Skip to content

Advertisement

  • Research
  • Open Access

\(L^{\varphi}\)-Embedding inequalities for some operators on differential forms

Journal of Inequalities and Applications20152015:392

https://doi.org/10.1186/s13660-015-0887-x

  • Received: 14 August 2015
  • Accepted: 6 November 2015
  • Published:

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.

Keywords

  • homotopy operator
  • Dirac operator
  • Green’s operator
  • \(L^{\varphi}\)-averaging domains

1 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 [110] 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 [69, 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
$$ \bigl\Vert TDG(u)-\bigl(TDG(u)\bigr)_{E}\bigr\Vert _{W^{1,\varphi}_{E}}\leq C\|u\|_{L^{1,\varphi}_{E}}, $$
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 l-forms 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 l-form on E. We use \(D'(E,\Lambda^{l})\) to denote the space of all differential l-forms on E, that is, \(u(x)\) belongs to \(D'(E,\Lambda ^{l}) \) if and only if there exist some lth-differential 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 Hodge-Dirac operator and the Euclidean Dirac operator. In this paper, the Dirac operator we choose is the Hodge-Dirac operator \(D=d^{*}+d\). The Green’s operator is a bound self-adjoint 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:
$$ Tu=\int_{D}\psi(y)K_{y}u\, dy, $$
where the linear operator \(K_{y}\) is denoted by \((K_{y}u)(x;\xi _{1},\ldots, \xi_{l-1})=\int_{0}^{1}t^{l-1}u(tx+y-ty;x-y,\xi_{1},\ldots , \xi_{l-1})\, dt\), 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:
$$ u=d(Tu)+T(du). $$
(1.1)
Meanwhile, for \(u_{\Omega}\in D'(\Omega, \Lambda^{l})\), we define
$$ u_{\Omega}=\left \{ \textstyle\begin{array}{l@{\quad}l} |\Omega|^{-1}\int_{\Omega}u(y)\, dy, & l=0, \\ dT(u) , & l=1,2,\ldots,n. \end{array}\displaystyle \right . $$
(1.2)
Then we know that, for any differential form \(u\in L^{s}_{\mathrm{loc}}(B,\Lambda ^{l})\), \(l=1,\ldots,n\), \(1< s<\infty\),
$$ \bigl\Vert \nabla(Tu)\bigr\Vert _{s,B}\leq C|B|\|u \|_{s,B} \quad \mbox{and} \quad \|Tu\| _{s,B}\leq C|B| \operatorname{diam}(B)\|u\|_{s,B} $$
(1.3)
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. (1)

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

     
  2. (2)

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

     
  3. (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:
$$ T_{\Omega}=\sum_{k=1}^{+\infty}T_{\Omega_{k}} \chi_{\Omega_{k}}, $$
where \(\chi_{\Omega_{k}}\) is the characteristic function defined on Ω.
Further, it is essential to recall some well-known results as regards the A-harmonic equation for differential forms, which appeared in [1]. To be more precise, we here consider the non-homogeneous A-harmonic equation as follows:
$$ d^{\star} A(x, du) = B(x, du), $$
(1.4)
where \(A: E \times\wedge^{l}(\mathbb{R}^{n}) \to \wedge^{l}(\mathbb{R}^{n})\) and \(B: E \times\wedge^{l}(\mathbb{R}^{n}) \to \wedge^{l-1}(\mathbb{R}^{n})\) satisfy the conditions
$$ \bigl\vert A(x, \xi)\bigr\vert \leq a\vert \xi \vert ^{p-1}, \qquad A(x, \xi) \cdot\xi \geq \vert \xi \vert ^{p} \quad \mbox{and} \quad \bigl\vert B(x, \xi)\bigr\vert \leq b \vert \xi \vert ^{p-1} $$
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 A-harmonic 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
$$ \|u\|_{L^{\varphi}_{E}}=\inf \biggl\{ \lambda>0: \int_{E} \varphi \biggl(\frac {|u|}{\lambda} \biggr)\, d\mu\leq1 \biggr\} , $$
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 Orlicz-Sobolev space of l-form is denoted by \(W^{1,\varphi}(E,\Lambda^{l})\), with the norm
$$ \|u\|_{W^{1,\varphi}_{E}}=\|u\|_{W^{1,\varphi}(E, \Lambda ^{l})}=\operatorname{diam}(E)^{-1} \|u\|_{L^{\varphi}_{E}}+\|\nabla u\| _{L^{\varphi}_{E}}. $$
(1.5)
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:
$$ \|u\|_{W^{1,s}_{E}}=\|u\|_{W^{1,s}(E, \Lambda^{l})}=\operatorname {diam}(E)^{-1} \|u\|_{L^{s}_{E}}+\|\nabla u\|_{L^{s}_{E}}. $$
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
$$ \frac{1}{\mu(E)}\int_{E}\varphi\bigl(\tau \vert u-u_{B_{0}}\vert \bigr)\, d\mu\leq C\sup_{4B\subset E} \frac{1}{\mu(B)}\int_{B}\varphi\bigl(\sigma \vert u-u_{B}\vert \bigr)\, d\mu, $$
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.

