Embedding theorems for composition of homotopy and projection operators

Abstract

We prove both local and global embedding theorems with $L^{\varphi}$-norms for the composition of the homotopy operator and projection operator applied to differential forms. We also establish some $L^{\varphi}$-norm inequalities for certain compositions of the related operators.

1 Introduction

The homotopy operator T and projection operator H defined on differential forms are two critical operators which have been very well studied and used in recent years, see [1–7]. In many situations, we need to deal with the composition $T\circ H$ of the homotopy operator T and the projection operator H. For example, when we consider the decomposition of $H(u)$, we have to face the composition $T\circ H$. The study of the composition $T \circ H$ of homotopy and projection operators was initiated by Ding and Liu in 2009 in [5] and [6], respectively, where they investigated singular integrals of this composite operator and established some $L^{p}$ inequalities for the composite operator $T \circ H$ with singular factors. Later, in 2011, Bi and Ding proved some $L^{\varphi}$-estimates for this composite operator $T \circ H$ in [7], where φ satisfies the $G(p, q, C)$ condition. The purpose of this paper is to establish the $L^{\varphi}$-embedding theorems for the composition $T \circ H$ applied to differential forms, here φ satisfies the $NG(p, q)$ condition. If we choose $\varphi(t)= t^{p}$, the $L^{\varphi}$-norm inequalities reduce to $L^{p}$-norm inequalities. Our main $L^{\varphi}$-embedding inequality for the composite operator can be simply stated as

$$ \bigl\| T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{\Omega} \bigr\| _{W^{1, \varphi} (\Omega)} \leq C \|u \|_{L^{\varphi}(\Omega )}, $$

(1.1)

where Ω is any bounded domain in $\mathbb{R}^{n}$, $n \geq2$, $\varphi: [0, \infty) \to[0, \infty)$ with $\varphi(0)= 0$ is a Young function satisfying certain conditions described later, and C is a constant independent of the differential form u. In order to establish the above main $L^{\varphi}$-embedding inequality, we also prove the Poincaré inequality and some inequalities with $L^{\varphi}$-norm for the related compositions of operators.

We keep using the traditional notations throughout this paper. Let B and σB be the balls with the same center and $\operatorname{diam}(\sigma B) = \sigma\operatorname{diam}(B)$. Let $|E|$ be the n-dimensional Lebesgue measure of a set E⊆ $\mathbb{R}^{n}$. In this paper, we treat a ball same as a cube and use $u_{B}=\frac{1}{|B|} \int_{B} u \,dx$ to denote the average of a function u. Let $\wedge^{l} = \wedge^{l} (\mathbb{R}^{n})$ be the set of all l-forms in $\mathbb{R}^{n}$, $D'(\Omega, \wedge^{l})$ be the space of all differential l-forms in Ω, and $L^{p}(\Omega, \wedge^{l})$ be the l-forms $u(x)= \sum_{I} u_{I}(x)\,dx_{I}$ in Ω satisfying $\int_{\Omega}|u_{I}|^{p} < \infty$ for all ordered l-tuples I, $l=1, 2, \ldots, n$. We denote the exterior derivative by d and the Hodge star operator by ⋆.

The definition of the operator $K_{y}$ with the case $y =0$ and its generalized version can be found in [8, 9]. To each $y \in\Omega$ there corresponds a linear operator $K_{y}: C^{\infty}(\Omega, \wedge^{l}) \to C^{\infty}(\Omega,\wedge^{l-1})$ defined by $(K_{y} \omega)(x; \xi_{1}, \ldots, \xi_{l-1})=\int_{0}^{1}t^{l-1}\omega (tx+y-ty;x-y, \xi_{1},\ldots,\xi_{l-1})\,dt$ and the decomposition $\omega= d(K_{y}\omega) + K_{y}(d \omega)$. A homotopy operator $T: C^{\infty}(\Omega, \wedge^{l}) \to C^{\infty}(\Omega ,\wedge^{l-1})$ is defined by averaging $K_{y}$ over all points $y \in \Omega$: $T\omega= \int_{\Omega}\phi(y)K_{y}\omega\,dy$, where $\phi\in C_{0}^{\infty}(\Omega)$ is normalized so that $\int\phi (y)\,dy=1$. For each differential form u, we have the decomposition

$$ u= d(Tu) + T(du) $$

(1.2)

and

$$ \bigl\| \nabla(T u) \bigr\| _{p, B} \leq C |B| \|u \|_{p, B}\quad \mbox{and}\quad \|Tu\|_{p, B} \leq C |B| \operatorname{diam} (B) \|u \|_{p, B}. $$

(1.3)

From [10], p.16, we know that any open subset Ω in $\mathbb{R}^{n}$ is the union of a sequence of cubes $Q_{k}$, whose sides are parallel to the axes, whose interiors are mutually disjoint, and whose diameters are approximately proportional to their distances from F, where F is the complement of Ω in $\mathbb{R}^{n}$. Specifically, (i) $\Omega= \bigcup_{k=1}^{\infty}Q_{k}$, (ii) $Q^{0}_{j} \cap Q^{0}_{k} = \phi$ if $j \neq k$, (iii) there exist two constants $c_{1}, c_{2} >0$ (we can take $c_{1}=1$, and $c_{2}=4$), so that $c_{1} \operatorname{diam} (Q_{k}) \leq\operatorname{distance} (Q_{k}, F) \leq c_{2}\operatorname{diam}(Q_{k})$. Thus, the definition of the homotopy operator T can be generalized to any domain Ω in $\mathbb{R}^{n}$: For any $x \in\Omega$, $x \in Q_{k}$ for some k. Let $T_{Q_{k}}$ be the homotopy operator defined on $Q_{k}$ (each cube is bounded and convex). Thus, we can define the homotopy operator $T_{\Omega}$ on any domain Ω by $T_{\Omega}= \sum_{k=1}^{\infty}T_{Q_{k}} {\chi_{Q_{k}}(x)}$. The nonlinear partial differential equation for differential forms

