In this section, we establish the weighted inequalities for the composite operator *T* ○ *H* in terms of Orlicz norms. To state our results, we need some definitions and lemmas.

We call a continuously increasing function Φ : [0, ∞) → [0, ∞) with Φ(0) = 0 an Orlicz function. If the Orlicz function Φ is convex, then Φ is often called a Young function. The Orlicz space

*L*^{Φ}(

*E*) consists of all measurable functions

*f* on

*E* such that ∫

_{
E
}Φ(

*|f |/λ*)

*dx <* ∞ for some

*λ* =

*λ*(

*f*)

*>* 0 with the nonlinear Luxemburg functional

$\parallel f{\parallel}_{\Phi ,E}=inf\left\{\lambda >0:{\int}_{E}\Phi \left(\frac{|f|}{\lambda}\right)dx\le 1\right\}.$

(2.1)

Moreover, if Φ is a restrictively increasing Young function, then *L*^{Φ}(*E*) is a Banach space and the corresponding norm || · || _{Φ,E}is called Luxemburg norm or Orlicz Norm. The following definition appears in [10].

**Definition 2.1**. *We say that an Orlicz function* Φ *lies in the class G*(*p*, *q*, *C*), 1 ≤ *p < q <* ∞ *and C* ≥ 1, *if (1)* 1/*C* ≤ Φ(*t*^{1/p})/*g*(*t*) ≤ *C and (2)* 1/*C* ≤ Φ(*t*^{1/q})/*h*(*t*) ≤ *C for all t* > 0, *where g*(*t*) *is a convex increasing function and h*(*t*) *is a concave increasing function on* [0, ∞).

We note from [

10] that each of Φ,

*g*, and

*h* mentioned in Definition 2.1 is doubling, from which it is easy to know that

${C}_{1}{t}^{q}\le {h}^{-1}\left(\Phi \left(t\right)\right)\le {C}_{2}{t}^{q},\phantom{\rule{2.77695pt}{0ex}}{C}_{1}{t}^{p}\le {g}^{-1}\left(\Phi \left(t\right)\right)\le {C}_{2}{t}^{p}$

(2.2)

for all *t* > 0, where *C*_{1} and *C*_{2} are constants.

We also need the following lemma which appears in [1].

**Lemma 2.2**.

*Let* $u\in {L}_{loc}^{s}\left(D,{\wedge}^{k}\right)$,

*k* = 1, 2,...,

*n*, 1

*< s <* ∞,

*be a smooth solution of the nonhomogeneous A-harmonic equation in a bounded convex domain D, H be the projection operator and T* :

*C*^{∞}(∧

^{
k
}*D*) →

*C*^{∞}(∧

^{k-1}*D*)

*be the homotopy operator. Then there exists a constant C, independent of u, such that*$\parallel T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}{\parallel}_{s,B}\le C\mathit{diam}\left(B\right)\parallel u{\parallel}_{s,\rho B}$

*for all balls B with ρB* ⊂ *D, where ρ* > 1 *is a constant*.

The *A*_{
r
} weights, *r* > 1, were first introduced by Muckenhoupt [11] and play a crucial role in weighted norm inequalities for many operators. As an extension of *A*_{
r
} weights, the following class was introduced in [2].

**Definition 2.3**.

*We call that a measurable function w*(

*x*)

*defined on a subset E* ⊂ ℝ

^{
n
} *satisfies the A*(

*α*,

*β*,

*γ*;

*E*)-

*condition for some positive constants α*,

*β*,

*γ*;

*write w*(

*x*) ∈

*A*(

*α*,

*β*,

*γ*;

*E*),

*if w*(

*x*)

*>* 0

*a.e. and*$\underset{B}{sup}\phantom{\rule{2.77695pt}{0ex}}\left(\frac{1}{|B|}{\int}_{B}{w}^{\alpha}dx\right){\left(\frac{1}{|B|}{{\int}_{B}\left(\frac{1}{w}\right)}^{\beta}dx\right)}^{\gamma \u2215\beta}={c}_{\alpha ,\beta ,\gamma}<\infty ,$

*where the supremum is over all balls B* ⊂ *E*.

