As generalizations of the functions, differential forms were widely used in many fields, including potential theory, partial differential equations, quasiconformal mappings and etc. During recent years a series of estimates and inequalities for differential forms, particularly, for the forms satisfying the homogeneous or nonhomogeneous A-harmonic equations, or the conjugate A-harmonic equations in $ \mathbb{R}^{n}(n \geq 2) $, were developed, see [1-8]. These estimates and inequalities are critical tools to investigate the properties of solutions to the nonlinear differential equations and to control oscillatory behavior in domains or on manifolds. However, the nonlinear PDE with the Hodge-Dirac operator for differential forms, that is, the Dirac-harmonic equation has yet to be further developed, where the Dirac operator was initiated by Paul Dirac in order to get a form of quantum theory compatible with special relativity, which was playing a critical role in some fields of mathematics and physics, such as quantum mechanics, Clifford analysis and partial differential equations, see [9-13].

The purpose of this paper is to introduce the non-homogeneous Dirac-harmonic equation $ d^{\star}A(x, D\omega) = B(x, D\omega) $ for differential forms and initiate the study of this new type of differential equations, where the Hodge-Dirac operator $ D $ is defined by $ D = d + d^{\star} $, $ d $ is the exterior differential operator, $ d^{\star} $ is the Hodge codifferential that is formal adjoint operator of $ d $, and $ A $ is an operator satisfying certain conditions. Specifically, we establish Poincaré-type inequalities, Caccioppoli-type inequalities and the weak reverse Hölder inequality for differential forms satisfying the non-homogeneous Dirac-harmonic equation. These basic inequalities will form the basis for the study of the $ L^{p} $-theory of the new introduced Dirac-harmonic equation for differential forms.

Now we introduce some notations and definitions. Let $ \Omega $ be an open subset of $ \mathbb{R}^n(n\geq 2) $ and $ B $ be a ball in $ \mathbb{R}^n $. Let $ \rho B $ denote the ball with the same center as $ B $ and diam$ (\rho B) = \rho $ diam$ (B)(\rho>0) $. $ |\Omega| $ is used to denote the Lebesgue measure of a set $ \Omega\subset\mathbb{R}^n $. Let $ \wedge^{l} = \wedge^{l}(\mathbb{R}^n), l = 0, 1, \ldots, n $, be the linear space of all $ l $-forms $ \omega(x) = \sum_{I}\omega_{I}(x)\mathrm{d}x_{I} = \sum_{I}\omega_{i_{1}i_{2}\cdots i_{l}}(x) \mathrm{d}x_{i_{1}}\wedge \mathrm{d}x_{i_{2}}\wedge\cdots\wedge \mathrm{d}x_{i_{l}} $ in $ \mathbb{R}^n $, where $ I = (i_{1}, i_{2}, \ldots , i_{l}), 1\leq i_{1}< i_{2}<\cdots<i_{l} \leq n $, are the ordered $ l $-tuples. The Grassman algebra $ \wedge = \wedge(\mathbb{R}^n) = \oplus^{n}_{l = 0}\wedge^{l}(\mathbb{R}^n) $ is a graded algebra with respect to the exterior products $ \wedge $. Moreover, if each of the coefficient $ \omega_{I}(x) $ of $ \omega(x) $ is differential on $ \Omega $, then we call $ \omega(x) $ a differential $ l $-form on $ \Omega $, use $ D^{'}(\Omega, \wedge^{l}) $ to denote the space of all differential $ l $-forms on $ \Omega $ and $ D'(\Omega, \wedge) = \oplus^{n}_{l = 0}D^{'}(\Omega, \wedge^{l}) $. Analogously $ C^{\infty}(\Omega, \wedge^{l}) $ denotes the space of smooth $ l $-forms on $ \Omega $. The exterior derivative $ d:D^{'}(\Omega, \wedge^{l})\rightarrow D^{'}(\Omega, \wedge^{l+1}) $, $ l = 0, 1, \ldots, n-1 $, is given by

$ \begin{equation} d\omega(x) = \sum\limits_{I}\sum^{n}_{j = 1}\frac{\partial\omega_{i_{1}i_{2}\cdots i_{l}}(x)}{\partial x_{j}}\mathrm{d}x_{j}\wedge \mathrm{d}x_{i_{1}}\wedge \mathrm{d}x_{i_{2}}\wedge\cdots\wedge \mathrm{d}x_{i_{l}} \end{equation} $

(1.1)

for all $ \omega\in D^{'}(\Omega, \wedge^{l}) $. The Hodge star operator $ \star:\wedge^{k}\rightarrow \wedge^{n-k} $ is defined as follows. If $ \omega = \omega_{i_{1}i_{2}\cdots i_{k}}(x_{1}, x_{2}, \ldots, x_{n})dx_{i_{1}}\wedge dx_{i_{2}}\wedge\cdots\wedge dx_{i_{k}} = \omega_{I}dx_{I} $, $ i_{1}<i_{2} <\cdots<i_{k} $, is a differential $ k $-form, then $ \star\omega = \star(\omega_{i_{1}i_{2}\cdots i_{k}}dx_{i_{1}}\wedge dx_{i_{2}} \wedge\cdots\wedge dx_{i_{k}}) = (-1)^{\sum(I)}\omega_{I}dx_{J} $, where $ I = (i_{1}, i_{2}, \cdots, i_{k}) $, $ J = \{1, 2, \ldots, n\}-I $, and $ \sum(I) = \frac{k(k+1)}{2}+\sum^{k}_{j = 1}i_{j} $. The Hodge codifferential operator $ d^{*}:D^{'}(\Omega, \wedge^{l+1})\rightarrow D^{'}(\Omega, \wedge^{l}) $is defined by $ d^{*} = (-1)^{nl+1}\star d\star $ on $ D'(\Omega, \wedge^{l+1}) $, $ l = 0, 1, \ldots , n-1 $. For all $ \omega\in D^{'}(\Omega, \wedge^{l}) $, we have $ d(d\omega) = d^{*}(d^{*}\omega) = 0 $. $ L^{p}(\Omega, \wedge^{l})(1\leq p<\infty) $ is a Banach space with the norm $ \|\omega\|_{p, \Omega} = (\int_{\Omega}|\omega(x)|^{p}\mathrm{d}x)^{1/p} = (\int_{\Omega}(\sum_{I}|\omega_{I}(x)|^{2})^{p/2}\mathrm{d}x)^{1/p}<\infty $ and $ L^{p}(\Omega, \wedge) = \oplus^{n}_{l = 0} L^{p}(\Omega, \wedge^{l}) $. Similarly, the notations $ L^{p}_{\text{loc}}(\Omega, \wedge^{l}) $ and $ W^{1, p}_{\text{loc}}(\Omega, \wedge^{l}) $ are self-explanatory.

From [14], $ \omega $ is a differential form in a bounded convex domain $ \Omega $, then there is a decomposition

$ \begin{equation} \omega = d(T\omega)+T(d\omega), \end{equation} $

(1.2)

where $ T $ is called a homotopy operator. For the homotopy operator $ T $, we know that

$ \begin{equation} \|T\omega\|_{p, B}\leq C|B|\text {diam}(B)\|\omega\|_{p, B} \end{equation} $

(1.3)

holds for any differential form $ \omega\in L^{p}_{\text{loc}}(\Omega, \wedge^{l}), l = 1, 2, \ldots, n, 1<p<\infty $. Furthermore, we can define the $ l $-form $ \omega_{\Omega} \in D^{'} (\Omega, \wedge^{l}) $ by

$ \begin{equation} \omega_{\Omega} = \left\{ \begin{array}{ll} |\Omega|^{-1}\int_{\Omega}\omega(x)\mathrm{d}x, & \hbox{$l = 0$}, \\ dT(\omega), & \hbox{$l = 1, 2, \ldots, n$} \end{array} \right. \end{equation} $

(1.4)

for all $ \omega \in L^{p}(\Omega, \wedge^{l}), 1\leq p<\infty $.