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

(1.4)

is called a non-homogeneous A-harmonic equation, where $A: \Omega\times\wedge^{l}(\mathbb{R}^{n}) \to\wedge^{l}(\mathbb{R}^{n})$ and $B: \Omega\times\wedge^{l}(\mathbb{R}^{n}) \to\wedge^{l-1}(\mathbb{R}^{n})$ satisfy the conditions:

$$ \bigl|A(x, \xi)\bigr| \leq a|\xi|^{p-1},\qquad A(x, \xi) \cdot \xi\geq| \xi|^{p} \quad\mbox{and} \quad\bigl|B(x, \xi)\bigr| \leq b | \xi|^{p-1} $$

(1.5)

for $x \in\Omega$ a.e. and all $\xi\in\wedge^{l} (\mathbb{R}^{n})$. Here $p>1$ is a constant related to the equation (1.4), and $a, b >0$. See [11–17] for recent results on the A-harmonic equations and related topics. Assume that $\wedge^{l} \Omega$ is the lth exterior power of the cotangent bundle, $C^{\infty}(\wedge^{l} \Omega)$ is the space of smooth l-forms on Ω and $\mathcal{W}(\wedge^{l} \Omega)= \{u \in L^{1}_{\mathrm{loc}}(\wedge^{l} \Omega):u\mbox{ has generalized gradient}\}$. The harmonic l-fields are defined by $\mathcal{H} (\wedge^{l} \Omega) = \{ u \in\mathcal{W}(\wedge^{l} \Omega): d u = d^{\star}u=0, u \in L^{p}\mbox{ for some }1 < p < \infty\}$. The orthogonal complement of $\mathcal{H}$ in $L^{1}$ is defined by $\mathcal{H}^{\perp} = \{u \in L^{1}: \langle u, h\rangle=0 \mbox{ for all }h \in\mathcal {H}\}$. Then the Green’s operator G is defined as $G: C^{\infty }(\wedge^{l} \Omega) \to\mathcal{H}^{\perp} \cap C^{\infty}(\wedge^{l} \Omega)$ by assigning $G(u)$ be the unique element of $\mathcal {H}^{\perp} \cap C^{\infty}(\wedge^{l} \Omega)$ satisfying Poisson’s equation $\Delta G(u)= u - H(u)$, where H is the harmonic projection operator that maps $C^{\infty }(\wedge^{l} \Omega)$ onto $\mathcal {H}$ so that $H(u)$ is the harmonic part of u. See [18] for more properties of these operators.

2 Local embedding theorem

The purpose of this section is to prove the local $L^{\varphi}$-embedding theorem and some related $L^{\varphi}$-norm inequalities that will be used to prove the global embedding theorem in the next section. We first recall the following subclass of Young functions that can be found in [19–21].

Definition 2.1

A Young function $\varphi: [0, \infty ) \longrightarrow[0, \infty)$ is said to be in the class $NG(p, q)$ if φ satisfies the nonstandard growth condition

$$ p\varphi(t)\leq t\varphi'(t)\leq q\varphi(t) , \quad1< p \leq q < \infty. $$

(2.1)

The first inequality in (2.1) is equivalent to that $\frac {\varphi(t)}{t^{p}}$ is increasing, and the second inequality in (2.1) is equivalent to △₂-condition, i.e., for each $t>0$, $\varphi (2t)\leq K \varphi(t)$, where $K>1$, and $\frac{\varphi(t)}{t^{q}}$ is decreasing with t. Also, condition (2.1) implies that $\varphi(t)$ satisfies

$$ c_{1} t^{p} -c_{2} \leq\varphi(t) \leq c_{3} \bigl(t^{q}+1\bigr). $$

(2.2)

Particularly, $\varphi(t)=t^{p}$ satisfies (2.1) because of $t\varphi '(t)=p\varphi(t)$, and this makes inequalities with the norm $\|\cdot \|_{p}$ become a special case of Theorem 2.5; for more details see [19] and [20].

An Orlicz function is a continuously increasing function $\varphi: [0, \infty) \to[0, \infty)$ with $\varphi(0)= 0$. The Orlicz space $L^{\varphi}(\Omega)$ consists of all measurable functions f on Ω such that $\int_{\Omega}\varphi(\frac{|f|}{\lambda} )\,dx < \infty$ for some $\lambda=\lambda(f) >0$. $L^{\varphi}(\Omega)$ is equipped with the nonlinear Luxemburg functional

$$\|f \|_{L^{\varphi}(\Omega)} = \inf\biggl\{ \lambda>0: \int _{\Omega}\varphi\biggl(\frac{|f|}{\lambda} \biggr)\,dx \leq1 \biggr\} . $$

A convex Orlicz function φ is often called a Young function. If φ is a Young function, then $\| \cdot\|_{L^{\varphi}(\Omega)}$ defines a norm in $L^{\varphi}(\Omega )$, which is called the Luxemburg norm or Orlicz norm. For any subset $E \subset \mathbb{R}^{n}$, we use $W^{1, \varphi}(E, \wedge^{l})$ to denote the Orlicz-Sobolev space of l-forms which equals $L^{\varphi}(E, \wedge^{l}) \cap L_{1}^{\varphi}(E, \wedge^{l})$ with norm

$$ \|u \|_{W^{1, \varphi}(E)} = \|u \|_{W^{1, \varphi}(E, \wedge^{l})} = \operatorname{diam} (E)^{-1} \|u \|_{L^{\varphi}(E)} + \|\nabla u \|_{L^{\varphi}(E)}. $$

(2.3)

If we choose $\varphi(t) = t^{p}$, $p >1$ in (2.3), we obtain the usual $L^{p}$-norm for $W^{1, p}(E, \wedge^{l})$

$$ \|u \|_{W^{1, p}(E)} = \|u \|_{W^{1, p}(E, \wedge^{l})} = \operatorname{diam} (E)^{-1} \|u \|_{p, E} + \|\nabla u \|_{p, E}. $$

(2.4)

Next, we recall some lemmas that will be used in this paper.

