Weak and strong type estimates for multilinear Calderón–Zygmund operators on differential forms

Abstract

In this paper, we define the multilinear Calderón–Zygmund operators on differential forms and prove the end-point weak type boundedness of the operators. Based on nonhomogeneous A-harmonic tensor, the Poincaré-type inequalities for multilinear Calderón–Zygmund operators on differential forms are obtained.

1 Introduction

The multilinear Calderón–Zygmund theory was originally introduced by Coifman and Meyer [1–3] in their study of certain singular integral operators, such as Calderón commutators, paraproducts, and pseudodifferential operators. Afterwards, the multilinear Calderón–Zygmund theory has been further developed by many scholars in the last few decades. For example, Grafakos and Torres studied systematically on the multilinear Calderón–Zygmund operators in [4]. They proved an end-point weak type estimate and obtained the strong type $L^{p_{1}}\times \cdots \times L^{p_{m}}\rightarrow L^{p}$ boundedness results for multilinear Calderón–Zygmund operators by the classical interpolation method. In [5] and [6], the maximal operator associated with multilinear Calderón–Zygmund singular integrals was introduced and used to obtain the weighted norm estimates for multilinear singular integrals. More recently, Stockdale and Wick [7] provided an alternative proof of the weak-type $(1,\ldots ,1;1/m)$ estimates for m-multilinear Calderón–Zygmund operators on $\mathbb{R}^{n}$ first proved by Grafakos and Torres. Subsequent results on end-point estimates of the bilinear Calderón–Zygmund operators can be found in [8]. Motivated by the research and considering that differential forms as the generalizations of functions are widely used in physics systems, differential geometry, and PDEs, we aim to establish the boundedness of multilinear Calderón–Zygmund operators on differential forms. In this paper, the definition of multilinear Calderón–Zygmund operators on differential forms is set forth. Moreover, by combining the Calderón–Zygmund decomposition with some skillful techniques, we establish the end-point weak type boundedness of multilinear Calderón–Zygmund operators on differential forms which includes the result of multilinear Calderón–Zygmund operators on functions in [4] as a special case. Unfortunately, it is difficult to apply the complex variable theory to differential forms. So the T1 theorem and strong type boundedness of multilinear Calderón–Zygmund operators on differential forms are still some open questions. But with the help of decomposition theorem of differential forms, we derive the Poincaré-type inequalities for multilinear Calderón–Zygmund operators on A-harmonic tensors which are the generalized solutions to A-harmonic equations on differential forms. Based on the Poincaré-type inequalities, we can make a further study of the multilinear Calderón–Zygmund operators in Orlicz spaces and establish the $L^{\varphi }$ norm inequalities. More results on the Poincaré-type inequalities and singular integral operators on differential forms can be found in [9–13].

This work is organized as follows. To state our results, we first recall some necessary notations and lemmas in Sect. 2. Then, in Sect. 3, we define the multilinear Calderón–Zygmund operators on differential forms and prove the end-point weak type boundedness of multilinear Calderón–Zygmund operators on differential forms in Theorem 1. Using the weak type inequality, we derive the Poincaré-type inequality for multilinear Calderón–Zygmund operator on a local domain in Theorem 2 in Sect. 4. Finally, the result is extended to obtain the Poincaré-type inequality for the multilinear Calderón–Zygmund operator on a bounded convex domain in Theorem 3.

2 Preliminaries

Throughout this paper, let $\Omega \subset \mathbb{R}^{n}$ be a bounded domain, $n\geq 2$, B and σB be the balls with the same center and $\operatorname{diam}(\sigma B)=\sigma \operatorname{diam}(B)$. We use $|E|$ to denote the Lebesgue measure of a set $E\subset \mathbb{R}^{n}$. Let $\Lambda ^{l}(\mathbb{R}^{n})=\Lambda ^{l}$, $l=1,2,\ldots ,n$, be the set of all l-forms $u(x)=\sum_{I}u_{I}(x)\,dx_{I}=\sum u_{i_{1} \cdots i_{l}}(x)\,dx_{i_{1}} \wedge \cdots \wedge {dx_{i_{l}}}$ with summation over all ordered l-tuples $I=(i_{1},i_{2},\ldots ,i_{l})$, $1\leq {i_{1}}<\cdots <i_{l}\leq {n}$. $D^{\prime }(\Omega ,\Lambda ^{l})$ is the space of all differential l-forms on Ω, namely, the coefficient of the l-forms is differential on Ω. The operator $\star :\Lambda ^{l}(\mathbb{R}^{n})\rightarrow \Lambda ^{n-l}( \mathbb{R}^{n})$ is the Hodge–Star operator as usual, and the linear operator $d:D^{\prime }(\Omega ,\Lambda ^{l})\rightarrow {D^{\prime }(\Omega ,\Lambda ^{l+1})}$, $0\leq l\leq n-1$, is called the exterior differential operator. The Hodge codifferential operator $d^{\star }:D^{\prime }(\Omega ,\Lambda ^{l+1})\rightarrow {D^{\prime }(\Omega , \Lambda ^{l})}$, the formal adjoint of d, is defined by $d^{\star }=(-1)^{nl+1}\star \,d\star$, see [14] for more introduction. We shall denote by $L^{p}(\Omega ,\Lambda ^{l})$ the space of differential l-forms with the coefficients in $L^{p}(\Omega ,\mathbb{R}^{n})$ and with the norm $\|u\|_{p,\Omega }= (\int _{\Omega } (\sum_{I}|u_{I}(x)|^{2} )^{ \frac{p}{2}}\,dx )^{\frac{1}{p}}$. The homotopy operator $T:C^{\infty }(\Omega ,\Lambda ^{l})\rightarrow C^{\infty }(\Omega , \Lambda ^{l-1})$ is a very important operator in the theory of differential forms, given by

$$ Tu= \int _{\Omega }\psi (y)K_{y}u\,dy, $$

where $\psi \in C^{\infty }_{0}(\Omega )$ is normalized by $\int _{\Omega }\psi (y)\,dy=1$, and $K_{y}$ is a linear operator defined by

$$ (K_{y}u) (x;\xi _{1},\ldots ,\xi _{l-1})= \int ^{1}_{0}t^{l-1}u(tx+y-ty;x-y; \xi _{1},\ldots ,\xi _{l-1})\,dt. $$

See [15] for more of the function ψ and the operator $K_{y}$. About the homotopy operator T, we have the following decomposition:

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

for any differential form $u\in L^{p}(\Omega ,\Lambda ^{l})$, $1\leq p<\infty$. We also call it the decomposition theorem for differential form which will be used repeatedly in the proof of this paper. A closed form $u_{\Omega }$ is defined by $u_{\Omega }=d(Tu)$, $l=1, \ldots , n$, and when u is a differential 0-form, $u_{\Omega }=|\Omega |^{-1}\int _{\Omega }u(y)\,dy$.