The theory of differential equations was very well developed during last several decades. Particularly, there was an increasing interest in different types of differential equations for differential forms, see [15-21]. Among these types of equations, the traditional A-harmonic equation for differential forms

$ \begin{equation} d^{*}A(x, d\omega) = 0 \end{equation} $

(1.5)

in $ \mathbb{R}^{n} $, and the corresponding nonhomogeneous A-harmonic equation for differential forms is a nonlinear elliptic equation of the form

$ \begin{equation} d^{*}A(x, d\omega) = B(x, d\omega) \end{equation} $

(1.6)

received much investigation in recent years. In [10], for the purpose of dealing with terms $ d\omega $ and $ d^{*}\omega $ simultaneously in many cases, such as in the case of Hodge decomposition of a differential form, Ding introduced the following Dirac-harmonic equation

$ \begin{equation} d^{*}A(x, D\omega) = 0 \end{equation} $

(1.7)

for differential forms. Similarly, we could introduce the corresponding nonhomogeneous Dirac-harmonic equation for differential forms is a nonlinear elliptic equation of the form

$ \begin{equation} d^{*}A(x, D\omega) = B(x, D\omega), \end{equation} $

(1.8)

where the Hodge-Dirac operator $ D $ is a Dirac operator defined by $ D = d + d^{*} $, $ d $ is the exterior differential operator, $ d^{*} $ is the Hodge codifferential that is formal adjoint operator of $ d $, $ A :\Omega\times\wedge(\mathbb{R}^n) \rightarrow \wedge(\mathbb{R}^n) $ and $ B :\Omega\times\wedge(\mathbb{R}^n) \rightarrow \wedge(\mathbb{R}^n) $ satisfy the following conditions

$ \begin{equation} |A(x, \xi)|\leq a|\xi|^{p-1}\quad , \quad \langle A(x, \xi), \xi\rangle \geq |\xi|^{p} \quad \text {and} \quad |B(x, \xi)|\leq b|\xi|^{p-1}\quad \end{equation} $

(1.9)

for almost every $ x\in \Omega $ and all $ \xi\in \wedge(\mathbb{R}^n) $, here $ a, b > 0 $ is a constant and $ 1 < p <\infty $ is a fixed exponent associated with (1.8). Let $ W^{1}_{p, \text{loc}} (\Omega, \wedge^{l-1}) = \bigcap W^{1}_{p} (\Omega', \wedge^{l-1}) $, where the intersection is for all $ \Omega' $ compactly contained in $ \Omega $. A solution to (1.2) is an element of the Sobolev space $ W^{1}_{p, \text{loc}} (\Omega, \wedge^{l-1}) $ such that

$ \begin{equation} \int_{\Omega}\langle A(x, D\omega), D\varphi\rangle+\langle B(x, D\omega), \varphi\rangle = 0 \end{equation} $

(1.10)

for all $ \varphi\in W^{1}_{p} (\Omega, \wedge^{l-1}) $ with compact support.

A solution $ \omega $ of the homogeneous and non-homogeneous A-harmonic equation (1.5) and (1.6) is called a nontrivial solution if $ d\omega\neq 0 $; otherwise, $ \omega $ is called a trivial solution of (1.5) and (1.6). Similarly, a solution $ \omega $ of the homogeneous and non-homogeneous Dirac-harmonic equation (1.7) and (1.8) is called a nontrivial solution if $ D\omega \neq 0 $; otherwise, $ \omega $ is called a trivial solution of (1.7) and (1.8). It should be noticed that the Dirac-harmonic equation can be considered as an extension of the traditional A-harmonic equations with operator $ d $ being replaced by the Dirac operator $ D = d + d^{\star} $. It is also easy to see that if $ \omega $ is a function ($ 0 $-form), both the traditional non-homogeneous A-harmonic equation $ d^{\star}A(x, d\omega) = B(x, d\omega) $and the non-homogeneous Dirac-harmonic equation $ d^{\star}A(x, D\omega) = B(x, D\omega) $ reduce to the usual non-homogeneous A-harmonic equation

$ \begin{equation} \text {div}A(x, \nabla\omega) = B(x, \nabla\omega). \end{equation} $

(1.11)

So far although the research on the non-homogeneous A-harmonic equation has gained great attention, we can find most of what we get is the solution of its degenerated equation, that is the special solution of the homogeneous A-harmonic equation, see [1, 3, 22-23] for more details. The non-homogeneous Dirac-harmonic equation (1.8), as a kind of more complicated equation compared to the non-homogeneous A-harmonic equation, now we can get some trivial special solutions, that is, the solutions satisfy $ D\omega = 0 $.

In order to find the trivial solution of equation (1.8), we need the lemmas below.

Lemma 1.1[9-11] Let $ \omega $ be any differential form. Then, $ D\omega = 0 $ if and only if $ d\omega = 0 $ and $ d^{\star}\omega = 0 $. Noticing that $ \Delta\omega = (dd^{\star}+d^{\star}d)\omega = (d+d^{\star})^{2}\omega = D^{2}\omega $, we have the proposition.

Proposition 1.1 Let $ \omega $ be any differential form. Then, $ \Delta\omega = 0 $ if and only if $ D\omega = 0 $.

According to Lemma 1.1 and the definitions of the operator $ d $ and $ d^{\star} = (-1)^{nl+1}\star d\star $, we can know that $ \omega = \sum\limits_{I}a_{I}dx_{I}, a_{I}\in\mathbb{R} $ is the special solution of equation (1.8). Besides, we can check the following two examples are also the trivial solutions of equation (1.8).

Example 1.1 Let

$ \begin{equation} \omega(x_{1}, x_{2}) = \frac{-x_{2}}{x^{2}_{1}+x^{2}_{2}}dx_{1}+\frac{x_{1}}{x^{2}_{1}+x^{2}_{2}}dx_{2} \end{equation} $

(1.12)

be a $ 1 $-form in $ \Omega \subset \mathbb{R}^2 $ which does not contain the origin $ (0, 0) $. Then $ \omega(x_{1}, x_{2}) $ is a trivial solution of the A-harmonic equation (1.5)–(1.8) in $ \Omega $.

Proof By simple calculation, we have

$ \begin{equation} \begin{split} d\omega& = \frac{\partial}{\partial x_{2}}(\frac{-x_{2}}{x^{2}_{1}+x^{2}_{2}})dx_{2}\wedge dx_{1}+\frac{\partial}{\partial x_{1}}(\frac{x_{1}}{x^{2}_{1}+x^{2}_{2}})dx_{1}\wedge dx_{2}\\ & = \frac{x^{2}_{2}-x^{2}_{1}}{x^{2}_{1}+x^{2}_{2}}(dx_{2}\wedge dx_{1}+dx_{1}\wedge dx_{2}) = 0, \\ \star(\omega)& = \frac{-x_{2}}{x^{2}_{1}+x^{2}_{2}}dx_{2}+\frac{-x_{1}}{x^{2}_{1}+x^{2}_{2}}dx_{1}, \\ d(\star(\omega))& = \frac{\partial}{\partial x_{1}}(\frac{-x_{2}}{x^{2}_{1}+x^{2}_{2}})dx_{1}\wedge dx_{2}+\frac{\partial}{\partial x_{2}}(-\frac{x_{1}}{x^{2}_{1}+x^{2}_{2}})dx_{2}\wedge dx_{1}\\ & = \frac{2x_{1}x_{2}}{x^{2}_{1}+x^{2}_{2}}(dx_{2}\wedge dx_{1}+dx_{1}\wedge dx_{2}) = 0, \end{split} \end{equation} $

(1.13)

$ \begin{align} d^{\star}(\omega) = &(-1)^{3}\star d\star(\omega) = 0, \cr D(\omega) = &(d+d^{\star})(\omega) = 0. \end{align} $

(1.14)

Example 1.2 Let

$ \begin{equation} \begin{split} \quad\omega(x_{1}, x_{2}, x_{3}) = &\omega_{1}dx_{1}\wedge dx_{2}+\omega_{2}dx_{2}\wedge dx_{3}+\omega_{3}dx_{3}\wedge dx_{1} \\ = &(x_{1}+x_{2}-2x_{3})dx_{1}\wedge dx_{2}+(x_{1}-x_{2}+x_{3})dx_{2}\wedge dx_{3}\\ &+(x_{1}+x_{2}-x_{3})dx_{3}\wedge dx_{1} \end{split} \end{equation} $

(1.15)

be a $ 2 $-form in $ \Omega \subset \mathbb{R}^3 $. Then $ \omega(x_{1}, x_{2}) $ is a trivial solution of the A-harmonic equation (1.5)–(1.8) in $ \Omega $.