2 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)

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

     
  2. (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 above-mentioned 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 non-homogeneous A-harmonic 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
$$ \bigl\Vert TDG(u)-\bigl(TDG(u)\bigr)_{B}\bigr\Vert _{L^{\varphi}_{B}} \leq C \operatorname {diam}(B)\|u \|_{L^{\varphi}_{\sigma B}}, $$
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 non-homogeneous A-harmonic 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
$$ \bigl\Vert TDG(u)-\bigl(TDG(u)\bigr)_{B}\bigr\Vert _{W^{1,\varphi}_{B}} \leq C \operatorname {diam}(B)\|u \|_{L^{1,\varphi}_{\sigma B}}, $$
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 Poincaré-type 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 non-homogeneous A-harmonic 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
$$ \bigl\Vert TDG(u)-\bigl(TDG(u)\bigr)_{E}\bigr\Vert _{W^{1,\varphi}_{E}} \leq C \|u \|_{L^{1,\varphi}_{E}}. $$

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 non-homogeneous A-harmonic 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
$$ \bigl\Vert TDG(u)-\bigl(TDG(u)\bigr)_{B_{0}}\bigr\Vert _{L^{\varphi}_{E}} \leq C \|u \|_{L^{\varphi }_{E}}, $$
where \(B_{0}\subseteq B \) is a fixed ball.

3 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
$$ \|fg\|_{t,I}\leq\|f\|_{p,I}\cdot\|g\|_{q,I} $$
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 non-homogeneous A-harmonic equation in a domain Ω, and \(0< s,t<\infty\), then there exists a constant C, independent of u, such that
$$ \|u\|_{s,B}\leq C|B|^{\frac{t-s}{st}}\|u\|_{t,\sigma B} $$
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 WRH-class.

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
$$ \bigl\Vert dd^{*}G(u)+d^{*}dG(u)+\bigl(d+d^{*} \bigr)G(u)+G(u)\bigr\Vert _{p,B}\leq C\|u\|_{p,B} $$
for any \(B\subset\Omega\).
Because \(\triangle=d^{*}d+dd^{*}\) and \(D=d^{*}+d\), by using Lemma 3.3, it implies that
$$ \bigl\Vert \triangle G(u)\bigr\Vert _{p,B}\leq C\Vert u\Vert _{p,B} \quad \mbox{and} \quad \bigl\Vert DG(u)\bigr\Vert _{p,B}\leq C\Vert u\Vert _{p,B}. $$

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
$$ \|u_{D}\|_{p,D}\leq A_{p}(n)\mu(D)\|u \|_{D}. $$

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 non-homogeneous A-harmonic 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
$$ \bigl\Vert TDG(u)-\bigl(TDG(u)\bigr)_{B}\bigr\Vert _{s,B}\leq C|B|\operatorname{diam}(B)\|u\|_{s,B} $$
for any \(B\subset E\), where \(s>1\).