Lemma 2.2

[9]

Let $u\in D'(B, \wedge^{l})$ and $du \in L^{p}(B, \wedge^{l+1})$. Then $u-u_{B} \in L^{\frac{np}{n-p}} (B, \wedge^{l})$, and

$$ \biggl(\int_{B} |u-u_{B}|^{\frac{np}{n-p}}\,dx \biggr)^{\frac{n-p}{np}} \leq c(n,p) \biggl( \int_{B} |du|^{p} \,dx \biggr)^{\frac{1}{p}} $$

(2.5)

for B is a ball or cube in Ω, $l=0,1,\ldots,n-1$ and $1< p< n$.

Lemma 2.3

[19]

Suppose φ is a continuous function in the class $NG(p, q)$ with $q(n-p)< np$, $1< p \leq q < \infty$. For any $t>0$, setting

$$ A(t)= \int_{0}^{t} \biggl(\frac{\varphi(s^{1/q})}{s} \biggr)^{\frac{n+q}{q}}\,ds, \qquad K(t)= \frac{( \varphi(t^{1/q}) )^{\frac{n+q}{q}}}{t^{n/q}}. $$

(2.6)

Then $A(t)$ is a concave function, and there exists a constant C, such that

$$ K(t)\leq A(t) \leq CK(t),\quad \forall t>0. $$

(2.7)

Lemma 2.4

[1]

Let u be a differential form satisfying the non-homogeneous A-harmonic equation (1.4) in Ω, $\sigma>1$ and $0 < s, t < \infty$. Then, there exists a constant C, independent of u, such that $\|u \|_{s, B}\leq C|B|^{(t-s)/st} \|u \|_{t,\sigma B}$ for all balls or cubes B with $\sigma B \subset\Omega$.

We are ready to state our main local $L^{\varphi}$-embedding theorem as follows, which will be used to prove the global $L^{\varphi}$-embedding theorem in the next section.

Theorem 2.5

Let φ be a Young function in the class $NG(p, q)$ with $q(n-p)< np$, $1< p \leq q < \infty$. Ω be a bounded domain, $u \in L^{p}(\Omega, \wedge^{l})$ be a solution of the non-homogeneous A-harmonic equation, T be the homotopy operator and H be the projection operator. If $\varphi (|u|) \in L^{1}_{\mathrm{loc}}(\Omega)$, then there exists a constant C, independent of u, such that

$$ \bigl\| T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B} \bigr\| _{W^{1, \varphi} (B, \wedge^{l})} \leq C |B| \| u \|_{L^{\varphi}(\sigma B)} $$

(2.8)

for all balls B with $\sigma B \subset\Omega$, where $\sigma>1$ is a constant.

In order to prove the above local $L^{\varphi}$-embedding theorem, we need to prove some local $L^{\varphi}$-norm inequalities. We begin with the following Poincaré-type inequality with $L^{\varphi}$-norm first.

Theorem 2.6

Let φ be a Young function in the class $NG(p, q)$ with $q(n-p)< np$, $1< p \leq q < \infty$, Ω be a bounded domain, $u \in L^{p}(\Omega, \wedge^{l})$, T be the homotopy operator and H be the projection operator. If $\varphi (|u|) \in L^{1}_{\mathrm{loc}}(\Omega)$, then there exists a constant C, independent of u, such that

$$ \bigl\| T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B} \bigr\| _{L^{\varphi}( B)} \leq C |B| \|u \| _{L^{\varphi}( B)} $$

(2.9)

for all balls $B\subset\Omega$.

Proof

First, we consider the case $1 < p <n$. By assumption, we have $q < \frac{np}{n-p}$. Using the Poincaré-type inequality, Lemma 2.2 to differential forms $T(H(u))$

$$ \biggl(\int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr) \bigr)_{B} \bigr| ^{np/(n-p)}\,dx \biggr)^{(n-p)/np} \leq C_{1} \biggl( \int_{ B} \bigl|d\bigl(T\bigl(H(u)\bigr) \bigr)\bigr|^{p} \,dx \biggr)^{1/p}, $$

(2.10)

we find that

$$ \biggl(\int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr) \bigr)_{B}\bigr| ^{q}\,dx \biggr)^{1 / q} \leq C_{2} \biggl( \int_{B} \bigl|d\bigl(T\bigl(H(u)\bigr) \bigr)\bigr|^{p} \,dx \biggr)^{1/p}. $$

(2.11)

It is well known that, for any differential form u, $d(T(u))=u_{B}$ and $\|u_{B} \|_{p, B} \leq C_{3} \|u \|_{p, B}$. Hence,

$$ \biggl( \int_{ B} \bigl|d\bigl(T\bigl(H(u)\bigr) \bigr)\bigr|^{p} \,dx \biggr)^{1/p} \leq C_{4} \biggl( \int _{ B} \bigl|H(u)\bigr|^{p} \,dx \biggr)^{1/p}. $$

(2.12)

Note that

$$\bigl\| \Delta G(u) \bigr\| _{p, B} = \bigl\| \bigl(d^{\star}d + d d^{\star}\bigr) G(u) \bigr\| _{p, B} \leq C_{5} \|u \|_{p, B}. $$

We have

$$\begin{aligned} \bigl\| H(u)\bigr\| _{p, B} &= \bigl\| u - \Delta G(u) \bigr\| _{p, B} \\ &\leq\|u \|_{p, B} + \bigl\| \Delta G(u) \bigr\| _{p, B} \\ &\leq C_{6} \|u\|_{p, B}. \end{aligned}$$

(2.13)

Combining (2.11), (2.12), and (2.13), we obtain

$$ \biggl(\int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr) \bigr)_{B}\bigr| ^{q}\,dx \biggr)^{1 / q} \leq C_{7} \biggl( \int_{ B} |u|^{p} \,dx \biggr)^{1/p} $$