Proof

$ \begin{align} d\omega = &(\frac{\partial(w_{1})}{\partial x_{3}}+\frac{\partial(w_{2})}{\partial x_{1}}+\frac{\partial(w_{3})}{\partial x_{2}})dx_{1}\wedge dx_{2}\wedge dx_{3} = 0, \end{align} $

(1.16)

$ \begin{align} \star\omega = &\omega_{1}dx_{3}+\omega_{2}dx_{1}-\omega_{3}dx_{2}, \cr d\star\omega = &(\frac{\partial(w_{1})}{\partial x_{1}}-\frac{\partial(w_{2})}{\partial x_{3}})dx_{1}\wedge dx_{3}-(\frac{\partial(w_{3})}{\partial x_{1}}+\frac{\partial(w_{2})}{\partial x_{2}})dx_{1}\wedge dx_{2}\cr &+(\frac{\partial(w_{1})}{\partial x_{2}}-\frac{\partial(w_{3})}{\partial x_{3}})dx_{2}\wedge dx_{3} = 0, \cr d^{\star}(\omega) = &(-1)^{7}\star d\star(\omega) = 0, \cr D(\omega) = &(d+d^{\star})(\omega) = 0. \end{align} $

(1.17)

In order to obtain the related inequalities with the solutions of the non-homogeneous Dirac-harmonic equation for differential forms, we need the following generalized Hölder inequality in this paper.

Lemma 1.2[1] Let $ 0 <p, q<\infty $ and $ s^{-1} = p^{-1} +q^{-1} $. If $ f $ and $ g $ are measurable functions on $ \mathbb{R}^n $, then

$ \begin{equation} \|fg\|_{s, \Omega}\leq \|f\|_{p, \Omega}\|g\|_{q, \Omega} \end{equation} $

(1.18)

for any $ \Omega\subset \mathbb{R}^n $.

If we select $ q = \frac{sp}{p-s} $ and $ g\equiv 1 $, we can get

$ \begin{equation} \|f\|_{s, Q}\leq |Q|^{\frac{1}{s}-\frac{1}{p}} \|f\|_{p, Q} \end{equation} $

(1.19)

or

$ \begin{equation} \|f\|_{s, Q}\leq |Q|^{\frac{p-s}{ps}} \|f\|_{p, Q}. \end{equation} $

(1.20)

Different versions of the classical Poincaré inequality were established in the study of the Sobolev space and differential forms, see [1, 6]. Susan proved the Poincaré inequality in $ L^{s} $-averaging domains in [15]. Tadeusz and Lutoborski proved a local Poincaré-type inequality in [14], which plays a crucial role in generalizing the theory of Sobolev functions to differential forms.

In [10], Ding Proved the Poincaré Sobolev and embedding inequalities with the Dirac operator. If appropriate substitution is made, ones can get the new forms of these inequalities.

Lemma 2.1[10] Let $ u\in D'(Q, \wedge^{l}) $ be a differential form and $ Du\in L^{p}(Q, \wedge) $. Then $ u-u_{Q} $ is in $ L^{\frac{np}{(n-p)}} $ and

$ \begin{equation} (\int_{Q}|u-u_{Q}|^{\frac{np}{n-p}}dx)^{\frac{(n-p)}{np}}\leq C_{p}(n)(\int_{Q}|Du|^{p}dx)^{\frac{1}{p}} \end{equation} $

(2.1)

for $ Q $ a cube or a ball in $ \mathbb{R}^n $, $ l = 0, 1, \ldots, n-1 $ and $ 1<p<n $.

Considering that the norms $ \|u\|_{p, Q} $ and $ \|u-u_{Q}\|_{p, Q} $ are comparable, see [4], namely,

$ \begin{equation} \|u-u_{Q}\|_{p, Q}\leq C_{1}\|u\|_{p, Q}\leq C_{2}\|u-u_{Q}\|_{p, Q}, \end{equation} $

(2.2)

we have

$ \begin{equation} (\int_{Q}|u|^{\frac{np}{n-p}}dx)^{\frac{(n-p)}{np}}\leq C_{p}(n)(\int_{Q}|Du|^{p}dx)^{\frac{1}{p}} \end{equation} $

(2.3)

for $ Q $ a cube or a ball in $ \mathbb{R}^n $, $ l = 0, 1, \ldots, n-1 $ and $ 1<p<n $.

Replacing $ u $ by $ Du $ in (2.1) and noting that $ \Delta = (d+d^{\star})^{2} = D^{2} $, we have the following inequality with the Dirac operator

$ \begin{equation} (\int_{Q}|Du-(Du)_{Q}|^{\frac{np}{n-p}}dx)^{\frac{(n-p)}{np}}\leq C_{p}(n)(\int_{Q}|\Delta u|^{p}dx)^{\frac{1}{p}}. \end{equation} $

(2.4)

Now, we give the new Poincaré inequality with the Dirac operator.

Theorem 2.1 Let $ u \in D'(Q, \wedge^{l}) $ be a differential form and $ Du \in L^{p}(Q, \wedge), p > 1 $. Then, $ u-u_{Q} $ is in $ L^{p}(Q, \wedge) $ and

$ \begin{equation} \|u-u_{Q}\|_{p, Q}\leq C|Q|\text {diam}(Q)\|Du\|_{p, Q} \end{equation} $

(2.5)

for all cubes or balls $ Q $ with $ Q\subset\mathbb{R}^n $, where $ C $ is a constant, independent of $ u $ and $ Du $.

Proof

$ \begin{equation} \|u-u_{Q}\|_{p, Q} = \|T(du)\|_{p, Q} \leq C_{1}|Q|\text {diam}(Q)\|du\|_{p, Q} \leq C_{1}|Q|\text {diam}(Q)\|Du\|_{p, Q}. \end{equation} $

(2.6)

Noticing that $ \text {diam}(Q)\leq C_{2}|Q|^{\frac{1}{n}} $, we have

$ \begin{equation} \|u-u_{Q}\|_{p, Q}\leq C_{3}|Q|^{1+\frac{1}{n}}\|Du\|_{p, Q}. \end{equation} $

(2.7)

Similarly, replacing $ u $ by $ Du $ in (2.5), we have

$ \begin{equation} \|Du-(Du)_{Q}\|_{p, Q}\leq C_{3}|Q|\text {diam}(Q)\|\Delta u\|_{p, Q}. \end{equation} $

(2.8)

Futhermore, considering that the norms $ \|Du\|_{p, Q} $ and $ \|Du-(Du)_{Q}\|_{p, Q} $ are comparable, we have

$ \begin{equation} \|Du\|_{p, Q}\leq C_{3}|Q|\text {diam}(Q)\|\Delta u\|_{p, Q}. \end{equation} $

(2.9)

The Caccioppoli-type estimates become powerful tools in analysis and related fields, for the Caccioppoli-type inequalities or estimates provide upper bounds for the norms of $ \nabla u $ or $ du $ in terms of the corresponding norm of $ u $ or $ u-c $, where $ u $ is a differential form or a function satisfying certain conditions. In recent years, different versions of Caccioppoli-type estimates were established; see [1, 3, 10].

In[10], Ding obtained the Caccioppoli-type inequality with solutions of the Dirac-harmonic equation (1.7) for differential forms. Similarly, we can also get the Caccioppoli-type inequality with solutions of the non-homogeneous Dirac-harmonic equation for differential forms. Now, we first introduce the product rules for the differentiations of exterior product.