The following lemma is the $L^{p}$ estimate for the homotopy operator T from [15].

Lemma 1

Let $u\in L^{p}_{loc}(\Omega ,\Lambda ^{l})$, $l=1,2,\ldots ,n,1< p<\infty$, be a differential form in Ω, and T be the homotopy operator defined on differential forms. Then there exists a constant C, independent of u, such that

$$ \Vert Tu \Vert _{p,\Omega }\leq C\operatorname{diam}(\Omega ) \Vert u \Vert _{p,\Omega }. $$

The following nonlinear partial differential equation for differential forms

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

(1)

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

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

for $x \in \Omega$ a.e. and all $\xi \in \Lambda ^{l} (\mathbb{R}^{n})$. Here, $p>1$ is a constant related to equation (1), and $a, b >0$.

In general, we call the differential form satisfying nonhomogeneous A-harmonic equation the nonhomogeneous A-harmonic tensor. In the proof of the strong type inequality for the multilinear Calderón–Zygmund operator on differential forms, we let the differential form be the nonhomogeneous A-harmonic tensor to get the desired result. See [16–20] for a list of recent results on the A-harmonic equations and related topics. We also need the following weak inverse Hölder inequality for nonhomogeneous A-harmonic tensor, see [21] for more introduction.

Lemma 2

Let $u\in \Omega$ satisfy equation (1), $\sigma >1$, $0 < s, t < \infty$. Then there exists a constant C, independent of u, such that

$$ \Vert u \Vert _{s,B} \leq C \vert B \vert ^{(t-s)/{st}} \Vert u \Vert _{t, \sigma B} $$

for $\sigma B \subset \Omega$.

3 End-point estimate

In the classical theory of singular integrals, it is important to prove the end-point weak type inequality which is the core link to get the strong boundedness of singular integral operators. So the focus of this section is to establish the end-point weak type inequality of multilinear Calderón–Zygmund operators on differential forms. Before giving the definition of multilinear Calderón–Zygmund operators on differential forms, we first define the kernel function in multilinear Calderón–Zygmund operators, see [4] for more introduction about the kernel function.

Definition 1

Let $K(x,y_{1},\ldots ,y_{m})$ be a locally integrable function which is defined on

$$ \bigl(\mathbb{R}^{n}\bigr)^{m+1}\setminus \bigl\{ \tau \in \bigl(\mathbb{R}^{n}\bigr)^{m+1},x=y_{1}= \cdots =y_{m}\bigr\} $$

and satisfy the following conditions:

$(1)$ For some $A>0$ and all points in the domain of definition, the function $K(x,y_{1},\ldots ,y_{m})$ satisfies

$$ \bigl\vert K(x,y_{1},\ldots ,y_{m}) \bigr\vert \leq \frac{A}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn}}; $$

$(2)$ For $\varepsilon >0$, we have

$$\begin{aligned} & \bigl\vert K(x,y_{1},\ldots ,y_{m})-K\bigl(x',y_{1},\ldots , y_{m}\bigr) \bigr\vert \\ &\quad \le \frac{A \vert x-x' \vert ^{\varepsilon }}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn+\varepsilon }}, \end{aligned}$$

(2)

where $|x-x'|\le \frac{1}{2}\max_{1\le j\le m}|x-y_{j}|$. For other $y_{j}$, we also have

$$\begin{aligned} & \bigl\vert K(x,y_{1},\ldots , y_{j},\ldots , y_{m})-K\bigl(x,y_{1},\ldots , y'_{j}, \ldots , y_{m}\bigr) \bigr\vert \\ &\quad \le \frac{A \vert y_{j}-y_{j}' \vert ^{\varepsilon }}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn+\varepsilon }}, \end{aligned}$$

where $|y_{j}-y'_{j}|\le \frac{1}{2}\max_{1\le j\le m}|x-y_{j}|$. For convenience, we call conditions $(1)$, $(2)$ for kernel function the $m-\operatorname{CZK}(A,\varepsilon )$ conditions.

Next, we give the definition of multilinear Calderón–Zygmund operators on differential forms.

Definition 2

The operator $\mathcal{L}:\Lambda ^{l}(\mathbb{R}^{n})\times \Lambda ^{l}( \mathbb{R}^{n})\times \cdots \times \Lambda ^{l}(\mathbb{R}^{n}) \rightarrow \Lambda ^{l}(\mathbb{R}^{n})$ is called multilinear operator on differential forms if

$$\begin{aligned} &\mathcal{L}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr) (x;{\xi }) \\ &\quad = \biggl( \int _{(\mathbb{R}^{n})^{m}}K(x,y_{1},\ldots ,y_{m})u^{(1)}(x;{ \xi })\cdots u^{(m)}(x;{\xi })\,\mathrm{d}y_{1}\cdots \mathrm{d}y_{m} \biggr), \end{aligned}$$

where $x\notin \bigcap^{m}_{i=1}\operatorname{supp} u^{(i)}_{I}$, ${\xi }=(\xi _{1},\xi _{2},\ldots ,\xi _{l})$.

Now, we establish the end-point weak type inequality for multilinear Calderón–Zygmund operators on differential forms.

