Research  Open  Published:
$L^{\varphi}$Embedding inequalities for some operators on differential forms
Journal of Inequalities and Applicationsvolume 2015, Article number: 392 (2015)
Abstract
In this paper, the local Poincaré inequality and embedding inequality are proved first. Then the global embedding inequality of composite operators for differential forms on $L^{\varphi}$averaging domains with $L^{\varphi}$norm is established. Some examples are also given to illustrate applications.
Introduction
The purpose of this paper is to establish the global Sobolev embedding inequality with $L^{\varphi}$norm for the composite operators applied to differential forms on $L^{\varphi}$averaging domains. As extensions of functions, differential forms have been deeply studied and widely applied in many fields, such as global analysis, nonlinear analysis, PDEs and differential geometry; see [1–10] for more information. The Sobolev embedding inequality, as a fundamental tool in the Sobolev space of functions, has been very well studied in [11]. The homotopy operator T, Dirac operator D, and Green’s operator G are three key operators acting on differential forms and each of them has received much attention recently. However, the compositions of these operators are so complicated that the results concerned on them are quite few. Even though they are very complicated, the study for the composite operators has been increasing during the recent years; see [6–9, 12]. In many cases, we need to study or to use the compositions of operators. For example, we have to deal with the composition of the Dirac operator D and Green’s operator G when we consider the Poisson equation $D^{2}(G(u)) = u H(u)$, where $D^{2}= \Delta$ and H is the projection operator. In 2013, Ding and Liu initiated the study of the composition $D\circ G$ and obtained some basic inequalities in [2]. During our recent investigation of the operator theory of differential forms, we realized that for applying the decomposition theorem to $D\circ G$, it is necessary to deal with the composition $T\circ D \circ G$. Hence, we are motivated to study the composition $T\circ D \circ G$. We obtain some basic estimates for $T\circ D \circ G$ and establish the global Sobolev embedding inequality with $L^{\varphi}$norm
where E is an $L^{\varphi}$averaging domain [13].
For convenience, we keep using the traditional notations and terminologies. Except for special instructions, $E\subseteq\mathbb {R}^{n}$ is a bounded domain, and $E$ denotes the Lebesgue measure of E, $n\geq2$. Suppose that $B_{x}^{r}$ is a ball with a radius r, centered at x. For any $\sigma>0$, $B\subseteq E$ and $\sigma B \subseteq E$ have the same center and satisfy diam $(\sigma B)=\sigma \operatorname{diam}(B)$. Let $\Lambda^{l}(\mathbb{R}^{n})$ be the space of all lforms in $\mathbb{R}^{n}$, which is expanded by the exterior product of $e^{\mathfrak{B}}=e^{i_{1}}\wedge e^{i_{2}} \wedge\cdots \wedge e^{i_{l}}$, where ${\mathfrak{B}}=(i_{1},\ldots, i_{l})$, $1\leq i_{1}< \cdots<i_{l} \leq n$, $l=1,2,\ldots,n$. $C^{\infty}(\Lambda^{l}E)$ is the space of a smooth lform on E. We use $D'(E,\Lambda^{l})$ to denote the space of all differential lforms on E, that is, $u(x)$ belongs to $D'(E,\Lambda ^{l}) $ if and only if there exist some lthdifferential functions $u_{\mathfrak{B}}$ in E such that $u(x)= \sum_{\mathfrak{B}} u_{\mathfrak{B}}(x)\,dx_{\mathfrak{B}} = \sum u_{i_{1}i_{2} \cdots i_{l}}(x) \,dx_{i_{1}}\wedge dx_{i_{2}}\wedge\cdots\wedge dx_{i_{l}} $. $L^{p}(E,\Lambda^{l})$ is a Banach space with the norm equipped by $\ u \_{p,E}= (\int_{E}u(x)^{p}\,dx )^{1/p} $, where $u(x)\in D'(E, \Lambda^{l})$ and every coefficient function $u_{\mathfrak{B}}\in L^{p}(E)$, $0< p< \infty$. In fact, $u(x)$ on E is the Schwarz distribution. If $w(x)> 0$ a.e. and $w(x)\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{m})$, $w(x)$ is called a weight. Let $d\mu =w(x)\,dx$, then $L^{p}(E,\Lambda^{l},w)$ is a weighted Banach space with the norm expressed by $\ u(x) \_{p,E,w}= (\int_{E}u(x)^{p}w(x)\,dx )^{1/p}$. In this notation, the exterior derivative is denoted by d and the Hodge codifferential operator is expressed by $d^{\star}$. Refer to [1] for more details. Moreover, the Dirac operator, designated by D, was initially set forth by Paul Dirac, and we get its extensions, such as the HodgeDirac operator and the Euclidean Dirac operator. In this paper, the Dirac operator we choose is the HodgeDirac operator $D=d^{*}+d$. The Green’s operator is a bound selfadjoint linear operator. It is commonly used to define the Poisson equation for differential forms; here it is represented by G. The homotopy operator T [10], advanced by Iwaniec, is defined on $C^{\infty}(D,\Lambda^{l})$ and constructed as follows:
where the linear operator $K_{y}$ is denoted by $(K_{y}u)(x;\xi _{1},\ldots, \xi_{l1})=\int_{0}^{1}t^{l1}u(tx+yty;xy,\xi_{1},\ldots , \xi_{l1})\, dt$, and ψ from $C^{\infty}_{0}(D)$ is normalized so that $\int\psi(y)\, dy=1$.