Lemma 3.1[10] (i) Product rule for $ d $ holds: let $ u = \sum\limits_{I}u_{I}dx_{I} $ be a $ k $-form and $ v = \sum\limits_{J}v_{J}dx_{J} $ be $ l $-form in $ D'(\Omega, \wedge) $, where $ u_{I} $ and $ v_{J} $ are differentiable functions in $ \Omega $. Then

$ \begin{equation} d(u\wedge v) = du\wedge v+(-1)^{k}u\wedge dv. \end{equation} $

(3.1)

(ii) Product rule for $ d^{\star} $ holds: let $ u\in D'(\Omega, \wedge) $ be a $ k $-form $ u = \sum\limits_{I}u_{I}dx_{I} $, where $ I = (i_{1}, i_{2}, \cdots , i_{k}) $ is a set of all $ k $ tuples with $ i_{1 }< i_{2 }< \cdots < i_{k} $ and $ 1 \leq i_{k}\leq n $. Let $ \eta $ be a differentiable function in $ \Omega $, then

$ \begin{equation} d^{\star}(u\eta) = (d^{\star}u)\eta+(-1)^{m}\sum\limits_{I-q}u_{I}\frac{\partial \eta}{\partial x_{q}}dx_{I-q}, \end{equation} $

(3.2)

$ \begin{equation} D(u\eta) = (Du)\eta+(-1)^{k}u(d\eta)+(-1)^{m}\sum\limits_{I-q}u_{I}\frac{\partial \eta}{\partial x_{q}}dx_{I-q}, \end{equation} $

(3.3)

where $ m $, $ q $ are integers, $ 1 \leq q \leq n $ and $ I-q $ is an abusive notation to represent an $ k-1 $ tuple with $ i_{q} $ is missing $ (i_{1}, \cdots , \hat{i_{q}}, \cdots , i_{k}) $ and $ q\notin J $ means $ q\neq j_{s} $ for any $ j_{s} $ in $ n-k $ tuples $ J $. Also, $ \sum\limits_{I} $ means the sum of all possible $ k $ tuples.

Theorem 3.1 Let $ u\in D'(\Omega, \wedge^{l}), l = 0, 1, \ldots , n $, be a solution of the non-homogeneous Dirac-harmonic equation (1.8) in a bounded domain $ \Omega\subset \mathbb{R}^n $, and assume that $ 1 < p <\infty $ is a fixed exponent associated with equation (1.8). Then there exists a constant $ C $, independent of $ u $ and $ Du $, such that

$ \begin{equation} \|Du\|_{p, B}\leq C |B|^{\frac{-1}{n}}\|u\|_{p, \sigma B} \end{equation} $

(3.4)

for all balls or cubes $ B $ with $ \sigma B\subset\Omega $.

Proof Let $ \eta\in C^{\infty}_{0}(\sigma B) $, $ 0\leq \eta\leq 1 $ with $ \eta\equiv 1 $ in $ B $, and $ |d\eta| = |\nabla\eta|\leq\frac{C_{1}}{\text {diam}(B)} $. Choosing the test form $ \varphi = -u\eta^{p} $. Then

$ \begin{equation} D\varphi = -(Du)\eta^{p}+(-1)^{l+1}up\eta^{p-1}d\eta+(-1)^{m+1}\sum\limits_{I-q}u_{I}p\eta^{p-1}\frac{\partial\eta}{\partial x_{q}}dx_{I-q}. \end{equation} $

(3.5)

From (1.10), we obtain

$ \begin{equation} \begin{split} &\int_{\sigma B}\langle A(x, Du), \eta^{p}Du\rangle = -\int_{\sigma B}\langle A(x, Du), (-1)^{l}up\eta^{p-1}d\eta\rangle\\ -&\int_{\sigma B}\langle A(x, Du), (-1)^{m}\sum\limits_{I-q}u_{I}p\eta^{p-1}\frac{\partial\eta}{\partial x_{q}}dx_{I-q}\rangle -\int_{\sigma B}\langle B(x, Du), u\eta^{p}\rangle. \end{split} \end{equation} $

(3.6)

Applying (1.9), we have

$ \begin{equation} \int_{\sigma B}|\langle A(x, Du), \eta^{p}Du\rangle|\geq \int_{\sigma B}|\eta|^{p}|Du|^{p}dx \end{equation} $

(3.7)

Notice that $ |\sum_{I-q}u_{I}p\eta^{p-1}\frac{\partial\eta}{\partial x_{q}}dx_{I-q}|\leq p|\eta|^{p-1}|u||d\eta| $ and $ \text {diam}(B)\leq \text {diam}(\Omega)<\infty $.

Using the Hölder inequality (1.18), we obtain

$ \begin{equation} \begin{split} &\int_{\sigma B}|\eta|^{p}|Du|^{p}dx \leq \int_{\sigma B}|\langle A(x, Du), \eta^{p}Du\rangle|\\ \leq& \int_{\sigma B} |A(x, Du)|(p|u||\eta|^{p-1}|d\eta|)dx+\int_{\sigma B} |A(x, Du)|(p|u||\eta|^{p-1}|d\eta|)dx +\int_{\sigma B}b|Du|^{p-1}(|u||\eta|^{p})dx\\ \leq& \frac{C_{2}}{\text {diam}(B)}\int_{\sigma B}|Du|^{p-1}|\eta|^{p-1}|u|dx+C_{3}\int_{\sigma B}|Du|^{p-1}|u||\eta|^{p}dx\\ \leq&\frac {C_{2}}{\text {diam}(B)}\|u\|_{p, \sigma B}(\int_{\sigma B}(|\eta||Du|)^{p}dx)^{\frac{p-1}{p}} +C_{3}\|\eta u\|_{p, \sigma B}(\int_{\sigma B}(|\eta||Du|)^{p}dx)^{\frac{p-1}{p}}\\ \leq&\frac{C_{2}+C_{3}\text {diam}(B)}{\text {diam}(B)}\|u\|_{p, \sigma B}(\int_{\sigma B}(|\eta||Du|)^{p}dx)^{\frac{p-1}{p}}\\ \leq&\frac{C_{4}}{\text {diam}(B)}\|u\|_{p, \sigma B}(\int_{\sigma B}(|\eta||Du|)^{p}dx)^{\frac{p-1}{p}}, \end{split} \end{equation} $

(3.8)

which is equivalent to

$ \begin{equation} \|\eta Du\|_{p, \sigma B}\leq \frac{C_{4}}{\text {diam}(B)}\|u\|_{p, \sigma B}. \end{equation} $

(3.9)

Thus, we have

$ \begin{equation} \|Du\|_{p, B} = \|\eta Du\|_{p, B}\leq \|\eta Du\|_{p, \sigma B}\leq\frac{C_{4}}{\text {diam}(B)}\|u\|_{p, \sigma B}\leq C_{5}|B|^{\frac{-1}{n}}\|u\|_{p, \sigma B}. \end{equation} $

(3.10)

Let $ u $ be a solution of equation (1.8) and $ c $ be a harmonic form. Then, $ w = u + c $ is also a solution of equation (1.8). Write $ u = \omega-c $, then $ Du = D\omega $. Thus, we have the following version of (3.10).

Corollary 3.1 Let $ c $ be a harmonic form and $ u $ is a solution of the non-homogeneous Dirac-harmonic equation (1.8). Then

$ \begin{equation} \|Du\|_{p, B}\leq C|B|^{-\frac{1}{n}}\|u-c\|_{p, \sigma B} \end{equation} $

(3.11)

for all balls or cubes $ B $ with $ \sigma B\subset\Omega $, $ \sigma > 1 $ be a constant.

It is known to all that a necessary condition for (1.20) to hold is that $ p\geq s $. When $ p, s $ are any two positive constants and an inequality weaker than (1.20) holds, which is known as the weak reverse Hölder inequality and plays an important role in establishing the weighted form of the related inequality, see [1] for more details.

In [15], if $ \omega $ is a solution of the A-harmonic equation (1.6), then the weak reverse Hölder inequality

$ \begin{equation} \|\omega\|_{s, Q}\leq C|Q|^{\frac{r-s}{rs}}\|\omega\|_{r, \sigma Q} \end{equation} $

(4.1)

holds for all cubes or balls $ Q $ with $ \sigma Q \subset\Omega $ and any $ 0<r, s<\infty $.

Recently, Ding proved that inequality (4.1) also holds for the solutions of the Dirac-harmonic equation (1.7), see [10]. Now we prove that for any solution $ \omega $ of the non-homogenious Dirac-harmonic equation (1.8), the weak reverse Hölder inequality still holds. Specifically, we have the following Theorem 4.3.

We define the differential forms $ \omega^{+} $ and $ \omega^{-} $ as follows. If $ \omega\in\wedge^{0}(R) $, let $ \omega^{+} = \max\{\omega, 0\} $ and $ \omega^{-} = \min\{\omega, 0\} $. Otherwise, let $ \omega^{+} = \sum\omega_{I}^{+}dx_{I} $ and $ \omega^{-} = \sum\omega_{I}^{-}dx_{I} $.