Theorem 1

Let $\mathcal{L}$ be a multilinear Calderón–Zygmund operator, the kernel function K satisfies the $m-\operatorname{CZK}(A,\varepsilon )$ conditions, and $u^{(1)},\ldots ,u^{(m)}\subset D'(\Omega ,\Lambda ^{l})$. Assume that, for $1\leq q_{1},q_{2},\ldots ,q_{m}\leq \infty$ and $0< q<\infty$ with

$$ \frac{1}{q}=\frac{1}{q_{1}}+\frac{1}{q_{2}}+\cdots + \frac{1}{q_{m}}, $$

if the operator $\mathcal{L}$ is weak-$(q_{1},\ldots ,q_{m},q)$, that is,

$$ \bigl\vert \bigl\{ x: \bigl\vert \mathcal{L}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr) \bigr\vert >\lambda \bigr\} \bigr\vert \leq \biggl( \frac{B}{\lambda } \bigl\Vert u^{(1)} \bigr\Vert _{q_{1}} \cdots \bigl\Vert u^{(m)} \bigr\Vert _{q_{m}} \biggr)^{q} $$

for all $\lambda >0$, then $\mathcal{L}$ is also a weak-$(1,1,\ldots ,1, \frac{1}{m})$ operator. Especially, we have

$$ [\mathcal{L}]_{L^{1}\times \cdots \times L^{1}\rightarrow L^{1/m}} \leq C\bigl(A+[\mathcal{L}]_{L^{q_{1}}\times \cdots \times L^{q_{m}} \rightarrow L^{q}} \bigr), $$

where $[\cdot ]$ means the weak norm of an operator and $C>0$ is a constant that depends only on the parameters n, m.

Proof

Without loss of generality, suppose

$$ \bigl\Vert u^{(1)} \bigr\Vert _{1}= \bigl\Vert u^{(2)} \bigr\Vert _{1}=\cdots = \bigl\Vert u^{(m)} \bigr\Vert _{1}=1. $$

In order to prove the conclusion, we need to show that there exists a constant C independent of $u^{(1)},\ldots ,u^{(m)}$ such that

$$\begin{aligned} \bigl(\mathcal{L}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr) \bigr)_{*}(\lambda ) =& \bigl\vert \bigl\{ x: \bigl\vert \mathcal{L}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr) \bigr\vert > \lambda \bigr\} \bigr\vert \\ \leq & \biggl\{ \frac{C}{\lambda }(A+B) \biggr\} ^{\frac{1}{m}}. \end{aligned}$$

(3)

According to the algorithms of differential forms, for $i_{1}< i_{2}<\cdots <i_{k}$ and $j_{1}< j_{2}<\cdots <j_{k}$, we obtain

$$ \mathrm{d}x_{i_{1}}\wedge \mathrm{d}x_{i_{2}}\wedge \cdots \wedge \mathrm{d}x_{i_{k}}(e_{j_{1}},e_{j_{2}},\ldots ,e_{j_{k}})=\textstyle\begin{cases} 1, & \text{if }i_{1}=j_{1},i_{2}=j_{2},\ldots ,i_{k}=j_{k}, \\ 0, & \text{others}, \end{cases} $$

where $e_{1},e_{2},\ldots ,e_{n}$ are orthogonal basis for the tangent space of $\mathbb{R}^{n}$. So, for a differential form

$$ u(x)=\sum_{I}{u_{I}(x)} \,\mathrm{d}x_{I} $$

with $I=(i_{1},i_{2},\ldots ,i_{l})$, $1\leq {i_{1}}<{i_{2}}<\cdots <{i_{l}}\leq n$, we have

$$ u_{I}=u(x) ({e}_{I}), $$

where ${e}_{I}=(e_{i_{1}},e_{i_{2}},\ldots ,e_{i_{l}})$. It is also a differential form for the image of the multilinear Calderón–Zygmund operator on a differential form. So, there exist $a_{I}, I=(i_{1},i_{2},\ldots ,i_{l})$, $1\leq {i_{1}}<{i_{2}}<\cdots <{i_{l}}\leq n$ satisfying

$$ \mathcal{L}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr)=\sum _{I}a_{I}\,\mathrm{d}x_{I}. $$

If we write $\mathrm{d}{y}=\mathrm{d}y_{1}\cdots \mathrm{d}y_{m}$, then it follows that

$$\begin{aligned} & \bigl\vert \mathcal{L}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr) (x) \bigr\vert = \biggl\vert \sum_{I}a_{I} \,\mathrm{d}x_{I} \biggr\vert \\ &\quad = \biggl(\sum_{I} \vert a_{I} \vert ^{2} \biggr)^{1/2}= \biggl(\sum _{I} \bigl\vert \mathcal{L}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr) (x;{e}_{I}) \bigr\vert ^{2} \biggr)^{1/2} \\ &\quad = \biggl(\sum_{I} \biggl\vert \biggl( \int _{(\mathbb{R}^{n})^{m}}K(x,y_{1}, \ldots ,y_{m})u^{(1)}(x;{e}_{I}) \cdots u^{(m)}(x;{e}_{I})\,\mathrm{d} {y} \biggr) \biggr\vert ^{2} \biggr)^{1/2} \\ &\quad = \biggl(\sum_{I} \biggl\vert \biggl( \int _{(\mathbb{R}^{n})^{m}}K(x,y_{1}, \ldots ,y_{m})u^{(1)}_{I} \cdots u^{(m)}_{I}\,\mathrm{d} {y} \biggr) \biggr\vert ^{2} \biggr)^{1/2}. \end{aligned}$$

Applying the Calderón–Zygmund decomposition to each function $u^{(i)}_{I}$ at height $\alpha =(\lambda \rho )^{1/m}$, we obtain a sequence of pairwise disjoint cubes $\{Q_{i,k_{i}}\}_{k_{i}=1}^{\infty }$ and a decomposition

$$ u^{(i)}_{I}=g^{(i)}_{I}+b^{(i)}_{I}=g^{(i)}_{I}+ \sum_{k_{i}}b^{(i,k_{i})}_{I} $$

such that, for all $i = 1, \ldots , m$,

$(a)$ $|g^{(i)}_{I}|\leq C_{1}\alpha$,

$(b)$ $\operatorname{supp}(b^{(i,k_{i})}_{I})\subset Q_{i,k_{i}}$, $\int _{Q_{i,k_{i}}} b^{(i,k_{i})}_{I}(x)\,\mathrm{d}x=0$, and

$$ \int _{Q_{i,k_{i}}} \bigl\vert b^{(i,k_{i})}_{I}(x) \bigr\vert \,\mathrm{d}x\leq C_{1}\alpha \vert Q_{i,k_{i}} \vert , $$

$(c)$ $\sum_{k_{i}}|Q_{i,k_{i}}|\leq \frac{C_{1}}{\alpha }\int _{ \mathbb{R}^{n}}|u^{(i)}_{I}(x)|\,dx$.