From [10], we obtain some good results. For any differential form u, the decomposition below holds:
Meanwhile, for $u_{\Omega}\in D'(\Omega, \Lambda^{l})$, we define
Then we know that, for any differential form $u\in L^{s}_{\mathrm{loc}}(B,\Lambda ^{l})$, $l=1,\ldots,n$, $1< s<\infty$,
holds for any ball B in Ω.
Concerning the homotopy operator T, according to [1], we can obtain a nice property. That is, $\Omega\subset\mathbb{R}^{n}$ is the union of a collection of cube $\Omega_{k}$, 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 $\mathbb{R}^{n}$. In detail, $\Omega \subset\mathbb{R}^{n}$ can be expressed as follows:

(1)
$\Omega=\bigcup^{\infty}_{k=1}\Omega_{k}$;

(2)
$\Omega_{i}^{0}\cap\Omega_{j}^{0}=\emptyset$, if $j\neq i$;

(3)
there exist two constants $C_{1}>0$ and $C_{2}>0$, such that
$$ C_{1}\operatorname{diam}(\Omega_{k})\leq \operatorname{distance}(\Omega_{k},F) \leq C_{2} \operatorname{diam}(\Omega_{k}). $$
Based on this fact, the homotopy operator T can be extended to any domain $\Omega\subset\mathbb{R}^{n}$. For any $x\in\Omega$, $x\in \Omega_{k}$ for some k, let $T_{\Omega_{k}}$ be the homotopy operator defined on $\Omega_{k}$, which is the convex and bounded set. So, the following definition for T on any domain Ω holds:
where $\chi_{\Omega_{k}}$ is the characteristic function defined on Ω.
Further, it is essential to recall some wellknown results as regards the Aharmonic equation for differential forms, which appeared in [1]. To be more precise, we here consider the nonhomogeneous Aharmonic equation as follows:
where $A: E \times\wedge^{l}(\mathbb{R}^{n}) \to \wedge^{l}(\mathbb{R}^{n})$ and $B: E \times\wedge^{l}(\mathbb{R}^{n}) \to \wedge^{l1}(\mathbb{R}^{n})$ satisfy the conditions
for almost every $x \in E$ and all $\xi\in\wedge^{l} (\mathbb{R}^{n})$. Meanwhile, the parameters $a, b >0$ and $1< p< \infty$ associated with (1.4) are constants and a fixed exponential, respectively. If $B=0$, the equation $d^{\star} A(x, du)=0$ is called a homogeneous Aharmonic equation.
In addition, for proving the global embedding inequality on the $L^{\varphi}$averaging domain, we also need the following definitions and notations.
A function $\varphi(x)$ is called an Orlicz function, if $\varphi(x)$ satisfies: (1) $\varphi(x)$ is continuously increasing; (2) $\varphi(0)=0$. Furthermore, if the Orlicz function $\varphi(x)$ is a convex function, $\varphi(x)$ is named a Young function. Therefore, based on this type of functions, the Orlicz norm for differential forms can be denoted as follows.