Theorem 4.1 Let $ \omega $ be a solution of (1.8) and $ \eta \in C^{\infty}_{0}(\Omega) $ with $ \eta\geq 0 $. There exists a constant $ C $, depending only on $ a, b $, and $ p $, such that

$ \begin{equation} \int_{\Omega}|D\omega^{+}|^{p}\eta^{p}dx\leq C(\int_{\Omega}|\omega^{+}|^{p}|d\eta|^{p}dx+\int_{\Omega}|\omega^{+}|^{p}|\eta|^{p}dx). \end{equation} $

(4.2)

The same is true for $ \omega^{-} $.

Proof Using the test form $ \varphi = -\omega^{+}\eta^{p} $ and using inequalities (1.9) and the Hölder inequality (1.18), we get

$ \begin{equation} \begin{split} \quad\int_{\Omega}\eta^{p}|D\omega^{+}|^{p}dx\leq& C_{1}\int_{\Omega}|D\omega^{+}|^{p-1}\eta^{p-1}|d\eta||\omega^{+}|dx +C_{2}\int_{\Omega}|D\omega^{+}|^{p-1}|\omega^{+}|\eta^{p}dx\\ \leq& C_{1}(\int_{\Omega}|\omega^{+}|^{p}|d\eta|^{p}dx)^{\frac{1}{p}} (\int_{\Omega}|D\omega^{+}|^{p}\eta^{p}dx)^{\frac{(p-1)}{p}}\\ &+C_{2}(\int_{\Omega}|\omega^{+}|^{p}\eta^{p}dx)^{\frac{1}{p}} (\int_{\Omega}|D\omega^{+}|^{p}\eta^{p}dx)^{\frac{(p-1)}{p}}. \end{split} \end{equation} $

(4.3)

By the following basic inequality

$ \sum\limits_{i = 1}^n | {a_i}{|^s} \le n{(\sum\limits_{i = 1}^n | {a_i}|)^s} \le {n^{s + 1}}\sum\limits_{i = 1}^n | {a_i}{|^s}, $

(4.4)

where $ s>0 $ is any constant, it follows that

$ \begin{equation} \begin{split} \quad\int_{\Omega}\eta^{p}|D\omega^{+}|^{p}dx&\leq (C_{1}(\int_{\Omega}|\omega^{+}|^{p}|d\eta|^{p}dx)^{\frac{1}{p}} +C_{2}(\int_{\Omega}|\omega^{+}|^{p}\eta^{p}dx)^{\frac{1}{p}})^{p}\\ &\leq C_{3}(\int_{\Omega}|\omega^{+}|^{p}|d\eta|^{p}dx+\int_{\Omega}|\omega^{+}|^{p}\eta^{p}dx). \end{split} \end{equation} $

(4.5)

Theorem 4.2 Let $ \omega $ be a solution of (1.8) in $ \Omega, q > 0 $. There exists a constant $ C $, depending only on $ a, p, q, n $ such that

$ \begin{equation} \int_{\Omega}|\omega^{+}|^{q}|D\omega^{+}|^{p}\eta^{p}dx\leq C\int_{\Omega}|\omega^{+}|^{p+q}(|d\eta|^{p}+|\eta|^{p})dx \end{equation} $

(4.6)

for all nonnegative $ \eta \in C^{\infty}_{0}(\Omega) $.

The proof of Theorem 4.2 is similar to [15]. For complete purpose and convenience to the readers, we state the proof as follows.

Proof Let $ t>0 $ be any constant, $ T = \sum\limits_{I}tdx_{I} $, then $ \omega-T $ is also a solution of (1.8) and satisfies (4.2) too. Consider the sets $ A = \cup_{I}\{x|(\omega_{I}-t)^{+} > 0\} $, $ B = \{x|(\omega-T)^{+}\neq 0\} $, $ C = \{x||\omega^{+}| > t\} $, $ D_{I} = \{x|\omega^{+}_{I}>t\} $ and $ B = \{x|(\omega-T)^{+}\neq 0\} $. Then $ D_{I}\subset A = B\subset C $ for all for all $ I $. Let $ dv = |D\omega^{+}|^{p}\eta^{p}dx $ and using (4.2) to get

$ \begin{equation} \begin{split} & \int_{\Omega}|\omega^{+}|^{q}dv \leq C_{1}\sum\limits_{I}\int_{\Omega}|\omega_{I}^{+}|^{q}dv = C_{1}\sum\limits_{I}\{q\int^{\infty}_{0}t^{q-1}\int_{D_{I}}dvdt\}\\ \leq& C_{1}\sum\limits_{I}\{q\int^{\infty}_{0}t^{q-1}\int_{A}|D(\omega^{+}_{I}-t)|^{p}\eta^{p}dxdt\} \leq C_{2}\int^{\infty}_{0}t^{q-1}\int_{B}|D(\omega-T)^{+}|^{p}\eta^{p}dxdt\\ \leq& C_{3}\int^{\infty}_{0}t^{q-1}\int_{B}|(\omega-T)^{+}|^{p}(|d\eta|^{p}+\eta^{p}) {dxdt} \leq C_{3}\int^{\infty}_{0}t^{q-1}\int_{C}|\omega^{+}|^{p}(|d\eta|^{p}+\eta^{p}){dxdt}\\ \leq& C_{3}\int_{\Omega}|\omega^{+}|^{q+p}|(|d\eta|^{p}+\eta^{p})dx. \end{split} \end{equation} $

(4.7)

Lemma 4.1[24] Let $ 0<s<p $ and $ |v|\in L^{p}_{\text{loc}}(\Omega) $, $ \sigma>1 $. If there exists a constant $ C_{1} $ such that

$ \begin{equation} (\int_{Q}|v|^{p}dx)^{\frac{1}{p}}\leq C_{1}|Q|^{\frac{(s-p)}{sp}}(\int_{2Q}|v|^{s}dx)^{\frac{1}{s}} \end{equation} $

(4.8)

for all cubes $ Q $ with $ 2Q\subset\Omega $, then for all $ r > 0 $, there exists a constant $ C_{2} $ depending only on $ \sigma, n, p, r $ and $ C_{1} $, such that

$ \begin{equation} (\int_{Q}|v|^{p}dx)^{\frac{1}{p}}\leq C_{2}|Q|^{\frac{(r-p)}{rp}}(\int_{\sigma Q}|v|^{r}dx)^{\frac{1}{r}} \end{equation} $

(4.9)

for all cubes $ Q $ with $ \sigma Q \subset \Omega $.

Now, we are ready to present and prove one of our main theorems, the weak reverse Hölder inequality as follows.

Theorem 4.3 Let $ \omega $ be a solution to the non-homogeneous Dirac-harmonic equation (1.8) associated with $ p>1 $ in $ \Omega $, $ \sigma>1 $ be some constant, and $ 0 < r, s <\infty $ be any constants. Then, there exists a constant $ C $, independent of $ \omega $, such that

$ \begin{equation} \|\omega\|_{s, B}\leq C|B|^{\frac{(r-s)}{rs}}\|\omega\|_{r, \sigma B} \end{equation} $

(4.10)

for all cubes or balls $ B $ with $ \sigma B\subset \Omega $.

Proof We prove this theorem in the following two steps. Step (i) is for the case $ 1 < p < n $ and Step (ii) is for the case $ p\geq n $.