Applying the Calderón–Zygmund decomposition above to the operator $\mathcal{L}$, we have

$$\begin{aligned} &\bigl(\mathcal{L}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr)\bigr)_{*}(\lambda ) \\ &\quad = \bigl\vert \bigl\{ x: \bigl\vert \mathcal{L}\bigl(u^{(1)}, \ldots ,u^{(m)}\bigr) \bigr\vert >\lambda \bigr\} \bigr\vert \\ &\quad = \biggl| \biggl\{ x: \biggl(\sum_{I}\bigl( \mathcal{L}\bigl(u^{(1)}_{I}, \ldots ,u^{(m)}_{I} \bigr)\bigr)^{2} \biggr)^{\frac{1}{2}}>\lambda \biggr\} \biggr| \\ &\quad \leq \sum_{I} \biggl\vert \biggl\{ x: \mathcal{L}\bigl(u^{(1)}_{I},\ldots ,u^{(m)}_{I} \bigr)> \frac{\lambda }{\mathrm{C}^{l}_{n}} \biggr\} \biggr\vert \\ &\quad \leq \sum_{I} \biggl( \biggl\vert \biggl\{ x:\mathcal{L}\bigl(g^{(1)}_{I}, \ldots ,g^{(m)}_{I} \bigr)>\frac{\lambda }{2^{m}\mathrm{C}^{l}_{n}} \biggr\} \biggr\vert \biggr) \\ &\qquad {}+\sum_{I} \biggl(\sum _{\exists h^{(i)}_{I}=b^{(i)}_{I}} \biggl\vert \biggl\{ x:\mathcal{L} \bigl(h^{(1)}_{I},\ldots ,h^{(m)}_{I} \bigr)> \frac{\lambda }{2^{m}\mathrm{C}^{l}_{n}} \biggr\} \biggr\vert \biggr), \end{aligned}$$

(4)

where $h^{(i)}_{I}\in \{b^{(i)}_{I},g^{(i)}_{I}\}$. Then, by properties $(a)$ and $(c)$, one has

$$\begin{aligned} \bigl\Vert g^{(i)}_{I} \bigr\Vert ^{q_{i}}_{q_{i}} =& \int _{(\bigcup Q_{i,k_{i}})^{c}} \bigl\vert g^{(i)}_{I}(x) \bigr\vert ^{q_{i}} \,\mathrm{d}x+ \int _{\bigcup Q_{i,k_{i}}} \bigl\vert g^{(i)}_{I}(x) \bigr\vert ^{q_{i}}\,\mathrm{d}x \\ =& \int _{(\bigcup Q_{i,k_{i}})^{c}} \bigl\vert g^{(i)}_{I}(x) \bigr\vert ^{q_{i}-1} \bigl\vert u^{(i)}_{I}(x) \bigr\vert \,\mathrm{d}x+ \int _{\bigcup Q_{i,k_{i}}} \bigl\vert g^{(i)}_{I}(x) \bigr\vert ^{q_{i}}\,\mathrm{d}x \\ \leq & C_{1}\alpha ^{q_{i}-1} \int \bigl\vert u^{(i)}_{I}(x) \bigr\vert \,\mathrm{d}x+C_{1} \alpha ^{q_{i}}\biggl|\biggl\{ \bigcup Q_{i,k_{i}}\biggr\} \biggr| \\ \leq & C_{1}\alpha ^{q_{i}-1} \bigl\Vert u^{(i)}_{I} \bigr\Vert _{1} \\ \leq & C_{1}\alpha ^{q_{i}-1} \bigl\Vert u^{(i)} \bigr\Vert _{1}. \end{aligned}$$

(5)

Using (5) and the weak $-(q_{1},q_{2},\ldots ,q_{m},q)$ boundedness to the first item of the right-hand side of inequality (4), we get

$$\begin{aligned} & \biggl\vert \biggl\{ x:\mathcal{L} \bigl(g^{(1)}_{I},\ldots ,g^{(m)}_{I} \bigr)> \frac{\lambda }{2^{m}\mathrm{C}^{l}_{n}} \biggr\} \biggr\vert \\ &\quad \leq \biggl(\frac{B2^{m}\mathrm{C}^{l}_{n}}{\lambda } \bigl\Vert g^{(1)}_{I} \bigr\Vert _{q_{1}} \cdots \bigl\Vert g^{(m)}_{I} \bigr\Vert _{q_{m}} \biggr)^{q} \\ &\quad \leq \frac{C_{1}B^{q}}{\lambda ^{q}}\prod^{m}_{i=1} \alpha ^{\frac{q(q_{i}-1)}{q_{i}}} \bigl\Vert u^{(i)} \bigr\Vert ^{q/q_{i}}_{1} \\ &\quad =\frac{C_{1}B^{q}}{\lambda ^{q}}(\lambda \rho )^{\frac{q}{m}(m- \frac{1}{q})} \\ &\quad =C_{1}B^{q}\lambda ^{-\frac{1}{m}}\rho ^{q-\frac{1}{m}}. \end{aligned}$$

(6)

Now we begin to estimate the measure of the following set which appeared in (4):

$$ \vert E_{s} \vert = \biggl\vert \biggl\{ x:\mathcal{L} \bigl(h^{(1)}_{I},\ldots ,h^{(m)}_{I} \bigr)> \frac{\lambda }{2^{m}\mathrm{C}^{l}_{n}} \biggr\} \biggr\vert , $$

where $h^{(i_{r})}_{I}=b^{(i_{r})}_{I}$, $1 \le r \le t$, and $h^{(i_{t+j})}_{I}=g^{(i_{t+j})}_{I}$, $1\le j\le m-t$, $1 \le t \le m$. For the sake of simplicity, there is no harm in the setting $b^{(i_{r})}_{I}$ appearing with superscript $1,\ldots ,t$. We set that $\Omega =\bigcup _{r=1}^{t}\Omega _{r}=\bigcup _{r=1}^{t}\bigcup _{k_{r}} \tilde{Q}_{r,k_{r}}$ with $\tilde{Q}_{r,k_{r}}$ is a concentric and double diameter cube to $Q_{r,k_{r}}$. Then we obtain

$$\begin{aligned} \vert E_{s} \vert \le & \vert \Omega \vert + \biggl\vert \biggl\{ x\in \mathbb{R}^{n}\setminus \Omega : \mathcal{L}\bigl(b^{(1)}_{I},\ldots ,b^{(t)}_{I},g^{(t+1)}_{I} ,\ldots ,g^{(m)}_{I}\bigr)>\frac{\lambda }{2^{m}\mathrm{C}^{l}_{n}} \biggr\} \biggr\vert \\ =& \vert \Omega \vert + \vert \tilde{E}_{s} \vert . \end{aligned}$$