Proof

Notice that \(TDG(u)\) is at least differential 1-form, therefore, by using (1.1) and replacing u by \(TDG(u)\), we have
$$ TDG(u)=Td\bigl(TDG(u)\bigr)+dT\bigl(TDG(u)\bigr). $$
It follows from (1.2) that \(dT(TDG(u))=(TDG(u))_{B}\), that is,
$$ \bigl\Vert TDG(u)-\bigl(TDG(u)\bigr)_{B}\bigr\Vert _{s,B}=\bigl\Vert Td\bigl(TDG(u)\bigr)\bigr\Vert _{s,B}. $$
Applying (1.3), Lemma 3.3, and Lemma 3.4, we obtain
$$\begin{aligned} \bigl\Vert TDG(u)-\bigl(TDG(u)\bigr)_{B}\bigr\Vert _{s,B} = & \bigl\Vert Td\bigl(TDG(u)\bigr)\bigr\Vert _{s,B} \\ \leq& C_{1}|B|\operatorname{diam}(B)\bigl\Vert d\bigl(TDG(u)\bigr) \bigr\Vert _{s,B} \\ \leq& C_{1}|B|\operatorname{diam}(B)\bigl\Vert \bigl(DG(u) \bigr)_{B}\bigr\Vert _{s,B} \\ \leq& C_{2}|B|\operatorname{diam}(B)\bigl\Vert DG(u)\bigr\Vert _{s,B} \\ \leq& C_{3}|B|\operatorname{diam}(B) \Vert u\Vert _{s,B}. \end{aligned}$$
Thus, we finish the proof of this lemma. □

Note

By observing the proving process, we can get a valuable result as follows:
$$ \bigl\Vert Td\bigl(TDG(u)\bigr)\bigr\Vert _{s,B}\leq C|B| \operatorname{diam}(B)\|u\|_{s,B}. $$

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 non-homogeneous A-harmonic 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
$$ \bigl\Vert TDG(u)\bigr\Vert _{L^{\varphi}_{B}}\leq C \operatorname{diam}(B) \Vert u\Vert _{L^{\varphi}_{\sigma B}}, $$
where the constant \(\sigma>1\) and \(\sigma B \subseteq E\).