(2.14)

for $1 < p <n$. Next, for the case of $p \geq n$, since the $L^{p}$-norm of $|T(H(u))- (T(H(u)))_{B}|$ increases with p and $\frac{np}{n-p} \to \infty$ as $p \to n$. Then, there exists $1 < p_{0} <n$ such that $q < \frac{np_{0} }{ n-p_{0}}$. Hence, it follows that

$$\begin{aligned} & \biggl(\int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T \bigl(H(u)\bigr)\bigr)_{B}\bigr| ^{q}\,dx \biggr)^{1 / q} \\ &\quad\leq\biggl(\int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T \bigl(H(u)\bigr)\bigr)_{B} \bigr| ^{n{p_{0}}/(n-{p_{0}} )}\,dx \biggr)^{(n-p_{0})/np_{0}} \\ & \quad\leq C_{8} \biggl( \int_{ B} \bigl|d\bigl(T \bigl(H(u)\bigr)\bigr)\bigr|^{p_{0}}\,dx \biggr)^{1/p_{0}} \\ &\quad\leq C_{9} \biggl( \int_{ B} \bigl|d\bigl(T \bigl(H(u)\bigr)\bigr)\bigr|^{p}\,dx \biggr)^{1/p} \\ &\quad\leq C_{10} \biggl( \int_{ B} |u|^{p}\,dx \biggr)^{1/p}. \end{aligned}$$

(2.15)

Hence, from (2.14) and (2.15), we obtain

$$ \biggl(\int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr) \bigr)_{B}\bigr| ^{q}\,dx \biggr)^{1 / q} \leq C_{10} \biggl( \int_{B} |u|^{p} \,dx \biggr)^{1/p} $$

(2.16)

for any $p>1$. Using the Hölder inequality with $1= \frac{q}{ n+q}+ \frac{n }{{n+q}}$, we obtain

$$\begin{aligned} &\int_{B} \varphi\bigl(\bigl|T\bigl(H(u)\bigr)- \bigl(T \bigl(H(u)\bigr)\bigr)_{B} \bigr| \bigr)\,dx \\ &\quad= \int_{B} \frac{ \varphi(|T(H(u))- (T(H(u)))_{B} | ) }{ |T(H(u))- (T(H(u)))_{B}|^{\frac{nq}{{n+q}}}} \bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B}\bigr|^{\frac{nq}{{n+q}}}\,dx \\ &\quad\leq\biggl( \int_{B} \frac{{\varphi (|T(H(u))-(T(H(u)))_{B}|)}^{\frac{n+q}{q}} }{|T(H(u))-(T(H(u)))_{B}|^{n}}\,dx \biggr)^{\frac{q}{{n+q}}}\\ &\qquad{}\times{ \biggl( \int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B}\bigr|^{q} \,dx \biggr)}^{\frac{n}{{n+q}}}. \end{aligned}$$

Applying Lemma 2.3 and noticing $A(t)$ is a concave function, we obtain

$$\begin{aligned} &\int_{B} \varphi\bigl(\bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B} \bigr| \bigr)\,dx \\ &\quad\leq\biggl(\int_{B} K\bigl(\bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B} \bigr|^{q}\bigr)\,dx \biggr)^{\frac{q}{n+q}} { \biggl( \int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B}\bigr|^{q} \,dx \biggr)}^{\frac{n}{{n+q}}} \\ &\quad\leq\biggl(\int_{B} A\bigl(\bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B} \bigr|^{q}\bigr)\,dx \biggr)^{\frac{q}{n+q}} { \biggl( \int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B}\bigr|^{q} \,dx \biggr)}^{\frac{n}{{n+q}}} \\ &\quad\leq A^{\frac{q}{n+q}} \biggl(\int_{B} \bigl(\bigl|T \bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B} \bigr|^{q} \bigr)\,dx \biggr) { \biggl( \int_{B} \bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B}\bigr|^{q} \,dx \biggr)}^{\frac{n}{{n+q}}} \\ &\quad\leq C_{1}(n, q) K^{\frac{q}{n +q}} \biggl(\int _{B} \bigl(\bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr) \bigr)_{B} \bigr|^{q}\bigr)\,dx \biggr) \\ &\qquad{}\times{\biggl( \int _{B} \bigl|T\bigl(H(u)\bigr)-\bigl(T\bigl(H(u)\bigr) \bigr)_{B}\bigr|^{q} \,dx \biggr)}^{\frac{n}{{n+q}}} \\ &\quad= C_{1}(n, q) \frac{\varphi( (\int_{B} (|T(H(u))- (T(H(u)))_{B} |^{q})\,dx )^{1/q} )}{ (\int_{B} (|T(H(u))- (T(H(u)))_{B} |^{q})\,dx )^{\frac{n}{ n+q} }} \\ &\qquad{}\times{ \biggl( \int _{B} \bigl|T\bigl(H(u)\bigr)-\bigl(T\bigl(H(u)\bigr) \bigr)_{B}\bigr|^{q} \,dx \biggr)}^{\frac{n }{{n+q} }} \\ & \quad= C_{1}(n, q) \varphi\biggl( \biggl(\int_{B} \bigl(\bigl|T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{B} \bigr|^{q}\bigr)\,dx \biggr)^{1/q} \biggr). \end{aligned}$$

(2.17)

Note that φ is increasing and satisfies $\Delta_{2}$-condition, substituting (2.16) into (2.17) gives

$$ \int_{B} \varphi\bigl(\bigl|T\bigl(H(u)\bigr)- \bigl(T \bigl(H(u)\bigr)\bigr)_{B} \bigr| \bigr)\,dx \leq C_{11} \varphi \biggl( \biggl( \int_{ B} |u|^{p} \,dx \biggr)^{1/p} \biggr). $$

(2.18)

Let $h(t)= \int_{0}^{t} \frac{\varphi(s) }{ s}\,ds$. From (2.1) we know that $\varphi(t)/t^{q}$ is decreasing with t, thus,

$$h(t) = \int_{0}^{t} \frac{\varphi(s) }{ s}\,ds = \int _{0}^{t} \frac{\varphi(s) }{ s^{q}}s^{q-1}\,ds \geq \varphi(t)/t^{q} \frac{1}{q} s^{q} \bigg|_{0}^{t} = \frac{1}{q} \varphi(t). $$

Similarly, using the fact that $\varphi(t)/t^{p}$ is increasing with t, we have $h(t) \leq\frac{1}{p} \varphi(t)$. Therefore,

$$ \frac{1}{q} \varphi(t) \leq h(t) \leq\frac{1}{p} \varphi(t). $$

(2.19)