(7)

Using properties $(a)$, $(b)$ and the Chebyshev inequality, we have

$$\begin{aligned} & \vert \tilde{E}_{s} \vert \\ &\quad \le \frac{2^{m}\mathrm{C}^{l}_{n}}{\lambda } \int _{\mathbb{R}^{n} \setminus \Omega } \bigl\vert \mathcal{L}\bigl(b^{(1)}_{I}, \ldots ,b^{(t)}_{I},g^{(t+1)}_{I} \cdots ,g^{(m)}_{I}\bigr) (x) \bigr\vert \,\mathrm{d}x \\ &\quad \le \frac{2^{m}\mathrm{C}^{l}_{n}}{\lambda } \int _{\mathbb{R}^{n} \setminus \Omega } \biggl\vert \sum_{k_{1},\ldots ,k_{t}} \int _{(\mathbb{R}^{n})^{m}}\bigl(K(x,y_{1}, \ldots ,y_{m})-K(x,c_{1,k_{1}},\ldots ,y_{m})\bigr) \\ &\qquad {} \times b^{(1,k_{1})}_{I} \cdots b^{(t,k_{t})}_{I} g^{(t+1)}_{I} \cdots g^{(m)}_{I} \biggr\vert \,\mathrm{d} {y}\,\mathrm{d}x \\ &\quad \le \frac{2^{m}\mathrm{C}^{l}_{n}}{\lambda }(\lambda \rho )^{\frac{m-t}{m}} \int _{\mathbb{R}^{n}\setminus \Omega }\sum_{k_{1}, \ldots ,k_{t}} \int _{(\mathbb{R}^{n})^{m}} \bigl\vert K(x,y_{1},\ldots ,y_{m})-K(x,c_{1,k_{1}}, \ldots ,y_{m}) \bigr\vert \\ &\qquad {} \times \bigl\vert b^{(1,k_{1})}_{I}\cdots b^{(t,k_{t})}_{I} \bigr\vert \,\mathrm{d} {y}\,\mathrm{d}x \\ &\quad \le \frac{2^{m}\mathrm{C}^{l}_{n}}{\lambda }(\lambda \rho )^{\frac{m-t}{m}} \int _{(\mathbb{R}^{n})^{m}}\sum_{k_{1},\ldots ,k_{t}} \biggl( \int _{\mathbb{R}^{n}\setminus \Omega } \bigl\vert K(x,y_{1},\ldots ,y_{m})-K(x,c_{1,k_{1}}, \ldots ,y_{m}) \bigr\vert \,\mathrm{d}x \biggr) \\ &\qquad {} \times \bigl\vert b^{(1,k_{1})}_{I}\cdots b^{(t,k_{t})}_{I} \bigr\vert \,\mathrm{d} {y} \\ &\quad \le \frac{2^{m}\mathrm{C}^{l}_{n}}{\lambda }(\lambda \rho )^{\frac{m-t}{m}} \int _{(\mathbb{R}^{n})^{m}}\sum_{k_{1},\ldots ,k_{t}} \biggl( \int _{\mathbb{R}^{n}\setminus \Omega } \biggl\vert \frac{A \vert y_{1}-c_{1,k_{1}} \vert ^{\varepsilon }}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn+\varepsilon }} \biggr\vert \,\mathrm{d}x \biggr) \\ &\qquad {} \times \bigl\vert b^{(1,k_{1})}_{I}\cdots b^{(t,k_{t})}_{I} \bigr\vert \,\mathrm{d} {y}, \end{aligned}$$

(8)

where $c_{1,k_{1}}$ is the center of cube $\tilde{Q}_{1,k_{1}}$. We represent the difference set $(2^{\tau _{r}+1}Q_{r,k_{r}})\setminus (2^{\tau _{r}}Q_{r,k_{r}})$ by $H^{\tau _{r}}_{r,k_{r}}$ for $\tau _{r}=1,2,\ldots$ , and $r=1,\ldots ,t$. Then the difference set of $\mathbb{R}^{n}$ and Ω satisfies

$$ \mathbb{R}^{n}\setminus \Omega \subset \bigcup ^{\infty }_{\tau _{1}, \ldots ,\tau _{t}=1}\bigcap^{t}_{r=1} \bigl(H^{\tau _{r}}_{r,k_{r}} \bigr). $$

Note that, for the arbitrary $x\in \bigcap^{t}_{r=1} (H^{\tau _{r}}_{r,k_{r}} )$ and $y_{r}\in Q_{r,k_{r}}$, we have

$$ \vert y_{1}-c_{1,k_{1}} \vert \leq 2^{-1} \tilde{l}(Q_{1,k_{1}}) $$

and

$$ 2^{\tau _{r}}\tilde{l}(Q_{r,k_{r}})\le \vert x-y_{r} \vert \le 2^{\tau _{r}+1} \tilde{l}(Q_{r,k_{r}}), $$

(9)

where $\tilde{l}(Q_{r,k_{r}})$ denotes the diameter of the cube $\tilde{Q}_{r,k_{r}}$. Hence, it follows that

$$\begin{aligned} \frac{ \vert y_{1}-c_{1,k_{1}} \vert ^{\varepsilon }}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{\varepsilon }} \le &\frac{ \vert y_{1}-c_{1,k_{1}} \vert ^{\varepsilon }}{( \vert x-y_{1} \vert )^{\varepsilon }} \\ \le & \biggl\vert \frac{2^{-1}\tilde{l}(Q_{1,k_{1}})}{ 2^{\tau _{1}}\tilde{l}(Q_{1,k_{1}})} \biggr\vert ^{\varepsilon } \\ =&\frac{1}{ 2^{\tau _{1}\varepsilon +\varepsilon }}. \end{aligned}$$