Step (i) Assume that $ 1 < p < n $. We first prove that (4.10) holds for any constants $ r, s $ with $ r > 0, 0 < s <\frac{np}{n-p} $. Let $ \max\left\{1, \frac{p}{s}, \frac{n}{s(n-1)}\right\}<\alpha<\frac{np}{s(n-p)} $, and $ q = \frac{\alpha ns}{(n+\alpha s)} $. Then

$ \begin{align} q = \frac{\alpha s}{n+\alpha s}\cdot n<n, \quad q-p = \frac{\alpha s(n-p)-np}{n+\alpha s}<0, \quad q-1 = \frac{\alpha s(n-1)-n}{n+\alpha s}>0, \end{align} $

(4.11)

that is, $ 1<q<p<n $, by (2.3), (1.19) and (3.4),

$ \begin{equation} \begin{split} (\int_B|\omega|^{\frac{nq}{n-q}}dx )^{\frac{n-q}{nq}}&\leq C_{1} (\int_{B}|D\omega|^{q}dx )^{\frac{1}{q}} \leq C_{1}|B|^{\frac{1}{q}-\frac{1}{p}} (\int_{B}|D\omega|^{p}dx )^{\frac{1}{p}}\\ &\leq C_{2}|B|^{\frac{(n-q)}{nq}-\frac{1}{p}} (\int_{\sigma B}|\omega|^{p}dx )^{\frac{1}{p}}. \end{split} \end{equation} $

(4.12)

Also, $ \frac{nq}{n-q} = \alpha s>p $, by Lemma 4.1 and (4.12), for any $ r>0 $, we have

$ \begin{equation} (\int_B|\omega|^{\frac{nq}{n-q}}dx )^{\frac{n-q}{nq}}\leq C_{3}|B|^{\frac{(n-q)}{nq}-\frac{1}{r}} (\int_{\sigma B}|\omega|^{r}dx )^{\frac{1}{r}}. \end{equation} $

(4.13)

Since $ \frac{nq}{n-q} = \alpha s>s $, by(1.19), it follows that

$ \begin{equation} (\int_{B}|\omega|^{s}dx )^{\frac{1}{s}}\leq |B|^{\frac{1}{s}-\frac{1}{\alpha s}} (\int_B|\omega|^{\frac{nq}{n-q}}dx )^{\frac{n-q}{nq}}. \end{equation} $

(4.14)

Combining (4.13) and (4.14) yields

$ \begin{equation} (\int_{B}|\omega|^{s}dx )^{\frac{1}{s}}\leq C_{3}|B|^{\frac{r-s}{rs}} (\int_{\sigma B}|\omega|^{r}dx )^{\frac{1}{r}}, \end{equation} $

(4.15)

that is (4.10) holds for any constants $ r, s $ with $ r > 0 $ and $ 0 <s <\frac{np}{n-p} $. Next, we show that (4.10) holds for any $ r>0, s\geq\frac{np}{(n-p)} $. We only need to prove that

$ \begin{equation} \|\omega^{+}\|_{s, B}\leq C_{4}|B|^{\frac{(r-s)}{rs}}\|\omega^{+}\|_{r, \sigma B}. \end{equation} $

(4.16)

The proof for $ \omega^{-} $ is similar. The proof follows Theorem 3.34 in [25] with $ \omega^{+} $ to replace $ u^{+} $ and $ D\omega^{+} $ in replace of $ du^{+} $. Also by using (4.6) and the Moser iteration technique to get (4.16).

Choose $ \alpha >\max\{1, \frac{n}{s(n-1)}\} $ and $ q = \frac{\alpha ns}{(n+\alpha s)} $. Then, $ \frac{nq}{n-q} = \alpha s>s $ and $ 1<q<n $. Let $ r_{m} = \lambda+(1-\lambda)2^{-m} $, $ m = 0, 1, \ldots; 0<\lambda< 1 $, then $ r_{0} = 1 $ and $ r_{m}\rightarrow \lambda $ as $ m \rightarrow \infty $. Let $ \eta_{m}\in C^{\infty} _{0}(r_{m}B) $ be a nonnegative function such that $ \eta_{m} = 1 $ in $ r_{m}B\setminus r_{m+1}B $ and that $ \eta_{m}\leq C_{0}|d\eta_{m}|\leq C_{0}b^{m}|B| $ over each ball, for some constant $ C_{0} $ and $ 1 < b \leq \frac{nq}{n-q} $. Let $ t_{m} \geq 0 $ be a sequence to be determined later and $ \omega_{m } = \omega^{+}|\omega^{+}|^{\frac{t_{m}}{q}}\eta_{m} $. Then by product rule (3.3) and exactly the same calculationas in [10], we get

$ \begin{eqnarray} (\int_{r_{m}B}|D\omega_{m}|^{q}dx )^{\frac{1}{q}}&\leq& (1+2\frac{t_{m}}{q}) (\int_{r_{m}B}|D\omega^{+}|^{q}|\omega^{+}|^{t_{m}}\eta_{ m }^{q}dx )^{\frac{1}{q}} +2 (\int_{r_{m}B}|\omega^{+}|^{q+t_{m}}|d\eta_{ m }|^{q}dx )^{\frac{1}{q}}\\ &\leq& [C_{5}(1+2\frac{t_{m}}{q})+2] (\int_{r_{m}B}|\omega^{+}|^{q+t_{m}}(|d\eta_{ m }|^{q}+|\eta_{ m }|^{q})dx )^{\frac{1}{q}}\\ &\leq& C_{6}(q+t_{m}) (\int_{r_{m}B}|\omega^{+}|^{q+t_{m}}|d\eta_{ m }|^{q}dx )^{\frac{1}{q}}. \end{eqnarray} $

(4.17)

Since the remaining derivation is exactly the same as in [10], we obtain

$ \begin{equation} (\int_{B}|\omega^{+}|^{s}dx )^{\frac{1}{s}}\leq C _{7}|B|^{\frac{r-s}{rs}} (\int_{\sigma B}|\omega^{+}|^{r}dx )^{\frac{1}{r}}. \end{equation} $

(4.18)

Similarly, let $ \omega_{m } = -\omega^{-}|\omega^{-}|^{\frac{t_{m}}{q}}\eta_{m} $ with the same method, we can get

$ \begin{equation} (\int_{B}|\omega^{-}|^{s}dx )^{\frac{1}{s}}\leq C _{7}|B|^{\frac{r-s}{rs}} (\int_{\sigma B}|\omega^{-}|^{r}dx )^{\frac{1}{r}}. \end{equation} $

(4.19)

Thus, by Minkowski inequality, (4.18) and (4.19), we have

$ \begin{equation} \begin{split} &\|\omega\|_{s, B} = \|\omega^{+}+\omega^{-}\|_{s, B} \leq \|\omega^{+}\|_{s, B}+\|\omega^{-}\|_{s, B}\\ \leq& C _{8}|B|^{\frac{r-s}{rs}}(\|\omega^{+}\|_{r, \sigma B}+\|\omega^{-}\|_{r, \sigma B} \leq C _{8}|B|^{\frac{r-s}{rs}}(\|\omega\|_{r, \sigma B}+\|\omega\|_{r, \sigma B})\\ \leq& C _{9}|B|^{\frac{r-s}{rs}}\|\omega\|_{r, \sigma B}, \end{split} \end{equation} $

(4.20)

that is, for any $ r > 0, s \geq \frac{np}{(n-p)} $.

$ \begin{equation} \|\omega\|_{s, B}\leq C _{9}|B|^{\frac{r-s}{rs}}\|\omega\|_{r, \sigma B}. \end{equation} $

(4.21)

Step (ii) Assume that $ p\geq n $. We prove that (4.10) holds for any two positive constants $ r, s $. Let $ \max\{1, \frac{p}{s}, \frac{n}{s(n-1)}\}<\alpha $, and $ q = \frac{\alpha ns}{(n+\alpha s)} $. we find that (4.11)–(4.14) still hold for such a choice of $ \alpha $. Thus, inequality (4.15) or (4.10) follows immediately for any constants $ r, s >0 $. We have completed the proof of Theorem 4.3.

In Theorem 4.3, considering that the norms $ \|\omega\|_{s, Q} $ and $ \|\omega- \omega_{Q}\|_{s, Q} $ are comparable, we have the following version of the weak reverse Hölder inequality for differential forms satisfies the non-homogeneous Dirac-harmonic equation (1.8)

$ \begin{equation} \|\omega-\omega_{Q}\|_{s, Q}\leq C|Q|^{\frac{r-s}{rs}}\|\omega-\omega_{Q}\|_{r, \sigma Q}. \end{equation} $

(4.22)

We know the weight functions were widely used in analysis and PDE, so as applications of the Poincaré inequality and weak reverse Hölder inequality established in previous sections, we prove the Poincaré inequality with the $ A(\varphi_{1}(x), \varphi_{2}(x), \tau, \Omega) $-weights. The $ A(\varphi_{1}(x), \varphi_{2}(x), \tau, \Omega) $-weight was introduced by Wen in [26].