Proof

From (1.3), we have
$$\begin{aligned} \bigl\Vert TDG(u)\bigr\Vert _{s,B} \leq& C_{1}\operatorname{diam}(B)|B|\bigl\Vert DG(u)\bigr\Vert _{s,B} \\ \leq& C_{2}\operatorname{diam}(B)|B|\Vert u\Vert _{s,B} . \end{aligned}$$
(3.1)
Because φ belongs to the \(G(p,q,C)\)-class, we have
$$\begin{aligned}& \int_{B}\varphi \bigl(\bigl\vert TDG(u)\bigr\vert \bigr)\,dx \\& \quad = g\cdot g^{-1} \biggl(\int_{B}\varphi \bigl(\bigl\vert TDG(u)\bigr\vert \bigr)\,dx \biggr) \\& \quad \leq g \biggl(\int_{B}g^{-1}\varphi\bigl( \bigl\vert TDG(u)\bigr\vert \bigr)\,dx \biggr) \\& \quad \leq g \biggl(\int_{B}C_{1}\bigl\vert TDG(u)\bigr\vert ^{q}\,dx \biggr) \\& \quad \leq C_{2}\varphi \biggl(\int_{B}C_{1} \bigl\vert TDG(u)\bigr\vert ^{q}\,dx \biggr)^{\frac{1}{q}}. \end{aligned}$$
Since φ is doubling, according to (3.1) and Lemma 3.2, we have
$$\begin{aligned}& \int_{B}\varphi \bigl(\bigl\vert TDG(u) \bigr\vert \bigr)\,dx \\& \quad \leq C_{2}\varphi \bigl(C_{3}\vert B\vert ^{1+\frac{1}{n}}\|u\|_{q,B} \bigr) \\& \quad \leq C_{4}\varphi \biggl(\vert B\vert ^{1+\frac{1}{n}} \vert B\vert ^{\frac{1}{p}-\frac {1}{q}} \biggl(\int_{\sigma B}\vert u \vert ^{p}\,dx \biggr)^{\frac{1}{p}} \biggr) \\& \quad \leq C_{5}f \biggl(\vert B\vert ^{p(1+\frac{1}{n})}\vert B \vert ^{\frac{p}{q}-1} \biggl(\int_{\sigma B}\vert u\vert ^{p}\,dx \biggr) \biggr) \\& \quad \leq C_{6} \biggl(\int_{\sigma B}f \bigl( \vert B\vert ^{p-1+p(\frac{1}{q}+\frac {1}{n})}+\vert u\vert ^{p} \bigr)\,dx \biggr) \\& \quad \leq C_{7} \biggl(\int_{\sigma B}C_{8} \varphi \bigl(\vert B\vert ^{1+\frac {1}{q}+\frac{1}{n}-\frac{1}{p}}\vert u\vert \bigr)\,dx \biggr) \\& \quad \leq C_{9}\vert B\vert ^{1+\frac{1}{q}+\frac{1}{n}-\frac{1}{p}}\int _{\sigma B}\varphi\bigl(\vert u\vert \bigr)\,dx \\& \quad \leq C_{10}\vert B\vert ^{\frac{1}{n}}\int _{\sigma B}\varphi\bigl(\vert u\vert \bigr)\,dx. \end{aligned}$$
(3.2)
Therefore, by using \(|B|^{\frac{1}{n}}=C_{11}\operatorname{diam}(B)\), we can get the following result:
$$ \bigl\Vert TDG(u)\bigr\Vert _{L^{\varphi}_{B}}\leq C\operatorname{diam}(B)\|u\| _{L^{\varphi}_{\sigma B}}, $$
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 non-homogeneous A-harmonic 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
$$ \bigl\Vert \nabla Td\bigl(TDG(u)\bigr)\bigr\Vert _{L^{\varphi}_{B}} \leq C |B| \operatorname{diam}(B) \|u \|_{L^{\varphi}_{\sigma B}} $$
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
$$ \bigl\Vert \nabla Td\bigl(TDG(u)\bigr)\bigr\Vert _{p, B} \leq C_{1}|B|\bigl\Vert d\bigl(TDG(u)\bigr)\bigr\Vert _{p,B}\leq C_{2}|B|\bigl\Vert DG(u)\bigr\Vert _{p,B}. $$
(3.3)
Applying (3.3) and Lemma 3.3, we get
$$ \bigl\Vert \nabla Td\bigl(TDG(u)\bigr)\bigr\Vert _{p, B} \leq C_{2}|B| \operatorname{diam}(B)\Vert u\Vert _{p,B}. $$
Using the same method as used in Lemma 3.6, the following inequality holds:
$$ \bigl\Vert \nabla Td\bigl(TDG(u)\bigr)\bigr\Vert _{L^{\varphi}_{B}} \leq C \operatorname{diam}(B) \Vert u \Vert _{L^{\varphi}_{\sigma B}}. $$
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
$$ \bigcup_{i} Q_{i} = \Omega,\quad \sum _{Q_{i} \in\mathcal{V}} \chi_{\sqrt{5 \over 4}Q_{i}} \leq N \chi_{\Omega}$$
and some \(N>1\), and if \(Q_{i} \cap Q_{j} \neq \emptyset\), then there exists a cube R (this cube 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 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 \(\rho= \rho(n, \delta)\).