(10)

Bringing formula (10) into (8) and combining with inequality (9), we get

$$\begin{aligned} & \int _{(\mathbb{R}^{n})^{m}} \biggl( \int _{\mathbb{R}^{n}\setminus \Omega } \biggl\vert \frac{A \vert y_{1}-c_{1,k_{1}} \vert ^{\varepsilon }}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn+\varepsilon }} \biggr\vert \,\mathrm{d}x \biggr) \bigl\vert b^{(1,k_{1})}_{I}\cdots b^{(t,k_{t})}_{I} \bigr\vert \,\mathrm{d} {y} \\ &\quad \le \int _{(\mathbb{R}^{n})^{m}} \int _{\mathbb{R}^{n}\setminus \Omega } \frac{A\frac{1}{ 2^{\tau _{1}\varepsilon +\varepsilon }}\,\mathrm{d}x}{(\sum_{j=1}^{m} \vert x-y_{j} \vert )^{mn}} \bigl\vert b^{(1,k_{1})}_{I} \cdots b^{(t,k_{t})}_{I} \bigr\vert \,\mathrm{d} {y} \\ &\quad \le \int _{(\mathbb{R}^{n})^{t}} \int _{\mathbb{R}^{n}\setminus \Omega } \frac{A\frac{1}{ 2^{\tau _{1}\varepsilon +\varepsilon }}\,\mathrm{d}x}{(\sum_{j=1}^{t} \vert x-y_{j} \vert )^{tn}} \bigl\vert b^{(1,k_{1})}_{I} \cdots b^{(t,k_{t})}_{I} \bigr\vert \,\mathrm{d}y_{1}\cdots \mathrm{d}y_{t} \\ &\quad \le \int _{Q_{1,k_{1}}}\cdots \int _{Q_{t,k_{t}}}\sum_{\tau _{1}, \ldots ,\tau _{t}=1}^{\infty } \int _{\bigcap ^{t}_{r=1} H^{\tau _{r}}_{r,k_{r}}} \frac{A\frac{1}{ 2^{\tau _{1}\varepsilon +\varepsilon }}\,\mathrm{d}x}{(\sum_{j=1}^{t} \vert x-y_{j} \vert )^{tn}} \bigl\vert b^{(1,k_{1})}_{I}\cdots b^{(t,k_{t})}_{I} \bigr\vert \,\mathrm{d}y_{1}\cdots \mathrm{d}y_{t} \\ &\quad \le \int _{Q_{1,k_{1}}}\cdots \int _{Q_{t,k_{t}}}\sum_{\tau _{1}, \ldots ,\tau _{t}=1}^{\infty } \int _{\bigcap ^{t}_{r=1} H^{\tau _{r}}_{r,k_{r}}} \frac{A\frac{1}{ 2^{\tau _{1}\varepsilon +\varepsilon }}\,\mathrm{d}x}{(\sum_{j=1}^{t} \vert 2^{\tau _{r}}\tilde{l}(Q_{r,k_{r}}) \vert )^{tn}} \bigl\vert b^{(1,k_{1})}_{I} \cdots b^{(t,k_{t})}_{I} \bigr\vert \,\mathrm{d}y_{1}\cdots \mathrm{d}y_{t} \\ &\quad \leq C_{2}\alpha ^{t}\sum _{\tau _{1},\ldots ,\tau _{t}=1}^{\infty } \int _{\bigcap ^{t}_{r=1} H^{\tau _{r}}_{r,k_{r}}} \frac{A\frac{1}{ 2^{\tau _{1}\varepsilon +\varepsilon }}\,\mathrm{d}x}{(\sum_{j=1}^{t} \vert 2^{\tau _{r}}\tilde{l}(Q_{r,k_{r}}) \vert )^{tn}} \prod _{r=1}^{t} \vert Q_{r,k_{r}} \vert \\ &\quad \le C_{2}\alpha ^{t}\sum _{\tau _{1},\ldots ,\tau _{t}=1}^{\infty }\frac{A}{ 2^{\tau _{1}\varepsilon +\varepsilon }} \int _{Q_{1,k_{1}}} \cdots \int _{Q_{t,k_{t}}} \int _{\bigcap ^{t}_{r=1} H^{\tau _{r}}_{r,k_{r}}} \frac{\mathrm{d}x}{(\sum_{j=1}^{t} \vert 2^{\tau _{r}}\tilde{l}(Q_{r,k_{r}}) \vert )^{tn}} \,\mathrm{d}y_{1}\cdots \mathrm{d}y_{t} \\ &\quad \le C_{3}\alpha ^{t}\sum _{\tau _{1},\ldots ,\tau _{t}=1}^{\infty }\frac{A}{ 2^{\tau _{1}\varepsilon +\varepsilon }} \int _{Q_{1,k_{1}}} \cdots \int _{Q_{t,k_{t}}} \int _{\bigcap ^{t}_{r=1} H^{\tau _{r}}_{r,k_{r}}} \frac{\mathrm{d}x}{(\sum_{j=1}^{t} \vert x-y_{j} \vert )^{tn}}\,\mathrm{d}y_{1} \cdots \mathrm{d}y_{t}. \end{aligned}$$

(11)

Owing to the fact that the sets $H^{\tau _{r}}_{r,k_{r}}$ and $\tau _{r}=1,\ldots ,\infty$ are disjoint, we know

$$\begin{aligned} &\sum_{\tau _{2},\ldots ,\tau _{t}=1}^{\infty } \int _{\bigcap ^{t}_{r=1} H^{\tau _{r}}_{r,k_{r}}} \frac{\mathrm{d}x}{(\sum_{j=1}^{t} \vert x-y_{j} \vert )^{tn}} \\ &\quad \leq \sum_{\tau _{2},\ldots ,\tau _{t-1}=1}^{\infty } \int _{\bigcap ^{t-1}_{r=1} H^{\tau _{r}}_{r,k_{r}}} \frac{\mathrm{d}x}{(\sum_{j=1}^{t} \vert x-y_{j} \vert )^{tn}} \\ &\quad \leq \int _{H^{\tau _{1}}_{1,k_{1}}} \frac{\mathrm{d}x}{(\sum_{j=1}^{t} \vert x-y_{j} \vert )^{tn}}. \end{aligned}$$

(12)