We also need the following reverse Hölder inequality for the solutions of the nonhomogeneous A-harmonic equation, which appears in [3].

**Lemma 2.4**.

*Let u be a solution of the nonhomogeneous A-harmonic equation, σ* > 1

*and* 0

*< s*,

*t <* ∞.

*Then there exists a constant C, independent of u and B, such that*$\parallel u{\parallel}_{s,B}\le C|B{|}^{\left(t-s\right)\u2215st}\parallel u{\parallel}_{t,\sigma B}$

*for all balls B with σB* ⊂ *E*.

**Theorem 2.5**.

*Assume that u is a smooth solution of the nonhomogeneous A-harmonic equation in a bounded convex domain D*, 1

*< p*,

*q <* ∞

*and* $w\left(x\right)\in A\left(\alpha ,\beta ,\frac{\alpha q}{p};D\right)$ *for some α* > 1

*and β* > 0.

*Let H be the projection operator and T* :

*C*^{∞}(∧

^{
k
}*D*) →

*C*^{∞} (∧

^{k-1}*D*),

*k* = 1, 2,...,

*n*,

*be the homotopy operator. Then there exists a constant C, independent of u, such that*${\left({\int}_{B}|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}{|}^{q}w\left(x\right)dx\right)}^{1\u2215q}\le C\mathit{diam}\left(B\right)|B{|}^{\left(p-q\right)\u2215pq}{\left({{\int}_{\sigma B}|u|}^{p}w\left(x\right)dx\right)}^{1\u2215p}$

*for all balls with σB* ⊂ *D for some σ* > 1.

**Proof**. Set

*s* =

*αq* and

*m* =

*βp/*(

*β* + 1). From Lemma 2.2 and the reverse Hölder inequality, we have

$\begin{array}{c}{\left({\int}_{B}|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}{|}^{q}w\left(x\right)dx\right)}^{1\u2215q}\\ \le {\left({\int}_{B}|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}{|}^{\frac{qs}{s-q}}dx\right)}^{\frac{s-q}{sq}}{\left({\int}_{B}{\left(w\left(x\right)\right)}^{\alpha}dx\right)}^{\frac{1}{\alpha q}}\\ \le {C}_{1}\mathit{diam}\left(B\right)|B{|}^{\frac{1}{q}-\frac{1}{s}-\frac{1}{m}}{\left({\int}_{\sigma B}|u{|}^{m}dx\right)}^{1\u2215m}{\left({\int}_{B}{\left(w\left(x\right)\right)}^{\alpha}dx\right)}^{1\u2215\alpha q}.\end{array}$

(2.3)

Let

$n=\frac{pm}{p-m}$, then

$\frac{1}{p}+\frac{1}{n}=\frac{1}{m}$. Thus, using the Hölder inequality, we obtain

$\begin{array}{c}{\left({\int}_{\sigma B}|u{|}^{m}dx\right)}^{1\u2215m}\\ ={\left({\int}_{\sigma B}|u{|}^{m}{\left({w}^{\frac{1}{p}}\cdot {w}^{\frac{-1}{p}}\right)}^{m}dx\right)}^{1\u2215m}\\ \le {\left({\int}_{\sigma B}|u{|}^{p}w\left(x\right)dx\right)}^{1\u2215p}{\left({\int}_{\sigma B}{w}^{\frac{-n}{p}}dx\right)}^{\frac{1}{n}}.\end{array}$

(2.4)

Note that

$w\left(x\right)\in A\left(\alpha ,\beta ,\frac{\alpha q}{p};D\right)$. It is easy to find that