4 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
$$ \bigl\Vert TDG(u)-\bigl(TDG(u)\bigr)_{B}\bigr\Vert _{L^{\varphi}_{B}}=\bigl\Vert Td\bigl(TDG(u)\bigr)\bigr\Vert _{L^{\varphi}_{B}}. $$
(4.1)
This means that the key point is to prove the following inequality:
$$\bigl\Vert Td\bigl(TDG(u)\bigr)\bigr\Vert _{L^{\varphi}_{B}}\leq C \Vert u \Vert _{L^{\varphi}_{\sigma B}}. $$
According to the properties of the \(G(p,q,C)\)-class, we have
$$\begin{aligned}& \int_{B}\varphi \bigl(\bigl\vert Td\bigl(TDG(u)\bigr) \bigr\vert \bigr)\,dx \\& \quad = g\cdot g^{-1} \biggl(\int_{B}\varphi \bigl(\bigl\vert Td\bigl(TDG(u)\bigr)\bigr\vert \bigr)\,dx \biggr) \\& \quad \leq g \biggl(\int_{B}g^{-1}\varphi\bigl( \bigl\vert Td\bigl(TDG(u)\bigr)\bigr\vert \bigr)\,dx \biggr) \\& \quad \leq g \biggl(\int_{B}C_{1}\bigl\vert Td \bigl(TDG(u)\bigr)\bigr\vert ^{q}\,dx \biggr) \\& \quad \leq C_{2}\varphi \biggl(\int_{B}C_{1} \bigl\vert Td\bigl(TDG(u)\bigr)\bigr\vert ^{q}\,dx \biggr)^{\frac{1}{q}}. \end{aligned}$$
Because φ is doubling, using Lemma 3.5, we have
$$\begin{aligned}& \int_{B}\varphi \bigl(\bigl\vert Td \bigl(TDG(u)\bigr)\bigr\vert \bigr)\,dx \\& \quad \leq C_{2}\varphi \bigl(C_{3}\vert B\vert ^{1+\frac{1}{n}}\|u\|_{q,B} \bigr) \\& \quad \leq C_{4}\varphi \biggl(\vert B\vert ^{1+\frac{1}{n}} \vert B\vert ^{\frac{1}{p}-\frac {1}{q}} \biggl(\int_{\sigma B}\vert u \vert ^{p}\,dx \biggr)^{\frac{1}{p}} \biggr) \\& \quad \leq C_{5}f \biggl(\vert B\vert ^{p(1+\frac{1}{n})}\vert B \vert ^{\frac{p}{q}-1} \biggl(\int_{\sigma B}\vert u\vert ^{p}\,dx \biggr) \biggr) \\& \quad \leq C_{6} \biggl(\int_{\sigma B}f \bigl( \vert B\vert ^{p-1+p(\frac{1}{q}+\frac {1}{n})}\vert u\vert ^{p} \bigr)\,dx \biggr) \\& \quad \leq C_{7} \biggl(\int_{\sigma B}C_{8} \varphi \bigl(\vert B\vert ^{1+\frac {1}{q}+\frac{1}{n}-\frac{1}{p}}\vert u\vert \bigr)\,dx \biggr) \\& \quad \leq C_{9}\vert B\vert ^{1+\frac{1}{q}+\frac{1}{n}-\frac{1}{p}}\int _{\sigma B}\varphi\bigl(\vert u\vert \bigr)\,dx \\& \quad \leq C_{10}\vert B\vert ^{\frac{1}{n}}\int _{\sigma B}\varphi\bigl(\vert u\vert \bigr)\,dx. \end{aligned}$$
(4.2)
Combining (4.1) with (4.2) and using \(|B|^{\frac {1}{n}}=C_{11}\operatorname{diam}(B)\) yield
$$ \int_{B}\varphi\bigl(\bigl\vert TDG(u)-\bigl(TDG(u) \bigr)_{B}\bigr\vert \bigr)\,dx\leq C_{12}\operatorname {diam}(B)\int_{\sigma B}\varphi\bigl(\vert u\vert \bigr)\,dx. $$
As a result, we obtain
$$ \int_{B}\varphi \biggl(\frac{|TDG(u)-(TDG(u))_{B}|}{\lambda} \biggr)\,dx \leq C \operatorname{diam}(B)\int_{\sigma B}\varphi \biggl( \frac {|u|}{\lambda} \biggr)\,dx $$
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:
$$ \bigl\Vert TDG(u)-\bigl(TDG(u)\bigr)_{B}\bigr\Vert _{L^{\varphi}_{B}}\leq C \operatorname{diam}(B)\Vert u\Vert _{L^{\varphi}_{\sigma B}}. $$
The proof of this theorem is finished. □