Definition 5.1 Let $ \varphi_{1}(x) $ and $ \varphi_{2}(x) $ be Young functions, a pair of weights $ (w_{1}(x), w_{2}(x)) $ satisfies the $ A(\varphi_{1}(x), \varphi_{2}(x), \tau, \Omega) $-condition, write $ (w_{1}(x), w_{2}(x))\in A(\varphi_{1}(x), \varphi_{2}(x), \tau, \Omega) $, if there exists a constant $ C $, such that

$ \begin{equation} \sup\limits_{Q\subset\Omega}\|w_{1}(x)\|_{\varphi_{1}(x), Q}\|w_{2}(x)^{-1}\|^{\tau}_{\varphi_{2}(x), Q} \leq C<\infty, \end{equation} $

(5.1)

where the normalized Luxemburg norm for any function $ w(x) $ on a cube or ball $ Q $ with Young functions $ \varphi:[0, \infty)\rightarrow[0, \infty) $ is defined by

$ \begin{equation} \|w\|_{\varphi, Q} = \inf\{\lambda>0:\frac{1}{|Q|}\int_{Q}\varphi(\frac{|w(x)|}{\lambda})dx\leq 1\}. \end{equation} $

(5.2)

Remark (i) If $ \varphi(t) = t^{p} $, then $ \|w(x)\|_{\varphi, Q} = (\frac{1}{|Q|}\int_{Q}|w|^{p}dx)^{\frac{1}{p}} $ and the Luxemburg norm reduce to the $ L^{p} $-norm. Given a Young function $ \varphi $, let $ \overline{\varphi} $ denote its associate function: the Young function with the property that $ t \leq\varphi^{-1}(t)\overline{\varphi}^{-1}(t)\leq 2t $, $ t>0 $. If $ \varphi(t) = t^{p} $, $ p>1 $, and $ \frac{1}{p}+\frac{1}{q} = 1 $, then $ \overline{\varphi}(t) = t^{q} $, and if $ \varphi(t) = t^{p}\log(e+t)^{\alpha} $, then $ \overline{\varphi}(t)\approx t^{q}\log(e+t)^{\frac{-\alpha q}{p}} $.

(ii) From [26], we know that $ A(\varphi_{1}(x), \varphi_{2}(x), \tau, \Omega) $-weight is the generation of some existing versions of the weight. For example, if $ \varphi_{1}(x) = x $, $ \varphi_{2}(x) = x^{\frac{1}{r-1}} $, $ \tau = 1 $, and $ w_{1} = w_{2} $ in above definition, then $ A(\varphi_{1}(x), \varphi_{2}(x), \tau, \Omega) $-weight reduces to the $ A_{r}(\Omega) $-weight, and if $ \varphi_{1}(x) = x^{\lambda} $, $ \varphi_{2}(x) = x^{\frac{\lambda}{r-1}} $, $ \tau = 1 $, then $ A(\varphi_{1}(x), \varphi_{2}(x), \tau, \Omega) $-weight reduces to the $ A_{r, \lambda}(\Omega) $-weight, see more details for [26].

Lemma 5.1 If $ \varphi $ is Young function, then for all functions $ f $ and $ g $ and any cube $ Q $,

$ \begin{equation} \frac{1}{|Q|}\int_{Q}|fg|dx\leq 2\parallel f\parallel_{\varphi, Q}\parallel g\parallel_{\overline{\varphi}, Q}. \end{equation} $

(5.3)

Theorem 5.1 Let $ \omega\in L^{s}_{\text{loc}}(\Omega, \wedge^{l}) $ be a solution to equation (1.8) in a bounded domain $ \Omega, l = 0, 1, \ldots, n-1 $, $ D\omega\in L^{s}_{\text{loc}}(\Omega, \wedge) $. and let $ \sigma > 1 $ be a constant. If $ \varphi_{1}(x) $ and $ \varphi_{2}(x) $ are Young functions with $ 1\in L^{\overline{\varphi}_{1}(x)}(\Omega)\cap L^{\overline{\varphi}_{2}(x)}(\Omega) $, and the pair of weights $ (w_{1}(x), w_{2}(x))\in A(\varphi_{1}(x), \varphi_{2}(x), \tau, \Omega) $. Then there exists a constant $ C $, independent of $ \omega $, such that

$ \begin{equation} \|\omega-\omega_{B}\|_{s, B, w^{\alpha}_{1}}\leq C |B|^{1+\frac{1}{n}}\|D\omega\|_{s, \sigma B, w^{\tau\alpha}_{2}} \end{equation} $

(5.4)

for all balls $ B $ with $ \sigma B \subset \Omega $ for some $ \sigma > 1 $ and $ \alpha $ is constant with $ 0 < \alpha< 1 $.

Proof Choose $ t = \frac{s}{1-\alpha} $, since $ \frac{1}{s} = \frac{1}{t}+ \frac{(t-s)}{ts} $ and $ t>s $, using the Hölder inequality (1.18), we have

$ \begin{equation} \begin{split} \|\omega-\omega_{B}\|_{s, B, w^{\alpha}_{1}}& = (\int_{B}(|\omega-\omega_{B}|w^{\frac{\alpha}{s}}_{1})^{s}dx )^{\frac{1}{s}} \leq (\int_{B}|\omega-\omega_{B}|^{t}dx )^{\frac{1}{t}} (\int_{B}w^{\frac{\alpha t}{t-s}}_{1}dx )^{\frac{t-s}{ts}}\\ & = \|\omega-\omega_{B}\|_{t, B} (\int_{B}w_{1}dx )^{\frac{\alpha}{s}} = \|\omega-\omega_{B}\|_{t, B}\|w_{1}\|^{\frac{\alpha}{s}}_{1, B}. \end{split} \end{equation} $

(5.5)

Taking $ m = \frac{s}{1+\alpha\tau} $, then $ m<s<t $. Using (4.22) and (2.7), we have

$ \begin{equation} \|\omega-\omega_{B}\|_{t, B}\leq C_{2}|B|^{\frac{(m-t)}{mt}}\|\omega-\omega_{B}\|_{m, \sigma B}\leq C_{3}|B|^{1+\frac{1}{n}}|B|^{\frac{(m-t)}{mt}}\|D\omega\|_{m, \sigma B}, \end{equation} $

(5.6)

where $ \sigma>1 $. Substituting (5.6) in (5.5), we have

$ \begin{equation} (\int_{B}(|\omega-\omega_{B}|w^{\frac{\alpha}{s}}_{1})^{s}dx )^{\frac{1}{s}} \leq C_{3}|B|^{1+\frac{1}{n}}|B|^{\frac{(m-t)}{mt}}\|D\omega\|_{m, \sigma B}\|w_{1}\|^{\frac{\alpha}{s}}_{1, B}. \end{equation} $

(5.7)

Using (1.18) again yields

$ \begin{equation} \begin{split} \|D\omega\|_{m, \sigma B}& = (\int_{\sigma B}|D\omega|^{m}dx )^{\frac{1}{m}} = (\int_{\sigma B}(|D\omega|w^{\frac{\tau\alpha}{s}}_{2}w^{\frac{-\tau\alpha}{s}}_{2})^{m}dx )^{\frac{1}{m}}\\ &\leq (\int_{\sigma B}(|D\omega|^{s}w^{\tau\alpha}_{2}dx )^{\frac{1}{s}} (\int_{\sigma B}(\frac{1}{w_{2}})^{\frac{\tau\alpha m}{s-m}}dx )^{\frac{s-m}{ms}}\\ & = (\int_{\sigma B}(|D\omega|^{s}w^{\tau\alpha}_{2}dx )^{\frac{1}{s}}\|\frac{1}{w_{2}}\|^{\frac{\tau\alpha}{s}}_{1, \sigma B} \end{split} \end{equation} $

(5.8)

for all balls $ B $ with $ \sigma B \subset \Omega $. Substituting (5.8) into (5.7), we find that

$ \begin{equation} \|\omega-\omega_{B}\|_{s, B, w^{\alpha}_{1}}\leq C_{3}|B|^{1+\frac{1}{n}}|B|^{\frac{(m-t)}{mt}}\|w_{1}\|^{\frac{\alpha}{s}}_{1, B}\|\frac{1}{w_{2}}\|^{\frac{\tau\alpha}{s}}_{1, \sigma B} (\int_{\sigma B}(|D\omega|^{s}w^{\tau\alpha}_{2}dx )^{\frac{1}{s}}. \end{equation} $