Let $g(t)=h(t^{1/p})$, then $( h(t^{1/p}) )' = \frac{1}{p} \frac{\varphi(t^{1/p})}{t}$ is increasing. Hence, g is a convex function. From the definitions of g and h and using Jensen’s inequality to g, we have

$$ h \biggl( \biggl(\int_{B} |u|^{p} \,dx \biggr)^{1/p} \biggr) = g \biggl(\int_{B} |u|^{p} \,dx \biggr) \leq\int_{B} g \bigl(|u|^{p}\bigr)\,dx = \int_{B} h\bigl(|u|\bigr)\,dx. $$

(2.20)

Combining (2.18), (2.19), and (2.20), we have

$$\begin{aligned} &\int_{B} \varphi\bigl(\bigl|T\bigl(H(u)\bigr)- \bigl(T \bigl(H(u)\bigr)\bigr)_{B} \bigr| \bigr)\,dx \\ &\quad \leq C_{11} \varphi\biggl( \biggl( \int_{ B} |u|^{p} \,dx \biggr)^{1/p} \biggr) \\ &\quad\leq C_{12} h \biggl( \biggl(\int_{B} |u|^{p} \,dx \biggr)^{1/p} \biggr) \\ &\quad\leq C_{13} \int_{B} h\bigl(|u|\bigr)\,dx \\ &\quad\leq C_{14} \int_{B} \varphi\bigl(|u|\bigr)\,dx, \end{aligned}$$

which indicates (2.9) holds. We have completed the proof of Theorem 2.6. □

Theorem 2.7

Let φ be a Young function in the class $NG(p, q)$ with $q(n-p)< np$, $1< p \leq q < \infty$, Ω be a bounded domain, $u \in L^{p}(\Omega, \wedge^{l})$ be a solution of the non-homogeneous A-harmonic equation, T be the homotopy operator and H be the projection operator. If $\varphi (|u|) \in L^{1}_{\mathrm{loc}}(\Omega)$, then there exists a constant C, independent of u, such that

$$\bigl\| TdTH(u)\bigr\| _{L^{\varphi}( B)} \leq C |B| \|u \|_{L^{\varphi}(\sigma B)} $$

for all balls B with $\sigma B \subset\Omega$, where $\sigma>1$ is a constant.

Proof

For any differential form u, we have $\|dT(u)\| _{q, B} = \|u_{B}\|_{q, B} \leq C_{1} \|u\|_{q, B}$. From (1.3) and (2.13), it follows that

$$\begin{aligned} \bigl\| TdTH(u) \bigr\| _{q, B} & \leq C_{2} |B| \operatorname{diam}(B) \bigl\| dTH(u) \bigr\| _{q, B} \\ &= C_{2} |B| \operatorname{diam}(B) \bigl\| H(u) \bigr\| _{q, B} \\ &= C_{3} |B| \operatorname{diam}(B) \|u \|_{q, B}. \end{aligned}$$

(2.21)

By Lemma 2.4 and $p, q >0$, we have

$$ \|u \|_{q, B} \leq C_{4} |B|^{(p-q)/pq} \|u \|_{p, \sigma B}, $$

(2.22)

where $\sigma>1$. Combining (2.21) and (2.22) yields

$$ \bigl\| TdTH(u) \bigr\| _{q, B} \leq C_{5} |B|^{(p-q)/pq} \|u \|_{p, \sigma B}. $$

(2.23)

From the Hölder inequality with $1= \frac{q}{n+q}+ \frac{n}{{n+q}}$, we find that

$$\begin{aligned} &\int_{B} \varphi\bigl(\bigl|TdTH(u)\bigr| \bigr)\,dx \\ &\quad= \int_{B} \varphi\bigl(\bigl|TdTH(u)\bigr| \bigr) \bigl(\bigl|TdTH(u)\bigr| \bigr)^{-\frac{nq}{{n+q}}} \bigl(\bigl|TdTH(u)\bigr| \bigr)^{\frac{nq}{{n+q}}}\,dx \\ &\quad\leq\biggl( \int_{B} \frac{{\varphi(|TdTH(u)|)}^{\frac {n+q}{q}}}{|TdTH(u)|^{n}}\,dx \biggr)^{\frac{q}{{n+q}}} { \biggl( \int_{B} \bigl|TdTH(u)\bigr|^{q} \,dx \biggr)}^{\frac{n}{{n+q}}}. \end{aligned}$$

Using Lemma 2.3 and noticing $A(t)$ is a concave function, we obtain

$$\begin{aligned} &\int_{B} \varphi\bigl(\bigl|TdTH(u) \bigr| \bigr)\,dx \\ &\quad\leq\biggl(\int_{B} K\bigl(\bigl|TdTH(u) \bigr|^{q} \bigr)\,dx \biggr)^{\frac{q}{n +q}} {\biggl( \int_{B} \bigl|TdTH(u)\bigr|^{q} \,dx \biggr)}^{\frac{n}{{n+q} }} \\ &\quad\leq\biggl(\int_{B} A\bigl(\bigl|TdTH(u) \bigr|^{q} \bigr)\,dx \biggr)^{\frac{q}{n +q}} { \biggl( \int_{B} \bigl|TdTH(u)\bigr|^{q} \,dx \biggr)}^{\frac{n}{{n+q}}} \\ &\quad\leq A^{\frac{q}{n +q}} \biggl(\int_{B} \bigl(\bigl|TdTH(u)\bigr|^{q}\bigr)\,dx \biggr) { \biggl( \int_{B} \bigl|TdTH(u)\bigr|^{q} \,dx \biggr)}^{\frac{n}{{n+q} }} \\ &\quad\leq C_{6} K^{\frac{q}{n +q}} \biggl(\int_{B} \bigl(\bigl|TdTH(u) \bigr|^{q}\bigr)\,dx \biggr) { \biggl( \int _{B} \bigl|TdTH(u)\bigr|^{q} \,dx \biggr)}^{\frac{n}{{n+q} }} \\ &\quad= C_{6} \frac{\varphi( (\int_{B} (\bigl|TdTH(u) \bigr|^{q})\,dx )^{1/q} )}{ (\int_{B} (|TdTH(u)|^{q})\,dx )^{\frac{n}{n+q}}} { \biggl( \int_{B} \bigl|TdTH(u)\bigr|^{q} \,dx \biggr)}^{\frac{n}{{n+q}}} \\ &\quad= C_{6} \varphi\biggl( \biggl(\int_{B} \bigl(\bigl|TdTH(u) \bigr|^{q}\bigr)\,dx \biggr)^{1/q} \biggr). \end{aligned}$$