Remark

From the proof of Theorem 2.1, it is easy to derive the following inequality:
$$ \bigl\Vert Td\bigl(TDG(u)\bigr)\bigr\Vert _{L^{\varphi}_{B}}\leq C \operatorname{diam}(B)\Vert u\Vert _{L^{\varphi}_{\sigma B}} . $$
(4.3)
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
$$ \bigl\Vert TDG(u)-\bigl(TDG(u)\bigr)_{B}\bigr\Vert _{W^{1,\varphi}_{B}}=\bigl\Vert Td\bigl(TDG(u)\bigr)\bigr\Vert _{W^{1,\varphi}_{B}} . $$
(4.4)
Combining (4.4) with (1.5), and applying Theorem 2.1 and Lemma 3.7, we have
$$\begin{aligned} \bigl\Vert TDG(u)-\bigl(TDG(u)\bigr)_{B}\bigr\Vert _{W^{1,\varphi}_{B}} =&\operatorname {diam}(B)^{-1}\bigl\Vert Td \bigl(TDG(u)\bigr)\bigr\Vert _{L^{\varphi}_{B}}+\bigl\Vert \nabla Td\bigl(TDG(u) \bigr)\bigr\Vert _{L^{\varphi}_{B}} \\ \leq& \bigl(\operatorname{diam}(B)\bigr)^{-1}\bigl(C_{1} \operatorname{diam}(B)\Vert u\Vert _{L^{\varphi}_{\sigma_{1} B}}\bigr)+C_{2} \Vert u\Vert _{L^{\varphi}_{\sigma_{2} B}} \\ \leq& C_{1}\Vert u\Vert _{L^{\varphi}_{\sigma_{1} B}}+C_{1}\Vert u\Vert _{L^{\varphi }_{\sigma_{2} B}} \\ \leq& C \Vert u\Vert _{L^{\varphi}_{\sigma B}}, \end{aligned}$$
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
$$\begin{aligned} \bigl\Vert \nabla Td \bigl(TDG(u)\bigr) \bigr\Vert _{L^{\varphi}_{E}} \leq&\sum_{B \in\mathcal{V}} \bigl\Vert \nabla Td \bigl(TDG(u)\bigr) \bigr\Vert _{L^{\varphi}_{B}} \\ \leq&\sum_{B \in\mathcal{V}} \bigl( C_{1} |B| \Vert u \Vert _{L^{\varphi}_{\sigma B}} \bigr) \\ \leq&C_{2} N \Vert u\Vert _{L^{\varphi}_{E}} \\ \leq&C_{3} \Vert u \Vert _{L^{\varphi}_{E}}. \end{aligned}$$
(4.5)
Similarly, using the covering lemma, Lemma 3.8, and (4.3) implies
$$\begin{aligned} \bigl\Vert Td \bigl(TDG(u)\bigr) \bigr\Vert _{L^{\varphi}_{E}} \leq& \sum_{B \in\mathcal{V}} \bigl\Vert Td \bigl(TDG(u)\bigr) \bigr\Vert _{L^{\varphi}_{B}} \\ \leq& \sum_{B \in\mathcal{V}} \bigl( C_{4} \operatorname{diam} (B) \Vert u \Vert _{L^{\varphi}_{\sigma B}} \bigr) \\ \leq& C_{5} \operatorname{diam} (E) N \Vert u\Vert _{L^{\varphi}_{E}} \\ \leq& C_{6} \operatorname{diam} (E) \Vert u\Vert _{L^{\varphi}_{E}}. \end{aligned}$$
(4.6)
Thus, from (1.5), (4.5), and (4.6), we obtain
$$\begin{aligned} \bigl\Vert TDG(u)- \bigl(TDG(u)\bigr)_{E} \bigr\Vert _{W^{1, \varphi}_{E}} =& \bigl\Vert Td\bigl(TDG(u)\bigr) \bigr\Vert _{W^{1, \varphi}_{E}} \\ =& \bigl(\operatorname{diam}(E)\bigr)^{-1} \bigl\Vert Td \bigl(TDG(u)\bigr) \bigr\Vert _{L^{\varphi}_{E}} + \bigl\Vert \nabla Td \bigl(TDG(u)\bigr) \bigr\Vert _{L^{\varphi}_{E}} \\ \leq& \bigl(\operatorname{diam}(E)\bigr)^{-1} \bigl(C_{6} \operatorname{diam} (E) \Vert u \Vert _{L^{\varphi}_{E}} \bigr) + C_{3} \Vert u \Vert _{L^{\varphi}_{E}} \\ \leq& C_{7} \Vert u \Vert _{L^{\varphi}_{E}}. \end{aligned}$$
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
$$\begin{aligned} \vert E\vert ^{-1}\int_{E}\varphi\bigl(\bigl\vert TDG(u)-\bigl(TDG(u)\bigr)_{B_{0}}\bigr\vert \bigr)\,dx \leq& \sup_{4B\subset E}\vert B\vert ^{-1}\int _{B}\varphi \bigl(\bigl\vert TDG(u)-\bigl(TDG(u) \bigr)_{B}\bigr\vert \bigr)\,dx \\ \leq& \sup_{4B\subset E}\vert B\vert ^{-1} C \operatorname{diam}(B)\int_{\sigma B}\varphi\bigl(\vert u\vert \bigr)\,dx. \end{aligned}$$
Because \(\sup_{B\subseteq E}\int_{E}\varphi(|u|)\,dx\) does not depend on B, we obtain
$$\vert E\vert ^{-1}\int_{E}\varphi\bigl(\bigl\vert TDG(u)-\bigl(TDG(u)\bigr)_{B_{0}}\bigr\vert \bigr)\,dx \leq \sup _{4B\subset E}\vert B\vert ^{-1} C \operatorname{diam}(B) \int_{E}\varphi\bigl(\vert u\vert \bigr)\,dx. $$
Therefore, we have
$$\int_{E}\varphi\bigl(\lambda^{-1}\bigl\vert TDG(u)-\bigl(TDG(u)\bigr)_{B_{0}}\bigr\vert \bigr)\,dx \leq C \int _{E}\varphi\bigl(\lambda^{-1}\vert u\vert \bigr)\,dx. $$
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 non-homogeneous A-harmonic 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
$$\bigl\Vert TDG(u)\bigr\Vert _{L^{\varphi}_{E}}\leq C \operatorname{diam}(B) \Vert u\Vert _{L^{\varphi}_{E}}. $$