$\begin{array}{c}{\left({\int}_{B}{\left(w\left(x\right)\right)}^{\alpha}dx\right)}^{1\u2215\alpha q}{\left({{\int}_{\sigma B}w}^{\frac{-n}{p}}dx\right)}^{\frac{1}{n}}\\ ={\left({\int}_{B}{\left(w\left(x\right)\right)}^{\alpha}dx\right)}^{1\u2215\alpha q}{\left({\int}_{\sigma B}{w}^{-\beta}dx\right)}^{\frac{1}{\beta p}}\\ \le \phantom{\rule{2.77695pt}{0ex}}|\sigma B{|}^{\frac{1}{s}+\frac{1}{n}}{\left[\left(\frac{1}{|\sigma B|}{\int}_{\sigma B}{\left(w\left(x\right)\right)}^{\alpha}dx\right){\left(\frac{1}{|\sigma B|}{\int}_{\sigma B}{w}^{-\beta}dx\right)}^{\frac{\alpha q}{\beta p}}\right]}^{1\u2215\alpha q}\\ \le {C}_{\alpha ,\beta ,\frac{\alpha q}{p}}^{1\u2215\alpha q}|\sigma B{|}^{\frac{1}{s}+\frac{1}{n}}.\end{array}$

(2.5)

Combining (2.3)-(2.5) immediately yields that

$\begin{array}{c}{\left({\int}_{B}|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}{|}^{q}w\left(x\right)dx\right)}^{1\u2215q}\\ \le {C}_{2}\mathit{diam}\left(B\right)|B{|}^{\frac{1}{q}-\frac{1}{s}-\frac{1}{m}}|\sigma B{|}^{\frac{1}{s}+\frac{1}{n}}{\left({\int}_{\sigma B}|u{|}^{p}w\left(x\right)dx\right)}^{1\u2215p}\\ \le {C}_{3}\mathit{diam}\left(B\right)|B{|}^{\left(p-q\right)\u2215pq}{\left({\int}_{\sigma B}|u{|}^{p}w\left(x\right)dx\right)}^{1\u2215p}.\end{array}$

This ends the proof of Theorem 2.5.

If we choose *p* = *q* in Theorem 2.5, we have the following corollary.

**Corollary 2.6**.

*Assume that u is a solution of the nonhomogeneous A-harmonic equation in a bounded convex domain D*, 1

*< q <* ∞

*and w*(

*x*) ∈

*A*(

*α*,

*β*,

*α*;

*D*)

*for some α* > 1

*and β* > 0.

*Let H be the projection operator and T* :

*C*^{∞}(∧

^{
k
}*D*) →

*C*^{∞}(∧

^{k-1}*D*),

*k* = 1, 2,...,

*n, be the homotopy operator. Then there exists a constant C, independent of u, such that*${\left({\int}_{B}|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}{|}^{q}w\left(x\right)dx\right)}^{1\u2215q}\le C\mathit{diam}\left(B\right){\left({\int}_{\sigma B}|u{|}^{q}w\left(x\right)dx\right)}^{1\u2215q}$

*for all balls with σB* ⊂ *D for some σ* > 1.

Next, we prove the following inequality, which is a generalized version of the one given in Lemma 2.2. More precisely, the inequality in Lemma 2.2 is a special case of the following result when *φ*(*t*) = *t*^{
p
}.

**Theorem 2.7**.

*Assume that φ is a Young function in the class G*(

*p*,

*q*,

*C*_{0}), 1

*< p < q <* ∞,

*C*_{0} ≥ 1

*and D is a bounded convex domain. If u* ∈

*C*^{∞}(∧

^{
k
}*D*),

*k* = 1, 2,...,

*n, is a solution of the nonhomogeneous A-harmonic equation in D*,

$\phi \left(|u|\right)\in {L}_{loc}^{1}\left(D,dx\right)$ *and* 1/

*p -* 1/

*q* ≤ 1/

*n, then there exists a constant C, independent of u, such that*${\int}_{B}\phi \left(|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}|\right)dx\le C{\int}_{\sigma B}\phi \left(|u|\right)dx$

*for all balls B with σB* ⊂ *D, where σ* > 1 *is a constant*.