(2.24)

Since φ is increasing and satisfies $\Delta_{2}$-condition, substituting (2.23) into (2.24), we have

$$ \int_{B} \varphi\bigl(\bigl|TdTH(u)\bigr| \bigr)\,dx \leq C_{7} \varphi\biggl( \biggl( \int_{ \sigma B} |u|^{p} \,dx \biggr)^{1/p} \biggr). $$

(2.25)

Starting from (2.25) and using the same discussion as we did in the proof of Theorem 2.6, we obtain

$$\int_{B} \varphi\bigl(\bigl|TdTH(u)\bigr| \bigr)\,dx \leq C \int _{\sigma B} \varphi\bigl(|u|\bigr)\,dx. $$

We have completed the proof of Theorem 2.7. □

From (1.3) and (2.13), we have

$$\begin{aligned} \bigl\| \nabla TdTH(u) \bigr\| _{q, B} &\leq C_{1} |B| \bigl\| dTH(u)\bigr\| _{q, B} \\ &= C_{1} |B| \bigl\| \bigl(H(u)\bigr)_{B}\bigr\| _{q, B} \\ & \leq C_{2} |B| \bigl\| H(u)\bigr\| _{q, B} \\ & \leq C_{3} |B| \|u\|_{q, B}. \end{aligned}$$

(2.26)

Using (2.26) and the similar techniques to the ones developed in the proof of Theorem 2.7, we obtain the following $L^{\varphi}$-norm estimate.

Theorem 2.8

Let φ be a Young function in the class $NG(p, q)$ with $q(n-p)< np$, $1< p \leq q < \infty$, Ω be a bounded domain, and $u \in L^{p}(\Omega, \wedge^{l})$ be a solution of the non-homogeneous A-harmonic equation, T be the homotopy operator and H be the projection operator. If $\varphi (|u|) \in L^{1}_{\mathrm{loc}}(\Omega)$, then there exists a constant C, independent of u, such that

$$ \bigl\| \nabla TdTH(u) \bigr\| _{L^{\varphi}( B)} \leq C |B| \|u \|_{L^{\varphi}( \sigma B)} $$

(2.27)

for all balls B with $\sigma B \subset\Omega$, where $\sigma>1$ is a constant.

Proof of Theorem 2.5

By definition (2.3), Theorem 2.7 and Theorem 2.8, we have

$$\begin{aligned} &\bigl\| T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr) \bigr)_{B} \bigr\| _{W^{1, \varphi} (B, \wedge^{l})} \\ &\quad= \bigl\| TdTH(u) \bigr\| _{W^{1, \varphi} (B, \wedge^{l})} \\ &\quad= \operatorname{diam} (B)^{-1} \bigl\| TdTH(u) \bigr\| _{L^{\varphi}(B)} + \bigl\| \nabla TdTH(u) \bigr\| _{L^{\varphi}(B)} \\ &\quad\leq\operatorname{diam} (B)^{-1} C_{1} |B| \operatorname{diam} (B) \|u \|_{L^{\varphi}( \sigma_{1} B)} + C_{2} |B| \|u \|_{L^{\varphi}(\sigma_{2} B)} \\ &\quad\leq C_{3}|B| \|u \|_{L^{\varphi}(\sigma B)}, \end{aligned}$$

(2.28)

where $\sigma=\max\{ \sigma_{1}, \sigma_{2} \}$. We have completed the proof of Theorem 2.5. □

3 Global embedding theorem

In this section, we prove our main result, the global $L^{\varphi}$-embedding theorem for the solutions of the non-homogeneous A-harmonic equation. We will use the following well-known covering lemma.

Lemma 3.1

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

$$\bigcup_{i} Q_{i} = \Omega,\qquad \sum _{Q_{i} \in\mathcal {V}} \chi_{\sqrt{\frac{5}{4}}Q_{i}} \leq N \chi _{\Omega}$$

and some $N>1$, and if $Q_{i} \cap Q_{j} \neq\emptyset$, then there exists a cube R (this cube need not be a member of $\mathcal {V}$) 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 $Q_{0}, Q_{1}, \ldots, Q_{k} = Q$ from $\mathcal {V}$ and such that $Q \subset\rho Q_{i}$, $i = 0, 1, 2, \ldots, k$, for some $\rho= \rho(n, \delta)$.

We are ready to prove the following global $L^{\varphi}$-embedding theorem with the $L^{\varphi}$-norm now.

Theorem 3.2

Let φ be a Young function in the class $NG(p, q)$ with $q(n-p)< np$, $1< p \leq q < \infty$, $u \in L^{p}(\Omega, \wedge^{l})$ be a solution of the non-homogeneous A-harmonic equation, T be the homotopy operator and H be the projection operator. If $\varphi(|u|) \in L^{1}(\Omega)$, then there exists a constant C, independent of u, such that

$$ \bigl\| T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{\Omega} \bigr\| _{W^{1, \varphi} (\Omega, \wedge^{l})} \leq C \|u \|_{L^{\varphi}(\Omega)} $$

(3.1)

for a bounded domain $\Omega\subset \mathbb{R}^{n}$.