Remark

If we choose \(\varphi(x)=x^{s}\), we have
$$\bigl\Vert TDG(u)\bigr\Vert _{s,E}\leq C \operatorname{diam}(B) \Vert u\Vert _{s,E}. $$

5 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 1-form
$$u(x_{1}, x_{2}, x_{3}) = {x_{1} \over 1 + (x_{1}^{2} + x_{2}^{2} + x_{3}^{2})^{2}} \,dx_{1} +{x_{2} \over 1 + (x_{1}^{2} + x_{2}^{2} + x_{3}^{2})^{2} } \,dx_{2} + {x_{3} \over 1 + (x_{1}^{2} + x_{2}^{2} + x_{3}^{2})^{2} } \,dx_{3} $$
defined in Ω. Then u is a solution of the non-homogeneous A-harmonic 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
$$ \bigl\vert u(x_{1},x_{2},x_{3})\bigr\vert ^{2}=\frac {x_{1}^{2}}{(1+(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})^{2})^{2}}+\frac {x_{2}^{2}}{(1+(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})^{2})^{2}} +\frac{x_{3}^{2}}{(1+(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})^{2})^{2}}. $$
Furthermore, because u is defined in Ω and \((x_{1}^{2}+x_{2}^{2}+x_{3}^{2})^{2}>0\), we get
$$ \bigl\vert u(x_{1},x_{2},x_{3}) \bigr\vert \leq1. $$
(5.1)
As a result, by using Theorem 2.3 and (5.1), we have
$$ \bigl\Vert TDG(u)-\bigl(TDG(u)\bigr)_{\Omega}\bigr\Vert _{W^{1,\varphi}_{\Omega}} \leq C_{1}\Vert u\Vert _{L^{\varphi}_{\Omega}} \leq C_{1}\Vert 1\Vert _{L^{\varphi}_{\Omega}} \leq C_{2}r^{3}. $$
The above example can be extended to the case of \(\mathbb{R}^{n}\). Particularly, we can check that the 1-form defined in \(\mathbb{R}^{n}\),
$$ u(x_{1}, \ldots, x_{n}) = \sum_{i=1}^{n} {x_{i} \over 1 + (x_{1}^{2} + \cdots+ x_{n}^{2})^{2}} \,dx_{i}, $$
is a solution of the non-homogeneous A-harmonic 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 3-form in \(\mathbb{R}^{3}\) as an example.