(5.9)

Since $ (w_{1}(x), w_{2}(x))\in A(\varphi_{1}(x), \varphi_{2}(x), \tau, \Omega), $

$ \begin{equation} \begin{split} &\|w_{1}\|^{\frac{\alpha}{s}}_{1, B}\|w^{-1}_{2}\|^{\frac{\alpha\tau}{s}}_{1, \sigma B} = (\|w_{1}\|_{1, B}\|w^{-1}_{2}\|^{\tau}_{1, \sigma B})^{\frac{\alpha}{s}}\\ \leq& (C_{2}|B|^{(1+\tau)} (\frac{1}{|\sigma B|}\int_{\sigma B}|w_{1}(x)|dx ) (\frac{1}{|\sigma B|} \int_{\sigma B}|\frac{1}{w_{2}(x)}|dx )^{\tau} )^{\frac{\alpha}{s}}\\ \leq& C_{3}|B|^{\frac{\alpha}{s}(1+\tau)}\|w_{1}(x)\|^{\frac{\alpha}{s}}_{\varphi_{1}(x), \sigma B}\|1\|^{\frac{\alpha}{s}}_{\overline{\varphi_{1}}(x), \sigma B}\|w^{-1}_{2}(x)\|^{\frac{\alpha\tau}{s}}_{\varphi_{2}(x), \sigma B}\|1\|^{\frac{\alpha\tau}{s}}_{\overline{\varphi_{2}}(x), \sigma B}\\ \leq& C_{4}|B|^{\frac{\alpha}{s}(1+\tau)}, \end{split} \end{equation} $

(5.10)

noting that

$ \begin{equation} \frac{\alpha}{s}(1+\tau)+\frac{1}{t}-\frac{1}{m} = 0. \end{equation} $

(5.11)

Combining (5.9)–(5.11), we conclude that $ \|\omega-\omega_{B}\|_{s, B, w^{\alpha}_{1}}\leq C |B|^{1+\frac{1}{n}}\|D\omega\|_{s, \sigma B, w^{\tau\alpha}_{2}} $ for all balls $ B $ with $ \sigma B \subset \Omega $. This ends the proof of Theorem 5.1.

If $ \varphi_{1}(x) = x $, $ \varphi_{2}(x) = x^{\frac{1}{r-1}} $, $ \tau = 1 $, and $ w_{1} = w_{2} $ in above definition, then $ A(\varphi_{1}(x), \varphi_{2}(x), $ $ \tau, \Omega) $-weight reduces to the $ A_{r}(\Omega) $-weight, then we have the following corollary.

Corollary 5.1 Let $ \omega\in L^{s}_{\text{loc}}(\Omega, \wedge^{l}) $ be a solution to equation(1.8) in a bounded domain $ \Omega, l = 0, 1, \ldots, n-1 $, $ D\omega\in L^{s}_{\text{loc}}(\Omega, \wedge) $, and let $ \sigma > 1 $ be a constant. If the weight $ w(x)\in A_{r}(\Omega) $. Then there exists a constant $ C $, independent of $ \omega $, such that

$ \begin{equation} \|\omega-\omega_{B}\|_{s, B, w^{\alpha}}\leq C |B|^{1+\frac{1}{n}}\|D\omega\|_{s, \sigma B, w^{\alpha}} \end{equation} $

(5.12)

for all balls $ B $ with $ \sigma B \subset \Omega $ for some $ \sigma > 1 $ and $ \alpha $ is constant with $ 0 < \alpha< 1 $.

As an applications of Corollary 5.1, we can prove the global result in $ L^{s}(\mu) $-averaging domain.

Definition 5.2[27] A proper subdomain $ \Omega\subset \mathbb{R}^n $ is called an $ L^{s}(\mu) $-averaging domain, $ s\geq 1 $, if $ \mu(\Omega)<\infty $ and there exists a constant $ C $ such that

$ \begin{equation} (\frac{1}{\mu(B_{0})}\int_{\Omega}|u-u_{B_{0}}|^{s}d\mu )^{\frac{1}{s}}\leq C\sup\limits_{2B\subset \Omega} (\frac{1}{\mu(B)}\int_{B}|u-u_{B}|^{s}d\mu )^{\frac{1}{s}} \end{equation} $

(5.13)

for some ball $ B_{0}\subset\Omega $ and all $ u\in L^{s}_{\text{loc}}(\Omega;\wedge^{l}) $. Here the measure $ \mu $ is defined by $ d\mu = w(x)dx $, where $ w(x) $ is a weight and $ w(x)>0 $ a.e., and the supremum is over all balls $ B\subset \Omega $.

Theorem 5.2 Let $ \omega\in L^{s}_{\text{loc}}(\Omega, \wedge^{l}) $ be a solution to equation (1.8) in a bounded domain $ \Omega, l = 0, 1, \ldots, n-1 $, $ D\omega\in L^{s}_{\text{loc}}(\Omega, \wedge) $. If $ 1<r<\infty $ and $ w\in A_{r}(\Omega) $ with $ w(x)\geq \varepsilon>0 $, then, there exists a constant $ C $, independent of $ \omega $ and $ D\omega $, such that

$ \begin{equation} (\frac{1}{\mu(\Omega)}\int_{\Omega}|\omega-\omega_{B_{0}}|^{s}d\mu )^{\frac{1}{s}}\leq C (\int_{\Omega}|D\omega|^{s}d\mu )^{\frac{1}{s}} \end{equation} $

(5.14)

for any $ L^{s}(\mu) $-averaging domain $ \Omega $ and some ball $ B_{0} $ with $ 2B_{0}\subset \Omega $. Here $ d\mu = w(x)^{\alpha}dx $.

Proof For any ball $ B\subset \Omega $, $ \mu(B) = \int_{B}w(x)^{\alpha}dx\geq\int_{B}\varepsilon^{\alpha} dx = \varepsilon^{\alpha}|B|, $ so that $ \frac{1}{\mu(B)}\leq\frac{C_{1}}{|B|}, $ where $ C_{1} = \frac{1}{\varepsilon^{\alpha}} $. By Corollary 5.1 and noticing that $ 1+\frac{1}{n}-\frac{1}{s}>0 $, we obtain

$ \begin{eqnarray} && (\frac{1}{\mu(B)}\int_{B}|\omega-\omega_{B}|^{s}d\mu )^{\frac{1}{s}} = (\mu_{B} )^{\frac{-1}{s}} (\int_{B}|\omega-\omega_{B}|^{s}d\mu )^{\frac{1}{s}}\\ &\leq& C_{2}|B|^{\frac{-1}{s}} (\int_{B}|\omega-\omega_{B}|^{s}d\mu )^{\frac{1}{s}} \leq C_{2}|B|^{\frac{-1}{s}}C_{3}|B|^{1+\frac{1}{n}} (\int_{B}|D\omega|^{s}d\mu )^{\frac{1}{s}} = C_{4} (\int_{B}|D\omega|^{s}d\mu )^{\frac{1}{s}}. \end{eqnarray} $

(5.15)

Thus, by (5.15) and the definition of $ L^{s}(\mu) $-averaging domain, we deduce that

$ \begin{equation} \begin{split} & (\frac{1}{\mu(\Omega)}\int_{\Omega}|\omega-\omega_{B_{0}}|^{s}d\mu )^{\frac{1}{s}}\leq (\frac{1}{\mu(B_{0})}\int_{\Omega}|\omega-\omega_{B_{0}}|^{s}d\mu )^{\frac{1}{s}}\\ \leq& C_{5}\sup\limits_{2B\subset \Omega} (\frac{1}{\mu(B)}\int_{B}|\omega-\omega_{B}|^{s}d\mu )^{\frac{1}{s}} \leq C_{5}\sup\limits_{2B\subset \Omega}(C_{4} (\int_{\sigma B}|D\omega|^{s}d\mu )^{\frac{1}{s}})\\ \leq& C_{6}\sup\limits_{2B\subset \Omega} (\int_{\Omega}|D\omega|^{s}d\mu )^{\frac{1}{s}}) = C_{6} (\int_{\Omega}|D\omega|^{s}d\mu )^{\frac{1}{s}}). \end{split} \end{equation} $

(5.16)

We complete the proof of Theorem 5.2.