_0\phi \left(x^{\frac{1}{t}}\right).$$

Thus, we obtain

$${\int }_{\mathrm{\Omega }}g\left(\frac{1}{k^t}{C}_1^t{|u|}^t\right)d\mu \le C_0{\int }_{\mathrm{\Omega }}\phi (\frac{1}{k}{C}_1|u|)d\mu .$$

(18)

Combining (17) and (18) yields

$$\begin{array}{r}\phi \left(\frac{1}{k}{\left({\int }_{\mathrm{\Omega }}{|{\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\mathcal{M}}_s^{\mathrm{♯}}{(G(u))}_{Q_0}|}^{t}d\mu \right)}^{1/t}\right)\\ \le C_0^2{\int }_{\mathrm{\Omega }}\phi (\frac{1}{k}{C}_1|u|)d\mu \\ =C_2{\int }_{\mathrm{\Omega }}\phi (\frac{1}{k}{C}_1|u|)d\mu .\end{array}$$

(19)

Now, using Jensen’s inequality for $h^{-1}$, (16) and (ii) in Definition 4, and noticing that φ is doubling, we see

$$\begin{array}{r}{\int }_{\mathrm{\Omega }}\phi \left(\frac{|{\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\mathcal{M}}_s^{\mathrm{♯}}{(G(u))}_{Q_0}|}{k}\right)d\mu \\ =h\left(h^{-1}\left({\int }_{\mathrm{\Omega }}\phi \left(\frac{|{\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\mathcal{M}}_s^{\mathrm{♯}}{(G(u))}_{Q_0}|}{k}\right)d\mu \right)\right)\\ \le h\left({\int }_{\mathrm{\Omega }}{h}^{-1}\left(\phi \left(\frac{|{\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\mathcal{M}}_s^{\mathrm{♯}}{(G(u))}_{Q_0}|}{k}\right)\right)d\mu \right)\\ \le h\left(C_3{\int }_{\mathrm{\Omega }}{\left(\frac{|{\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\mathcal{M}}_s^{\mathrm{♯}}{(G(u))}_{Q_0}|}{k}\right)}^{t}d\mu \right)\\ \le C_0\phi \left({\left(C_3{\int }_{\mathrm{\Omega }}{\left(\frac{|{\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\mathcal{M}}_s^{\mathrm{♯}}{(G(u))}_{Q_0}|}{k}\right)}^{t}d\mu \right)}^{\frac{1}{t}}\right)\\ =C_0\phi \left(\frac{1}{k}{\left(C_3{\int }_{\mathrm{\Omega }}{\left(|{\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\mathcal{M}}_s^{\mathrm{♯}}{(G(u))}_{Q_0}|\right)}^{t}d\mu \right)}^{\frac{1}{t}}\right)\\ \le C_4\phi \left(\frac{1}{k}{\left({\int }_{\mathrm{\Omega }}{\left(|{\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\mathcal{M}}_s^{\mathrm{♯}}{(G(u))}_{Q_0}|\right)}^{t}d\mu \right)}^{\frac{1}{t}}\right).\end{array}$$

(20)

Substituting (19) into (20) and using the fact that φ is doubling, we get

$${\int }_{\mathrm{\Omega }}\phi \left(\frac{|{\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\mathcal{M}}_s^{\mathrm{♯}}{(G(u))}_{Q_0}|}{k}\right)d\mu$$

(21)

$$\begin{array}{r}\le C_5{\int }_{\mathrm{\Omega }}\phi (\frac{1}{k}{C}_1|u|)d\mu \\ \le C_6{\int }_{\mathrm{\Omega }}\phi (\frac{1}{k}|u|)d\mu .\end{array}$$

(22)

Therefore, from Definition 3, we have

$${\parallel {\mathcal{M}}_s^{\mathrm{♯}}(G(u))-{\mathcal{M}}_s^{\mathrm{♯}}{(G(u))}_{Q_0}\parallel }_{L(\phi ,\mathrm{\Omega },\mu )}\le C_6{\parallel u\parallel }_{L(\phi ,\mathrm{\Omega },\mu )}.$$

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