Take $\varphi(x)$ as a Orlicz function, and $E\subset\mathbb{R}^{n}$ as a bounded domain, for any $u(x)\in L^{p}_{\mathrm{loc}}(E,\Lambda^{l})$, $l=0,1,2,\ldots, n$, the Orlicz norm for the differential form is equipped with
where the measure μ is expressed by $d\mu=w(x)\,dx$, $w(x)$ is a weight.
It is easy to prove that $L^{\varphi}_{E}$ is a Banach space. Actually, provided that $\varphi(x)$ is taken as $\varphi(x)=x^{s}$ ($s>0$), then $\varphi(x)$ is an Orlicz function, and it trivially corresponds to a $L^{s}(\mu)$ space. As a result of that, we can say that $L^{\varphi}_{E}$ is the generalization of $L^{s}_{E}$.
Similarly, we intend to give $L^{\varphi}$averaging domains. Based on the Orlicz norm above, the OrliczSobolev space of lform is denoted by $W^{1,\varphi}(E,\Lambda^{l})$, with the norm
From the definition of $W^{1,\varphi}_{E}$, we know that $W^{1,\varphi }_{E}$ is equal to $L^{\varphi}_{E}\cap L^{\varphi}_{1,E}$. In detail, if $\varphi(x)$ is expressed by $\varphi(x)=t^{s}$ ($s>0$), we can conclude the norm of $W^{1,s}_{E}$ for the differential form:
Besides that, we will show another definition, which was initially introduced by Ding.
Definition 1.1
[13]
Let $\varphi(x)$ be a Young function, the proper domain $E\subseteq\mathbb{R}^{n}$ is called the $L^{\varphi}$averaging domain, if $\mu(E)<\infty$ and there exists a constant $C>0$ such that for any $B_{0}\subseteq E$ and $\varphi(u)\in L^{1}_{\mathrm{loc}}(E,\mu)$, u satisfies
where the measure μ is denoted by $d\mu=w(x)\,dx$, $w(x)$ is a weight, σ and τ are constants with $0<\tau,\sigma<1$, and the supremum is over all balls $B\subset E$ with $4B \subset E$.
Using the same analysis method as of the $L^{\varphi}_{E}$norm, we can conclude that $L^{\varphi}$averaging domains are the generalization of $L^{s}$averaging domains.
Main results
Before the main results given, we will make some restrictions to the Young function $\varphi(x)$. Here, we let the Young function $\varphi (x)$ belong to the $G(p,q,C)$class ($1\leq p < q<\infty$, $C\geq1$), that is, for any $t>0$, the Young function $\varphi(x)$ satisfies:

(1)
$1/C \leq{\varphi(t^{1/p})}/{f(t)}\leq C$;

(2)
$1/C\leq{\varphi(t^{1/q})}/{g(t)}\leq C$,
where f and g are increasingly convex and concave functions defined on $[0,\infty]$, respectively.
Now, we establish four important theorems based on the abovementioned conditions.
Theorem 2.1
Let T be the homotopy operator, D be the Dirac operator, and G be the Green’s operator. Meanwhile, we assume that $\varphi(u)\in L^{1}_{\mathrm{loc}}(E)$, $u\in C^{\infty}(\Lambda^{l}E)$ is a solution of the nonhomogeneous Aharmonic equation, the Young function imposed a doubling property $\varphi(x)$ belongs to the $G(p, q, C)$class, and the bounded subset $E\subseteq\mathbb{R}^{n}$ is the $L^{\varphi}$averaging domain. Then, for any ball $B\subseteq E$, we get
where $\sigma B \subseteq E$ and $\sigma>1$ is a constant.
Theorem 2.2
Let T be the homotopy operator, D be the Dirac operator, and G be the Green’s operator. Meanwhile, we assume that $\varphi(u)\in L^{1}_{\mathrm{loc}}(E)$, $u\in C^{\infty}(\Lambda^{l}E)$, is a solution of the nonhomogeneous Aharmonic equation, the Young function $\varphi(x)$ having imposed a doubling property belongs to the $G(p, q, C)$class and the bounded subset $E\subseteq\mathbb{R}^{n}$ is the $L^{\varphi}$averaging domain. Then, for any ball $B\subseteq E$, we get
where $\sigma B \subseteq E$ and $\sigma>1$ is a constant.