Similarly, we have

$$\begin{aligned} &\sum_{k_{2},\ldots ,k_{t}} \int _{Q_{1,k_{1}}}\cdots \int _{Q_{t,k_{t}}} \frac{\mathrm{d}y_{1}\cdots \mathrm{d}y_{t}}{(\sum_{j=1}^{t} \vert x-y_{j} \vert )^{tn}} \\ &\quad \leq \sum_{k_{2},\ldots ,k_{t-1}} \int _{Q_{1,k_{1}}}\cdots \int _{Q_{t-1,k_{t-1}}} \int _{\mathbb{R}^{n}} \frac{\mathrm{d}y_{1}\cdots \mathrm{d}y_{t}}{(\sum_{j=1}^{t} \vert x-y_{j} \vert )^{tn}} \\ &\quad \leq \sum_{k_{2},\ldots ,k_{t-1}} \int _{Q_{1,k_{1}}}\cdots \int _{Q_{t-1,k_{t-1}}} \frac{\mathrm{d}y_{1}\cdots \mathrm{d}y_{t-1}}{(\sum_{j=1}^{t-1} \vert x-y_{j} \vert )^{(t-1)n}} \\ &\quad \leq \int _{Q_{1,k_{1}}}\frac{\mathrm{d}y_{1}}{( \vert x-y_{1} \vert )^{n}}. \end{aligned}$$

(13)

Combining (8), (11), (12), and (13), we obtain

$$\begin{aligned} \vert \tilde{E}_{s} \vert \le &C_{3} \frac{2^{m}\mathrm{C}^{l}_{n}}{\lambda } \alpha ^{t}(\lambda \rho )^{\frac{m-t}{m}}\sum _{\tau _{1}=1}^{\infty }\sum _{k_{1}} \int _{Q_{1,k_{1}}} \int _{H^{\tau _{1}}_{1,k_{1}}} \frac{\frac{A}{ 2^{\tau _{1}\varepsilon +\varepsilon }}\,\mathrm{d}y_{1}\,\mathrm{d}x}{( \vert x-y_{1} \vert )^{n}} \\ \le &C_{3}\frac{2^{m}\mathrm{C}^{l}_{n}}{\lambda } \alpha ^{t}(\lambda \rho )^{\frac{m-t}{m}}\sum_{\tau _{1}=1}^{\infty } \frac{A}{ 2^{\tau _{1}\varepsilon +\varepsilon }}\sum_{k_{1}} \int _{Q_{1,k_{1}}} \int _{H^{\tau _{1}}_{1,k_{1}}} \frac{\mathrm{d}x}{ \vert x-y_{1} \vert ^{n}} \,\mathrm{d}y_{1} \\ \le &C_{3}\frac{2^{m}\mathrm{C}^{l}_{n}}{\lambda } \alpha ^{t}(\lambda \rho )^{\frac{m-t}{m}}\sum_{\tau _{1}=1}^{\infty } \frac{A}{ 2^{\tau _{1}\varepsilon +\varepsilon }}\sum_{k_{1}} \int _{Q_{1,k_{1}}} \int _{H^{\tau _{1}}_{1,k_{1}}} \frac{\mathrm{d}x}{ \vert 2^{\tau _{r}}\tilde{l}(Q_{r,k_{r}}) \vert ^{n}} \,\mathrm{d}y_{1} \\ \le &C_{3}\frac{2^{m}\mathrm{C}^{l}_{n}}{\lambda } \alpha ^{t}(\lambda \rho )^{\frac{m-t}{m}}\sum_{\tau _{1}=1}^{\infty } \frac{A}{ 2^{\tau _{1}\varepsilon +\varepsilon }}\sum_{k_{1}} \int _{Q_{1,k_{1}}} \int _{2^{\tau _{r}+1}Q_{r,k_{r}}} \frac{\mathrm{d}x}{ \vert 2^{\tau _{r}}\tilde{l}(Q_{r,k_{r}}) \vert ^{n}} \,\mathrm{d}y_{1} \\ \le &C_{3}\frac{2^{m}\mathrm{C}^{l}_{n}}{\lambda } \alpha ^{t}(\lambda \rho )^{\frac{m-t}{m}}\sum_{\tau _{1}=1}^{\infty } \frac{A}{ 2^{\tau _{1}\varepsilon +\varepsilon }}\sum_{k_{1}} \vert Q_{1,k_{1}} \vert \\ \le &C_{3}\frac{2^{m}\mathrm{C}^{l}_{n}}{\lambda } \alpha ^{t}(\lambda \rho )^{\frac{m-t}{m}}\sum_{\tau _{1}=1}^{\infty } \frac{A}{ 2^{\tau _{1}\varepsilon +\varepsilon }}\frac{c}{\alpha } \int _{ \mathbb{R}^{n}} \bigl\vert u^{(1)}_{I}(y_{1}) \bigr\vert \,\mathrm{d}y_{1} \\ \le &C_{3}A(\lambda \rho )^{\frac{m-1}{m}}\frac{1}{\lambda } \int _{ \mathbb{R}^{n}} \bigl\vert u^{(1)}_{I}(y_{1}) \bigr\vert \,\mathrm{d}y_{1} \\ =&C_{3}A\rho \frac{1}{\alpha } \int _{\mathbb{R}^{n}} \bigl\vert u^{(1)}_{I}(y_{1}) \bigr\vert \,\mathrm{d}y_{1} \\ \leq &C_{3}A\rho \frac{1}{\alpha } \int _{\mathbb{R}^{n}} \bigl\vert u^{(1)}(y_{1}) \bigr\vert \,\mathrm{d}y_{1}. \end{aligned}$$

(14)