**Proof**. From Lemma 2.2, we know that

$\parallel T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}{\parallel}_{s,B}\phantom{\rule{2.77695pt}{0ex}}\le {C}_{1}\mathit{diam}\left(B\right)\parallel u{\parallel}_{s,\sigma B}$

for 1

*< s <* ∞. Note that

*u* is a solution of the nonhomogeneous A-harmonic equation. Hence, by the reverse Hölder inequality, we have

$\begin{array}{c}{\left({\int}_{B}|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}{|}^{q}dx\right)}^{1\u2215q}\\ \le {C}_{1}\mathit{diam}\left(B\right){\left({\int}_{{\sigma}_{1}B}|u{|}^{q}dx\right)}^{1\u2215q}\\ \le {C}_{2}\mathit{diam}\left(B\right)|{\sigma}_{1}B{|}^{\left(p-q\right)\u2215pq}{\left({\int}_{{\sigma}_{2}B}|u{|}^{p}dx\right)}^{1\u2215p},\end{array}$

(2.6)

where

*σ*_{2} *> σ*_{1} *>* 1 are some constants. Thus, using that

*φ* and

*g* are increasing functions as well as Jensen's inequality for

*g*, we deduce that

$\begin{array}{c}\phi \left({\left({\int}_{B}|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}{|}^{q}dx\right)}^{1\u2215q}\right)\\ \le \phi \left({C}_{2}\mathit{diam}\left(B\right)|{\sigma}_{1}B{|}^{\left(p-q\right)\u2215pq}{\left({\int}_{{\sigma}_{2}B}|u{|}^{p}dx\right)}^{1\u2215p}\right)\\ \le \phi \left({\left({C}_{2}^{p}{\left(\mathit{diam}\left(B\right)\right)}^{p}|{\sigma}_{1}B{|}^{\left(p-q\right)\u2215q}{\int}_{{\sigma}_{2}B}|u{|}^{p}dx\right)}^{1\u2215p}\right)\\ \le {C}_{3}g\left({C}_{2}^{p}{\left(\mathit{diam}\left(B\right)\right)}^{p}|{\sigma}_{1}B{|}^{\left(p-q\right)\u2215q}{\int}_{{\sigma}_{2}B}|u{|}^{p}dx\right)\\ ={C}_{3}g\left({\int}_{{\sigma}_{2}B}{C}_{2}^{p}{\left(\mathit{diam}\left(B\right)\right)}^{p}|{\sigma}_{1}B{|}^{\left(p-q\right)\u2215q}|u{|}^{p}dx\right)\\ \le {C}_{3}{\int}_{{\sigma}_{2}B}g\left({C}_{2}^{p}{\left(\mathit{diam}\left(B\right)\right)}^{p}|{\sigma}_{1}B{|}^{\left(p-q\right)\u2215q}|u{|}^{p}\right)dx.\end{array}$

(2.7)

Since 1/

*p -* 1/

*q* ≤ 1/

*n*, we have

$\mathit{diam}\left(B\right)|{\sigma}_{1}B{|}^{\frac{p-q}{pq}}\le {C}_{4}|D{|}^{\frac{1}{n}+\frac{1}{q}-\frac{1}{p}}\le {C}_{5}.$

(2.8)

Applying (2.7) and (2.8) and noting that

*g*(

*t*) ≤

*C*_{0}*φ*(

*t*^{1/p}), we have

$\begin{array}{c}{\int}_{{\sigma}_{2}B}g\left({C}_{2}^{p}{\left(\mathit{diam}\left(B\right)\right)}^{p}|{\sigma}_{1}B{|}^{\left(p-q\right)\u2215q}|u{|}^{p}\right)dx\\ \le {C}_{0}{\int}_{{\sigma}_{2}B}\phi \left({C}_{2}\mathit{diam}\left(B\right)|{\sigma}_{1}B{|}^{\left(p-q\right)\u2215pq}|u|\right)dx\\ \le {C}_{0}{\int}_{{\sigma}_{2}B}\phi \left({C}_{6}|u|\right)dx.\end{array}$