Based on the above theorem, we cannot only establish the following global embedding inequality on $L^{\varphi}$averaging domains, but we also get the global Poincarétype inequality fortunately.
Theorem 2.3
Let T be the homotopy operator, D be the Dirac operator, and G be the Green’s operator. Additionally, we assume that $\varphi(u)\in L^{1}_{\mathrm{loc}}(E)$, $u\in C^{\infty}(\Lambda^{l}E)$, is a solution of the nonhomogeneous Aharmonic equation, the Young function $\varphi(x)$ having imposed a doubling property belongs to the $G(p, q, C)$class and the bounded subset $E\subseteq\mathbb{R}^{n}$ is the $L^{\varphi}$averaging domain. Then, for any ball $B\subseteq E$, we have
Theorem 2.4
Let T be the homotopy operator, D be the Dirac operator, and G be the Green’s operator. Additionally, we assume that $\varphi(u)\in L^{1}_{\mathrm{loc}}(E)$, $u\in C^{\infty}(\Lambda^{l}E)$, is a solution of the nonhomogeneous Aharmonic equation, the Young function $\varphi(x)$ having imposed a doubling property belongs to the $G(p, q, C)$class and the bounded subset $E\subseteq\mathbb{R}^{n}$ is the $L^{\varphi}$averaging domain. Then, for any ball $B\subseteq E$, we have
where $B_{0}\subseteq B $ is a fixed ball.
Preliminary results
For proving the theorems in Section 2, we shall show and demonstrate some lemmas in this section.
Lemma 3.1
[1]
Let $0< p,q<\infty$, and $\frac {1}{t}=\frac{1}{p}+\frac{1}{q}$, if f and g are the measurable functions defined on $\mathbb{R}^{n}$, then
for any $I\subseteq\mathbb{R}^{n}$.
The inequality in Lemma 3.1 is actually the generalized Hölder inequality. Specifically, if $t=1$ and $1< p,q<\infty$, the inequality above is the classical Hölder inequality.
Lemma 3.2
[1]
Let u be a solution of the nonhomogeneous Aharmonic equation in a domain Ω, and $0< s,t<\infty$, then there exists a constant C, independent of u, such that
for all balls B with $\sigma B \subset E$, where $\sigma>1$ is some constant.
In fact, we can get a valuable result; if u satisfies the inequality in Lemma 3.2, we say u belongs to the WRHclass.
Lemma 3.3
[2]
Suppose that the differential form $u\in L^{p}(E,\Lambda^{l})$, $l=1,\ldots,n$, and G is the Green’s operator, then there exists a constant C, independent of u, such that
for any $B\subset\Omega$.
Because $\triangle=d^{*}d+dd^{*}$ and $D=d^{*}+d$, by using Lemma 3.3, it implies that
Lemma 3.4
[10]
If $u\in L^{p}(D, \Lambda^{l})$, $1 < p < \infty$, then $u_{D}\in L^{p}(D, \Lambda^{l})$ and
According to the above results, we can prove a very useful lemma as follows.
Lemma 3.5
Let T be the homotopy operator, D be the Dirac operator, and G be the Green’s operator. Additionally, we assume that $u\in C^{\infty}(\Lambda^{l}E)$ is a solution of the nonhomogeneous Aharmonic equation, the Young function $\varphi(x)$ belongs to the $G(p, q, C)$class, and the bounded subset $E\subseteq\mathbb{R}^{n}$ is the $L^{\varphi}$averaging domain. Then, for any ball $B\subseteq E$, we have
for any $B\subset E$, where $s>1$.
Proof
Notice that $TDG(u)$ is at least differential 1form, therefore, by using (1.1) and replacing u by $TDG(u)$, we have
It follows from (1.2) that $dT(TDG(u))=(TDG(u))_{B}$, that is,
Applying (1.3), Lemma 3.3, and Lemma 3.4, we obtain
Thus, we finish the proof of this lemma. □
Note
By observing the proving process, we can get a valuable result as follows:
Lemma 3.6
Let T be the homotopy operator, D be the Dirac operator, and G be the Green’s operator. Additionally, we assume that $u\in C^{\infty}(\Lambda^{l}E)$ is a solution of the nonhomogeneous Aharmonic equation, the Young function $\varphi(x)$ having imposed a doubling property belongs to the $G(p, q, C)$class and the bounded subset $E\subseteq \mathbb{R}^{n}$ is the $L^{\varphi}$averaging domain. Then, for any ball $B\subseteq E$, we have
where the constant $\sigma>1$ and $\sigma B \subseteq E$.