Proof

From the Lemma 3.1 and Theorem 2.8, we have

$$\begin{aligned} \bigl\| \nabla Td \bigl(T\bigl(H(u)\bigr)\bigr) \bigr\| _{L^{\varphi}(\Omega)} & \leq\sum_{B \in\mathcal {V}} \bigl\| \nabla Td \bigl(T\bigl(H(u)\bigr) \bigr) \bigr\| _{L^{\varphi}(B)} \\ & \leq\sum_{B \in\mathcal {V}} \bigl( C_{1} |B| \|u \|_{L^{\varphi}(\sigma B)} \bigr) \\ & \leq C_{2} N \|u\|_{L^{\varphi}(\Omega)} \\ & \leq C_{3} \|u \|_{L^{\varphi}(\Omega)}. \end{aligned}$$

(3.2)

Similarly, from Theorem 2.7 and Lemma 3.1, it follows that

$$\begin{aligned} \bigl\| Td \bigl(T\bigl(H(u)\bigr)\bigr) \bigr\| _{L^{\varphi}(\Omega)} & \leq\sum _{B \in\mathcal{V}} \bigl\| Td \bigl(T\bigl(H(u)\bigr)\bigr) \bigr\| _{L^{\varphi}(B)} \\ & \leq\sum_{B \in\mathcal{V}} \bigl( C_{4} \operatorname{diam} (B) \|u \|_{L^{\varphi}(\sigma B)} \bigr) \\ & \leq C_{5} \operatorname{diam} (\Omega) N \|u\|_{L^{\varphi}(\Omega)} \\ & \leq C_{6} \operatorname{diam} (\Omega) \|u \|_{L^{\varphi}(\Omega)}. \end{aligned}$$

(3.3)

Using (2.3), (3.2), and (3.3), we find that

$$\begin{aligned} & \bigl\| T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr) \bigr)_{\Omega}\bigr\| _{W^{1, \varphi} (\Omega)} \\ & \quad= \bigl\| Td\bigl(T\bigl(H(u)\bigr)\bigr) \bigr\| _{W^{1, \varphi} (\Omega)} \\ &\quad= \bigl(\operatorname{diam}(\Omega)\bigr)^{-1} \bigl\| Td \bigl(T \bigl(H(u)\bigr)\bigr) \bigr\| _{L^{\varphi}(\Omega)} + \bigl\| \nabla Td \bigl(T\bigl(H(u)\bigr) \bigr) \bigr\| _{L^{\varphi}(\Omega)} \\ &\quad\leq\bigl(\operatorname{diam}(\Omega)\bigr)^{-1} \bigl(C_{6} \operatorname{diam} (\Omega) \|u \|_{L^{\varphi}(\Omega)} \bigr) + C_{3} \|u \|_{L^{\varphi}(\Omega)} \\ &\quad\leq C_{7} \|u \|_{L^{\varphi}(\Omega)}. \end{aligned}$$

(3.4)

We have completed the proof of Theorem 3.2. □

Choosing $\varphi(t) = t^{p} \log_{+}^{\alpha}t$ in Theorems 3.2, we have the following embedding inequality with the $L^{p}(\log_{+}^{\alpha}L)$-norms.

Corollary 3.3

Let $\varphi(t) = t^{p} \log_{+}^{\alpha}t$, $p \geq1$, $\alpha\in \mathbb{R}$, $u \in L^{p}(\Omega, \wedge^{l})$ be a solution of the non-homogeneous A-harmonic equation, T be the homotopy operator, and H be the projection operator. If $\varphi (|u|) \in L^{1}(\Omega)$, then there exists a constant C, independent of u, such that

$$ \bigl\| T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{\Omega}\bigr\| _{W^{1, t^{p} \log_{+}^{\alpha}t} (\Omega)} \leq C \|u \|_{L^{t^{p} \log _{+}^{\alpha}t} (\Omega)} $$

(3.5)

for any bounded domain Ω.

Let $\varphi(t) = t^{p}$ in Theorem 3.2. Then, we obtain the following version of the embedding inequality with $L^{p}$-norms.

Corollary 3.4

Let $\varphi(t) = t^{p}$, $p \geq1$, $u \in L^{p}(\Omega, \wedge^{l})$ be a solution of the non-homogeneous A-harmonic equation, T be the homotopy operator, and H be the projection operator. If $\varphi(|u|) \in L^{1}(\Omega)$, then there exists a constant C, independent of u, such that

$$\bigl\| T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{\Omega}\bigr\| _{W^{1, p} (\Omega)} \leq C \|u \|_{p, \Omega} $$

holds for any bounded domain Ω.

Similarly, Theorem 2.6 can be extended into the following global Poincaré-type inequality with $L^{\varphi}$-norm.

Theorem 3.5

Let φ be a Young function in the class $NG(p, q)$ with $q(n-p)< np$, $1< p \leq q < \infty$, $u \in L^{p}(\Omega, \wedge^{l})$ be a solution of the non-homogeneous A-harmonic equation, T be the homotopy operator and H be the projection operator. If $\varphi(|u|) \in L^{1}(\Omega)$, then there exists a constant C, independent of u, such that

$$ \bigl\| T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{\Omega}\bigr\| _{L^{\varphi}(\Omega)} \leq C \operatorname{diam}(\Omega) \|u \| _{L^{\varphi}(\Omega)} $$

(3.6)

for any bounded domain Ω.

Proof

From (2.3) and Theorem 3.2, we have

$$\begin{aligned} & \bigl\| T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{\Omega}\bigr\| _{L^{\varphi}(\Omega)} \\ &\quad= \operatorname{diam}(\Omega) \bigl( \bigl(\operatorname {diam}(\Omega) \bigr)^{-1} \bigl\| T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr) \bigr)_{\Omega}\bigr\| _{L^{\varphi}(\Omega)} \bigr) \\ &\quad \leq\operatorname{diam}(\Omega) \bigl( \bigl(\operatorname {diam}(\Omega) \bigr)^{-1} \bigl\| T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr) \bigr)_{\Omega}\bigr\| _{L^{\varphi}(\Omega)} \\ &\qquad{}+ \bigl\| \nabla\bigl(T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{\Omega}\bigr)\bigr\| _{L^{\varphi}(\Omega)} \bigr) \\ &\quad =C_{1} \operatorname{diam} (\Omega) \bigl\| T\bigl(H(u)\bigr)- \bigl(T \bigl(H(u)\bigr)\bigr)_{\Omega} \bigr\| _{W^{1, \varphi} (\Omega, \wedge^{l})} \\ &\quad\leq C_{2} \operatorname{diam}(\Omega) \|u \|_{L^{\varphi}(\Omega)}. \end{aligned}$$

We have completed the proof of Theorem 3.5. □

4 Applications

As applications of our main results established in the previous sections, we consider the following examples.