(2.9)

It follows from (2.7) and (2.9) that

$\begin{array}{c}\phi \left({\left({\int}_{B}|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}{|}^{q}dx\right)}^{1\u2215q}\right)\\ \le {C}_{7}{\int}_{{\sigma}_{2}B}\phi \left({C}_{6}|u|\right)dx.\end{array}$

(2.10)

Applying Jensen's inequality once again to

*h*^{-1} and considering that

*φ* and

*h* are doubling, we have

$\begin{array}{c}{\int}_{B}\phi \left(|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}|\right)dx\\ =h\left({h}^{-1}\left({\int}_{B}\phi \left(|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}|\right)dx\right)\right)\\ \le h\left({\int}_{B}{h}^{-1}\left(\phi \left(|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}|\right)dx\right)\right)\\ \le h\left({C}_{8}{\int}_{B}|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}{|}^{q}dx\right)\\ \le {C}_{0}\phi \left({\left({C}_{8}{\int}_{B}|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}{|}^{q}dx\right)}^{1\u2215q}\right)\\ \le {C}_{9}{\int}_{{\sigma}_{2}B}\phi \phantom{\rule{2.77695pt}{0ex}}\left({C}_{6}|u|\right)dx\\ \le {C}_{10}{\int}_{{\sigma}_{2}B}\phi \phantom{\rule{2.77695pt}{0ex}}\left(|u|\right)dx.\end{array}$

This ends the proof of Theorem 2.7.

To establish the weighted version of the inequality obtained in the above Theorem 2.7, we need the following lemma which appears in [4].

**Lemma 2.8**.

*Let u be a solution of the nonhomogeneous A-harmonic equation in a domain E and* 0

*< p*,

*q <* ∞.

*Then, there exists a constant C, independent of u, such that*${\left({\int}_{B}|u{|}^{q}d\mu \right)}^{1\u2215q}\le C{\left(\mu \left(B\right)\right)}^{\frac{p-q}{pq}}{\left({\int}_{\sigma B}|u{|}^{p}d\mu \right)}^{1\u2215p}$

*for all balls B with σB* ⊂ *E for some σ* > 1, *where the Radon measure µ is defined by dµ* = *w*(*x*)*dx and w* ∈ *A*(*α*, *β*, *α*; *E*), *α* > 1, *β* > 0.

**Theorem 2.9**.

*Assume that φ is a Young function in the class G*(

*p*,

*q*,

*C*_{0}), 1

*< p < q <* ∞,

*C*_{0} ≥ 1

*and D is a bounded convex domain. Let dµ* =

*w*(

*x*)

*dx, where w*(

*x*) ∈

*A*(

*α*,

*β*,

*α*;

*D*)

*for α* > 1

*and β* > 0.

*If u* ∈

*C*^{∞}(∧

^{
k
}*D*),

*k* = 1, 2,...,

*n, is a solution of the nonhomogeneous A-harmonic equation in D*,

$\phi \left(|u|\right)\in {L}_{loc}^{1}\left(D,d\mu \right)$,

*then there exists a constant C, independent of u, such that*${\int}_{B}\phi \left(|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}|\right)d\mu \le C{\int}_{\sigma B}\phi \left(|u|\right)d\mu $

*for all balls B with σB* ⊂ *D and |B|* ≥ *d*_{0} *>* 0, *where σ* > 1 *is a constant*.