Proof
From (1.3), we have
Because φ belongs to the $G(p,q,C)$class, we have
Since φ is doubling, according to (3.1) and Lemma 3.2, we have
Therefore, by using $B^{\frac{1}{n}}=C_{11}\operatorname{diam}(B)$, we can get the following result:
where $\sigma>1$.
This is the end of the proof of Lemma 3.6. □
Lemma 3.7
Let T be the homotopy operator, D be the Dirac operator, and G be the Green’s operator. Additionally, we assume that $\varphi(u)\in L^{1}_{\mathrm{loc}}(E)$ and $u\in C^{\infty}(\Lambda^{l}E)$ is a solution of the nonhomogeneous Aharmonic equation, the Young function $\varphi(x)$ having imposed a doubling property belongs to the $G(p, q, C)$class and E is a bounded domain. Then we get
for all balls B with $\sigma B \subset E$, where $\sigma>1$ is some constant.
Proof
According to the elementary inequality (1.3) of the homotopy operator T and Lemma 3.4, we have
Applying (3.3) and Lemma 3.3, we get
Using the same method as used in Lemma 3.6, the following inequality holds:
The proof of Lemma 3.7 is completed. □
Lemma 3.8
(Covering lemma) [1]
Each domain Ω has a modified Whitney cover of cubes ${\mathcal{V}} = \{Q_{i}\}$ such that
and some $N>1$, and if $Q_{i} \cap Q_{j} \neq \emptyset$, then there exists a cube R (this cube does not need to 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 δJohn, 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 $\rho= \rho(n, \delta)$.
Demonstration of main results
According to the above definitions and lemmas, we will prove four theorems in detail. First of all, let us prove Theorem 2.1.
Proof of Theorem 2.1
First of all, similar to the proof of Lemma 3.5, we have
This means that the key point is to prove the following inequality:
According to the properties of the $G(p,q,C)$class, we have
Because φ is doubling, using Lemma 3.5, we have
Combining (4.1) with (4.2) and using $B^{\frac {1}{n}}=C_{11}\operatorname{diam}(B)$ yield
As a result, we obtain
for any B with $\sigma B\subset E$ and any constant $\lambda>0$. It means that the following inequality on $L^{\varphi}_{E}$averaging domains holds:
The proof of this theorem is finished. □
Remark
From the proof of Theorem 2.1, it is easy to derive the following inequality:
We shall prove Theorem 2.2 by using Theorem 2.1 and Lemma 3.7.
Proof for Theorem 2.2
Using (1.1) repeatedly, we have
Combining (4.4) with (1.5), and applying Theorem 2.1 and Lemma 3.7, we have
where $\sigma=\max\{\sigma_{1}, \sigma_{2}\}$ and $\sigma>1$, for all balls B with $\sigma B \subset E$.
This is the end of the proof of Theorem 2.2. □
Proof for Theorem 2.3
From the covering lemma, Lemma 3.8, and Lemma 3.7, we have
Similarly, using the covering lemma, Lemma 3.8, and (4.3) implies
Thus, from (1.5), (4.5), and (4.6), we obtain
We have completed the proof of Theorem 2.3. □
Next, we will prove Theorem 2.4 by using Definition 1.1 and Theorem 2.1.
Proof of Theorem 2.4
According to Definition 1.1, we have
Because $\sup_{B\subseteq E}\int_{E}\varphi(u)\,dx$ does not depend on B, we obtain
Therefore, we have
We finish the proof of Theorem 2.4. □
In addition, we can obtain a global estimate about the composite operators using the same method as of Theorem 2.1.
Corollary 4.1
Let T be the homotopy operator, D be the Dirac operator, and G be the Green’s operator. Additionally, we assume that $u\in C^{\infty }(\Lambda^{l}E)$ is a solution of the nonhomogeneous Aharmonic equation, the Young function $\varphi(x)$ belongs to the $G(p, q, C)$class, and the bounded subset $E\subseteq\mathbb{R}^{n}$ is the $L^{\varphi}$averaging domain. Then, for any ball $B\subseteq E$, we have
Remark
If we choose $\varphi(x)=x^{s}$, we have
Applications
In this section, we will discuss the applications of the results obtained.