Example 5.2

Let \(k, m>0\) be constants. Consider a differential 3-form in \(\mathbb{R}^{3}\),
$$ u(x, y, z)={k \over m + x^{2} +y^{2} +z^{2}} \,dx \wedge dy \wedge dz. $$
Then u is a solution of the non-homogeneous A-harmonic 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 \(\Omega\subset\mathbb{R}^{3} \):
$$ \bigl\vert u(x,y,z)\bigr\vert \leq\frac{k}{m} $$
and
$$ \bigl\Vert TDG(u)-\bigl(TDG(u)\bigr)_{\Omega}\bigr\Vert _{W^{1,\varphi}_{\Omega}} \leq C \Vert u\Vert _{L^{\varphi}_{\Omega}} \leq C \Vert k/m \Vert _{L^{\varphi}_{\Omega}}. $$

Declarations

Acknowledgements

The authors express their deep appreciation to the referees’ efforts and suggestions, which highly improved the quality of this paper.

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.

Authors’ Affiliations

(1)
Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, P.R. China
(2)
Department of Mathematics, Seattle University, Seattle, WA 98122, USA

References

  1. Agarwal, RP, Ding, S, Nolder, C: Inequalities for Differential Forms. Springer, New York (2009) MATHView ArticleGoogle Scholar
  2. Ding, S, Liu, B: Norm inequalities for composition of the Dirac and Green’s operator. J. Inequal. Appl. (2013). doi:10.1186/1029-242X-2013-436 MathSciNetGoogle Scholar
  3. Wang, Y, Wu, C: Sobolev embedding theorems and Poincaré inequalities for Green’s operator on solution of the non-homogeneous A-harmonic equation. Comput. Math. Appl. 47, 1545-1554 (2004) MATHMathSciNetView ArticleGoogle Scholar
  4. Liu, B: \(A^{\lambda}_{r}(\Omega)\)-Weighted embedding inequalities for A-harmonic tensors. J. Math. Anal. Appl. 273, 667-676 (2002) MATHMathSciNetView ArticleGoogle Scholar
  5. Xing, Y: Two-weight embedding inequalities for solutions to the A-harmonic equation. J. Math. Anal. Appl. 307, 555-564 (2005) MATHMathSciNetView ArticleGoogle Scholar
  6. Bi, H: Weighted inequalities for potential operators on differential forms. J. Inequal. Appl. 2010, Article ID 713625 (2010) View ArticleGoogle Scholar
  7. Bi, H, Ding, S: Some strong \((p, q)\)-type inequalities for the homotopy operator. Comput. Math. Appl. 62, 1780-1789 (2011) MATHMathSciNetView ArticleGoogle Scholar
  8. Ding, S, Liu, B: A singular integral of the composite operator. Appl. Math. Lett. 22, 1271-1275 (2009) MATHMathSciNetView ArticleGoogle Scholar
  9. Ding, S, Liu, B: Global estimates for singular integrals of the composite operator. Ill. J. Math. 53, 1173-1185 (2009) MATHMathSciNetGoogle Scholar
  10. Iwaniec, T, Lutoborski, A: Integral estimates for null Lagrangians. Arch. Ration. Mech. Anal. 125, 25-79 (1993) MATHMathSciNetView ArticleGoogle Scholar
  11. Adams, RA, Fourier, JJF: Sobolev Spaces, 2nd edn. Elsevier, Amsterdam (2003) MATHGoogle Scholar
  12. Scott, C: \(L^{p}\)-Theory of differential forms on manifolds. Trans. Am. Math. Soc. 347, 2075-2096 (1995) MATHGoogle Scholar
  13. Ding, S: \(L^{\varphi}(\mu)\)-Averaging domains and the quasi-hyperbolic metric. Comput. Math. Appl. 47, 1611-1618 (2004) MATHMathSciNetView ArticleGoogle Scholar

Copyright

Advertisement