**Proof**. From Corollary 2.6 and Lemma 2.8, we have

$\begin{array}{l}{\left({\displaystyle {\int}_{B}|T(H(u))}-{(T(H(u)))}_{B}{|}^{q}d\mu \right)}^{1/q}\\ \le {C}_{1}\mathit{diam}(B){\left({\displaystyle {\int}_{{\sigma}_{1}B}}|u{|}^{q}d\mu \right)}^{1/q}\\ \le {C}_{2}\mathit{diam}(B)(\mu (B){)}^{(p-q)/pq}{\left({\displaystyle {\int}_{{\sigma}_{2}B}}|u{|}^{p}d\mu \right)}^{1/p},\end{array}$

(2.11)

where

*σ*_{2} *> σ*_{1} *>* 1 is some constant. Note that

*φ* and

*g* are increasing functions and

*g* is convex in

*D*. Hence by Jensen's inequality for

*g*, we deduce that

$\begin{array}{l}\phi \left({\left({\displaystyle {\int}_{B}|T(H(u))}-{(T(H(u)))}_{B}{|}^{q}d\mu \right)}^{1/q}\right)\\ \le \phi \left({C}_{2}\mathit{diam}(B)(\mu (B{))}^{(p-q)/pq}{\left({\displaystyle {\int}_{{\sigma}_{2}B}|u{|}^{p}}d\mu \right)}^{1/p}\right)\\ =\phi \left({\left({C}_{2}^{p}{(\mathit{diam}(B))}^{p}{(\mu (B))}^{(p-q)/q}{\displaystyle {\int}_{{\sigma}_{2}B}|u{|}^{p}}d\mu \right)}^{1/p}\right)\\ \le {C}_{3}g\left({C}_{2}^{p}{(\mathit{diam}(B))}^{p}{(\mu (B))}^{(p-q)/q}{\displaystyle {\int}_{{\sigma}_{2}B}|u{|}^{p}}d\mu \right)\\ ={C}_{\text{3}}g\left({\displaystyle {\int}_{{\sigma}_{2}B}{C}_{2}^{p}}{(\mathit{diam}(B))}^{p}{(\mu (B))}^{(p-q)/q}|u{|}^{p}d\mu \right)\\ \le {C}_{3}{\displaystyle {\int}_{{\sigma}_{2}B}g}\left({C}_{2}^{p}{(\mathit{diam}(B))}^{p}{(\mu (B))}^{(p-q)/q}|u{|}^{p}\right)d\mu .\end{array}$

(2.12)

Set

*D*_{1} = {

*x* ∈

*D* : 0

*< w*(

*x*)

*<* 1} and

*D*_{2} = {

*x* ∈

*D* :

*w*(

*x*) ≥ 1}. Then

*D* =

*D*_{1} ∪

*D*_{2}. We let

$\stackrel{\u0303}{w}\left(x\right)=1$, if

*x* ∈

*D*_{1} and

$\stackrel{\u0303}{w}\left(x\right)=w\left(x\right)$, if

*x* ∈

*D*_{2}. It is easy to check that

*w*(

*x*) ∈

*A*(

*α*,

*β*,

*α*;

*D*) if and only if

$\stackrel{\u0303}{w}\left(x\right)\in A\left(\alpha ,\beta ,\alpha ;D\right)$. Thus, we may always assume that

*w*(

*x*) ≥ 1 a.e. in

*D*. Hence, we have

*µ*(

*B*) = ∫

_{
B
} *w*(

*x*)

*dx* ≥

*|B|* for all balls

*B* ⊂

*D*. Since

*p < q* and

*|B| = d*_{0} *>* 0, it is easy to find that

$\mathit{diam}\left(B\right)\mu {\left(B\right)}^{\left(p-q\right)\u2215pq}\le \mathit{diam}\left(D\right){d}_{0}^{\left(p-q\right)\u2215pq}\le {C}_{3}.$

(2.13)

It follows from (2.13) and

*g*(

*t*) ≤

*C*_{0}*φ*(

*t*^{1/p}) that

$\begin{array}{l}{\displaystyle {\int}_{{\sigma}_{2}B}g}({C}_{2}^{p}(\mathit{diam}(B){)}^{p}(\mu (B){)}^{(p-q)/q}|u{|}^{p})d\mu \\ \le {C}_{0}{\displaystyle {\int}_{{\sigma}_{2}B}\phi}({C}_{2}\mathit{diam}(B)(\mu (B){)}^{(p-q)/pq}\left|u\right|)d\mu \\ \le {C}_{0}{\displaystyle {\int}_{{\sigma}_{2}B}\phi}({C}_{4}|u|)d\mu .\end{array}$