Example 4.1

Assume that $r>0$ and $k>0$ are any constants and $\Omega= \{ (x_{1}, x_{2}, x_{3}): x_{1}^{2} + x_{2}^{2} + x_{3}^{2} \leq r^{2}\} \subset \mathbb{R}^{3}$. Consider the 1-form

$$ u(x_{1}, x_{2}, x_{3}) = \frac{x_{1}}{ k + x_{1}^{2} + x_{2}^{2} + x_{3}^{2}}\,dx_{1} +\frac {x_{2}}{k + x_{1}^{2} + x_{2}^{2} + x_{3}^{2}}\,dx_{2} +\frac{x_{3}}{ k + x_{1}^{2} + x_{2}^{2} + x_{3}^{2}}\,dx_{3} $$

(4.1)

defined in Ω. It is easy to check that $du=0$. Hence, u is a solution of the non-homogeneous A-harmonic equation (1.4) for any operators A and B satisfying (1.5). Also, it can be calculated that

$$\begin{aligned} |u| &= \biggl( \biggl( \frac{x_{1}}{k + x_{1}^{2} + x_{2}^{2} + x_{3}^{2}} \biggr)^{2} + \biggl( \frac{x_{2}}{k + x_{1}^{2} + x_{2}^{2} + x_{3}^{2}} \biggr)^{2} + \biggl( \frac{x_{3}}{k + x_{1}^{2} + x_{2}^{2} + x_{3}^{2}} \biggr)^{2} \biggr)^{1/2} \\ &= \biggl( \frac{x_{1}^{2} + x_{2}^{2} + x_{3}^{2} }{(k + x_{1}^{2} + x_{2}^{2} + x_{3}^{2})^{2} } \biggr)^{1/2} < 1. \end{aligned}$$

(4.2)

Using (4.2) and (3.1), we find that

$$\bigl\| T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{\Omega}\bigr\| _{W^{1,\varphi}(\Omega)} \leq C_{1}\|u\| _{L^{\varphi}(\Omega)} \leq C_{2} \|1\|_{L^{\varphi}(\Omega)}, $$

that is,

$$\bigl\| T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{\Omega}\bigr\| _{W^{1,\varphi}(\Omega)} \leq C_{2}\int_{\Omega}{\varphi \biggl(\frac{1}{\lambda} \biggr)\,dx} \leq C_{3} r^{3}. $$

We should notice that the above example can be extended to the case of $\mathbb{R}^{n}$. Specifically, we can check that the 1-form defined in $\mathbb{R}^{n}$

$$ u(x_{1}, \ldots,x_{n}) = \sum_{i=1}^{n} \frac{x_{i}}{ k + x_{1}^{2} + \cdots+ x_{n}^{2}}\,dx_{i},\quad k>0 $$

(4.3)

is a solution of the non-homogeneous A-harmonic equation (1.4) for any operators A and B satisfying (1.5). Hence, Theorem 3.2 is applicable to $u(x_{1}, \ldots,x_{n})$ as we did in Example 4.1. Finally, we consider the following example in $\mathbb{R}^{3}$.

Example 4.2

Let $\Omega= \{ (x, y, z): x^{2} +y^{2}+z^{2} \leq1 \} \subset \mathbb{R}^{3}$ and $u(x, y, z)$ be defined in $\mathbb{R}^{3}$ by

$$u(x, y, z)= e^{x^{2} +y^{2} +z^{2}} (x\,dx +y\,dy + z\,dz ). $$

It is easy to check that $du=0$. Hence, u is a solution of the non-homogeneous A-harmonic equation (1.4) for any operators A and B satisfying (1.5) in $\mathbb{R}^{3}$. For any bounded domain Ω in $\mathbb{R}^{3}$, it would be very complicated if we calculate the integral $\|T(H(u))- (T(H(u)))_{\Omega}\|_{W^{1,\varphi}(\Omega)}$ directly. However, using the embedding inequality (3.1), we can easily obtain the upper bound of the norm $\|T(H(u))- (T(H(u)))_{\Omega}\|_{W^{1,\varphi}(\Omega)}$ as follows. We notice that

$$\begin{aligned} \bigl|u(x, y, z)\bigr| & = \bigl( \bigl(e^{x^{2} +y^{2} +z^{2}} x \bigr)^{2} + \bigl(e^{x^{2} +y^{2} +z^{2}} y\bigr)^{2} + \bigl(e^{x^{2} +y^{2} +z^{2}} z\bigr)^{2} \bigr)^{1/2} \\ &= \bigl(e^{2(x^{2} +y^{2} +z^{2})} \bigl(x^{2} +y^{2} +z^{2}\bigr) \bigr)^{1/2} \\ &\leq\bigl( e^{2} \cdot1 \bigr)^{1/2} = e. \end{aligned}$$

(4.4)

From (4.4) and (3.1), we have

$$\bigl\| T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{\Omega}\bigr\| _{W^{1,\varphi}(\Omega)} \leq C_{1}\|u\| _{L^{\varphi}(\Omega)} \leq C_{2} \|e\|_{L^{\varphi}(\Omega)}, $$

which is equal to

$$\bigl\| T\bigl(H(u)\bigr)- \bigl(T\bigl(H(u)\bigr)\bigr)_{\Omega}\bigr\| _{W^{1,\varphi}(\Omega)} \leq C_{2}\int_{\Omega}{\varphi \biggl(\frac{e}{\lambda} \biggr)\,dx} \leq C_{3} . $$

Remark

(i) From inequalities (3.1), (3.2), and (3.3), we find that the compositions TH, $TdTH$ and $\nabla TdTH$ are bounded operators on $L^{p}(\Omega, \wedge^{l})$. (ii) Note that our global embedding theorem holds on any bounded domains. Hence, the theorem is true if Ω is one of the bounded John domains, $L^{p}$-averaging domains or $L^{\varphi}(\mu)$-averaging domains. See [1, 22, 23] for more properties of these kinds of domains.