Example 5.1
Let $r>0$ 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 1form
defined in Ω. Then u is a solution of the nonhomogeneous Aharmonic equation for any operators A and B satisfying the required conditions.
As the application of the obtained theorems, our goal is to get the upper bound of $TDG(u)$ satisfying the above conditions. Normally, we would consider calculating the integral of $TDG(u)$, however, one will see that $TDG(u)$ with the $L^{\varphi}$norm is very complicated. In this case, we can use Theorem 2.3 to get the estimate of $TDG(u)$ with the $L^{\varphi}$norm in $W^{1,\varphi}_{\Omega}$.
Initially, according to the condition and the expression of u in Example 5.1, we see that
Furthermore, because u is defined in Ω and $(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})^{2}>0$, we get
As a result, by using Theorem 2.3 and (5.1), we have
The above example can be extended to the case of $\mathbb{R}^{n}$. Particularly, we can check that the 1form defined in $\mathbb{R}^{n}$,
is a solution of the nonhomogeneous Aharmonic equation for any operators A and B satisfying the required conditions. So, we can also apply Theorem 2.3 to this extended case as we did in Example 5.1.
To end this section, we take a 3form in $\mathbb{R}^{3}$ as an example.
Example 5.2
Let $k, m>0$ be constants. Consider a differential 3form in $\mathbb{R}^{3}$,
Then u is a solution of the nonhomogeneous Aharmonic equation for any operators A and B satisfying the required conditions.
Similarly, applying the same method, we can obtain the following result as regards $TDG(u)$ in any bounded set $\Omega\subset\mathbb{R}^{3} $:
and
References
 1.
Agarwal, RP, Ding, S, Nolder, C: Inequalities for Differential Forms. Springer, New York (2009)
 2.
Ding, S, Liu, B: Norm inequalities for composition of the Dirac and Green’s operator. J. Inequal. Appl. (2013). doi:10.1186/1029242X2013436
 3.
Wang, Y, Wu, C: Sobolev embedding theorems and Poincaré inequalities for Green’s operator on solution of the nonhomogeneous Aharmonic equation. Comput. Math. Appl. 47, 15451554 (2004)
 4.
Liu, B: $A^{\lambda}_{r}(\Omega)$Weighted embedding inequalities for Aharmonic tensors. J. Math. Anal. Appl. 273, 667676 (2002)
 5.
Xing, Y: Twoweight embedding inequalities for solutions to the Aharmonic equation. J. Math. Anal. Appl. 307, 555564 (2005)
 6.
Bi, H: Weighted inequalities for potential operators on differential forms. J. Inequal. Appl. 2010, Article ID 713625 (2010)
 7.
Bi, H, Ding, S: Some strong $(p, q)$type inequalities for the homotopy operator. Comput. Math. Appl. 62, 17801789 (2011)
 8.
Ding, S, Liu, B: A singular integral of the composite operator. Appl. Math. Lett. 22, 12711275 (2009)
 9.
Ding, S, Liu, B: Global estimates for singular integrals of the composite operator. Ill. J. Math. 53, 11731185 (2009)
 10.
Iwaniec, T, Lutoborski, A: Integral estimates for null Lagrangians. Arch. Ration. Mech. Anal. 125, 2579 (1993)
 11.
Adams, RA, Fourier, JJF: Sobolev Spaces, 2nd edn. Elsevier, Amsterdam (2003)
 12.
Scott, C: $L^{p}$Theory of differential forms on manifolds. Trans. Am. Math. Soc. 347, 20752096 (1995)
 13.
Ding, S: $L^{\varphi}(\mu)$Averaging domains and the quasihyperbolic metric. Comput. Math. Appl. 47, 16111618 (2004)
Acknowledgements
The authors express their deep appreciation to the referees’ efforts and suggestions, which highly improved the quality of this paper.
Author information
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All results and investigations of this manuscript were due to the joint efforts of all authors. GS mainly studied the theoretical proofs and drafted the manuscript. SD and YX conceived of the initial idea of this manuscript and improved the final version. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Received
Accepted
Published
DOI
Keywords
 homotopy operator
 Dirac operator
 Green’s operator
 $L^{\varphi}$averaging domains