(2.14)

Applying Jensen's inequality to

*h*^{-1} and considering that

*φ* and

*h* are doubling, we have

$\begin{array}{c}{\int}_{B}\phi \left(|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}|\right)d\mu \\ =h\left({h}^{-1}\left({\int}_{B}\phi \left(|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}|\right)d\mu \right)\right)\\ \le h\left({\int}_{B}{h}^{-1}\left(\phi \left(|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}|\right)d\mu \right)\right)\\ \le h\left({C}_{8}{\int}_{B}|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}{|}^{q}d\mu \right)\\ \le {C}_{0}\phi \left({\left({C}_{8}{\int}_{B}|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}{|}^{q}d\mu \right)}^{1\u2215q}\right)\\ \le {C}_{9}{\int}_{{\sigma}_{2}B}\phi \left({C}_{6}|u|\right)d\mu \\ \le {C}_{10}{\int}_{{\sigma}_{2}B}\phi \left(|u|\right)d\mu .\end{array}$

This ends the proof of Theorem 2.9.

Note that if we remove the restriction on balls *B*, then we can obtain a weighted inequality in the class $A\left(\alpha ,\beta ,\frac{\alpha q}{p};D\right)$, for which the method of proof is analogous to the one in Theorem 2.9. We now give the statement as follows.

**Theorem 2.10**.

*Assume that φ is a Young function in the class G*(

*p*,

*q*,

*C*_{0}), 1

*< p < q <* ∞,

*C*_{0} ≥ 1

*and D is a bounded convex domain. Let dµ* =

*w*(

*x*)

*dx, where* $w\left(x\right)\in A\left(\alpha ,\beta ,\frac{\alpha q}{p};D\right)$ *for α* > 1

*and β* > 0.

*If u* ∈

*C*^{∞}(∧

^{
k
}*D*),

*k* = 1, 2,...,

*n, is a solution of the nonhomogeneous A-harmonic equation in D*,

$\phi \left(|u|\right)\in {L}_{loc}^{1}\left(D,d\mu \right)$ *and* 1/

*p -* 1/

*q* ≤ 1/

*n, then there exists a constant C, independent of u, such that*${\int}_{B}\phi \left(|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}|\right)d\mu \le C{\int}_{\sigma B}\phi \left(|u|\right)d\mu $

*for all balls B with σB* ⊂ *D, where σ* > 1 *is a constant*.

Directly from the proof of Theorem 2.7, if we replace |

*T*(

*H*(

*u*))-(

*T*(

*H*(

*u*)))

_{
B
}| by

$\frac{1}{\lambda}|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}|$, then we immediately have

${\int}_{B}\phi \left(\frac{|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}|}{\lambda}\right)dx\le C{\int}_{\sigma B}\phi \left(\frac{|u|}{\lambda}\right)dx$

(2.15)

for all balls *B* with *σB* ⊂ *D* and *λ* > 0. Furthermore, from the definition of the Orlicz norm and (2.15), the following Orlicz norm inequality holds.

**Corollary 2.11**.

*Assume that φ is a Young function in the class G*(

*p, q, C*_{0}), 1

*< p < q <* ∞,

*C*_{0} ≥ 1

*and D is a bounded convex domain. If u* ∈

*C*^{∞}(∧

^{
k
}*D*),

*k* = 1, 2,...,

*n, is a solution of the nonhomogeneous A-harmonic equation in D*,

$\phi \left(|u|\right)\in {L}_{loc}^{1}\left(D,dx\right)$ *and* 1/

*p -* 1/

*q* ≤ 1/

*n, then there exists a constant C, independent of u, such that*$\parallel T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}{\parallel}_{\phi ,B}\phantom{\rule{2.77695pt}{0ex}}\le C\parallel u{\parallel}_{\phi ,\sigma B}$

(2.16)

*for all balls B with σB* ⊂ *D, where σ* > 1 *is a constant*.

Next, we extend the local Orlicz norm inequality for the composite operator to the global version in the *L*^{
φ
}(*µ*)-averaging domains.

In [12], Staples introduced *L*^{
s
}-averaging domains in terms of Lebesgue measure. Then, Ding and Nolder [6] developed *L*^{
s
}-averaging domains to weighted versions and obtained a similar characterization. At the same time, they also established a global norm inequality for conjugate A-harmonic tensors in *L*^{
s
}(*µ*)-averaging domains. In the following year, Ding [5] further generalized *L*^{
s
}-averaging domains to *L*^{
φ
}(*µ*)-averaging domains, for which *L*^{
s
}(*µ*)-averaging domains are special cases when *φ*(*t*) = *t*^{
s
}. The following definition appears.

**Definition 2.12**.

*Let φ be an increasing convex function defined on* [0, ∞)

*with φ*(0) = 0.

*We say a proper subdomain* Ω ⊂ ℝ

^{
n
} *an L*^{
φ
}(

*µ*)-

*averaging domain, if µ*(Ω)

*<* ∞

*and there exists a constant C such that*${\int}_{\Omega}\phi \left(\tau |u-{u}_{{B}_{0}}|\right)d\mu \le C\underset{B}{sup}{\int}_{B}\phi \left(\sigma |u-{u}_{B}|\right)d\mu $

*for some balls B*_{0} ⊂ Ω *and all u such that* $\phi \left(|u|\right)\in {L}_{loc}^{1}\left(\Omega ,d\mu \right)$, *where* 0 *< τ, σ <* ∞ *are constants and the supremum is over all balls B* ⊂ Ω.

**Theorem 2.13**.

*Let φ be a Young function in the class G*(

*p*,

*q*,

*C*_{0}), 1

*< p < q <* ∞,

*C*_{0} ≥ 1

*and D is a bounded convex L*^{
φ
}(

*dx*)-

*averaging domain. Suppose that φ*(

*|u|*) ∈

*L*^{1}(

*D*,

*dx*),

*u* ∈

*C*^{∞}(∧

^{1}*D*)

*is a solution of the nonhomogeneous A-harmonic equation in D and* 1/

*p -* 1/

*q* ≤ 1/

*n. Then there exists a constant C, independent of u, such that*${\int}_{D}\phi \left(|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{{B}_{0}}|\right)dx\le C{\int}_{D}\phi \left(|u|\right)dx,$

(2.17)

*where B*_{0} ⊂ *D is a fixed ball*.

**Proof**. Since

*D* is an

*L*^{
φ
}(

*dx*)-averaging domain and

*φ* is doubling, from Theorem 2.7, we have

$\begin{array}{c}{\int}_{D}\phi \left(|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{{B}_{0}}|\right)dx\\ \le {C}_{1}\underset{B\subset D}{sup}{\int}_{B}\phi \left(|T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{B}|\right)dx\\ \le {C}_{1}\underset{B\subset D}{sup}\left({C}_{2}{\int}_{\sigma B}\phi \left(|u|\right)dx\right)\\ \le {C}_{3}{\int}_{D}\phi \left(|u|\right)dx.\end{array}$

We have completed the proof of Theorem 2.13.

Clearly, (2.17) implies that

$\parallel T\left(H\left(u\right)\right)-{\left(T\left(H\left(u\right)\right)\right)}_{{B}_{0}}{\parallel}_{\phi ,D}\le C\parallel u{\parallel}_{\phi ,D}.$

(2.18)

Similarly, we also can develop the inequalities established in Theorems 2.9 and 2.10 to *L*^{
φ
}(*µ*)-averaging domains, for which *dµ* = *w*(*x*)*dx* and *w*(*x*) ∈ *A*(*α*, *β*, *α*; *D*) and $A\left(\alpha ,\beta ,\frac{\alpha q}{p};D\right)$, respectively.