Choose $\rho =\frac{1}{A+B}$, then inequality (6) shows that

$$\begin{aligned} & \biggl\vert \biggl\{ x:\mathcal{L} \bigl(g^{(1)}_{I},\ldots ,g^{(m)}_{I} \bigr)> \frac{\lambda }{2^{m}\mathrm{C}^{l}_{n}} \biggr\} \biggr\vert \\ &\quad \leq C_{1}B^{q}\lambda ^{-\frac{1}{m}}\rho ^{q-\frac{1}{m}} \bigl\Vert u^{(i)} \bigr\Vert ^{q/q_{i}}_{1} \\ &\quad =C_{1}B^{q}\lambda ^{-\frac{1}{m}}\biggl( \frac{1}{A+B}\biggr)^{q-\frac{1}{m}} \bigl\Vert u^{(i)} \bigr\Vert ^{q/q_{i}}_{1} \\ &\quad =C_{1} \biggl(\frac{B}{A+B} \biggr)^{q}\lambda ^{-\frac{1}{m}} \biggl( \frac{1}{A+B} \biggr)^{-\frac{1}{m}} \bigl\Vert u^{(i)} \bigr\Vert ^{q/q_{i}}_{1} \\ &\quad =C_{1} \biggl(\frac{A+B}{\lambda } \biggr)^{\frac{1}{m}} \bigl\Vert u^{(i)} \bigr\Vert ^{q/q_{i}}_{1}. \end{aligned}$$

(15)

On the other hand, property $(c)$ implies that

$$\begin{aligned} \vert \Omega \vert =& \biggl\vert \bigcup _{r}\Omega _{r} \biggr\vert \\ =& \biggl\vert \bigcup _{r}\bigcup _{k_{i}}\tilde{B}_{i,k_{i}} \biggr\vert \\ \leq &\frac{C_{4}}{\alpha } \int _{\mathbb{R}^{n}} \bigl\vert u^{(s)}_{I}(x) \bigr\vert \,\mathrm{d}x \\ =&C_{4} \biggl(\frac{A+B}{\lambda } \biggr)^{\frac{1}{m}} \int _{ \mathbb{R}^{n}} \bigl\vert u^{(s)}_{I}(x) \bigr\vert \,\mathrm{d}x, \end{aligned}$$

(16)

where

$$ \int _{\mathbb{R}^{n}} \bigl\vert u^{(s)}_{I}(x) \bigr\vert =\max \biggl\{ \int _{\mathbb{R}^{n}} \bigl\vert u^{(1)}_{I}(x) \bigr\vert , \ldots , \int _{\mathbb{R}^{n}} \bigl\vert u^{(m)}_{I}(x) \bigr\vert \biggr\} . $$

And it should be noted that

$$\begin{aligned} \vert \tilde{E}_{s} \vert \leq &C_{3}A\rho \frac{1}{\alpha } \int _{\mathbb{R}^{n}} \bigl\vert u^{(1)}(y_{1}) \bigr\vert \,\mathrm{d}y_{1} \\ =&C_{3} \biggl(\frac{A}{A+B} \biggr) \biggl(\frac{A+B}{\lambda } \biggr)^{ \frac{1}{m}} \int _{\mathbb{R}^{n}} \bigl\vert u^{(1)}(y_{1}) \bigr\vert \,\mathrm{d}y_{1} \\ \leq &C_{5} \biggl(\frac{A+B}{\lambda } \biggr)^{\frac{1}{m}} \int _{ \mathbb{R}^{n}} \bigl\vert u^{(1)}(y_{1}) \bigr\vert \,\mathrm{d}y_{1}. \end{aligned}$$

(17)

Finally, combining (15), (16), and (17), we have

$$\begin{aligned} &\bigl(\mathcal{L}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr) \bigr)_{*}(\lambda ) \\ &\quad = \bigl\vert \bigl\{ x: \bigl\vert \mathcal{L}\bigl(u^{(1)}, \ldots ,u^{(m)}\bigr) \bigr\vert >\lambda \bigr\} \bigr\vert \\ &\quad \leq \sum_{I} \biggl( \biggl\vert \biggl\{ x:\mathcal{L}\bigl(g^{(1)}_{I},\ldots ,g^{(m)}_{I} \bigr)> \frac{\lambda }{2^{m}\mathrm{C}^{l}_{n}}\biggr\} \biggr\vert \biggr) \\ &\qquad {}+\sum_{I} \biggl(\sum _{\exists h^{(i)}_{I}=b^{(i)}_{I}} \biggl\vert \biggl\{ x: \mathcal{L} \bigl(h^{(1)}_{I},\ldots ,h^{(m)}_{I} \bigr)> \frac{\lambda }{2^{m}\mathrm{C}^{l}_{n}}\biggr\} \biggr\vert \biggr) \\ &\quad \leq \sum_{I} \biggl(C_{1} \biggl(\frac{A+B}{\lambda } \biggr)^{ \frac{1}{m}} \bigl\Vert u^{(i)} \bigr\Vert ^{q/q_{i}}_{1} \biggr)+\sum _{I} \biggl(\sum_{ \exists h^{(i)}_{I}=b^{(i)}_{I}}\bigl( \vert \Omega \vert + \vert \tilde{E}_{s} \vert \bigr) \biggr) \\ &\quad \leq \sum_{I} \biggl(C_{1} \biggl(\frac{A+B}{\lambda } \biggr)^{ \frac{1}{m}} \bigl\Vert u^{(i)} \bigr\Vert ^{q/q_{i}}_{1} \biggr) \\ &\qquad {}+\sum_{I} \biggl(\sum _{\exists h^{(i)}_{I}=b^{(i)}_{I}} \biggl(C_{4} \biggl(\frac{A+B}{\lambda } \biggr)^{\frac{1}{m}} \int \bigl\vert u^{(s)}_{I}(x) \bigr\vert \,\mathrm{d}x \biggr) \biggr) \\ &\qquad {}+\sum_{I} \biggl(\sum _{\exists h^{(i)}_{I}=b^{(i)}_{I}} \biggl(C_{5} \biggl(\frac{A+B}{\lambda } \biggr)^{\frac{1}{m}} \int \bigl\vert u^{(1)}(y_{1}) \bigr\vert \,\mathrm{d}y_{1} \biggr) \biggr) \\ &\quad \leq C_{6} \biggl(\frac{A+B}{\lambda } \biggr)^{\frac{1}{m}}. \end{aligned}$$

(18)

This ends the proof of Theorem 1. □

4 Poincaré-type inequalities

In this section, we establish the Poincaré-type inequalities for multilinear Calderón–Zygmund operators on differential forms in the local and global domain. So we first define the operator on differential l-form in a local domain as follows:

$$\begin{aligned} &\widetilde{\mathcal{L}}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr) (x;{\xi }) \\ &\quad = \biggl( \int _{(B)^{m}}K(x,y_{1},\ldots ,y_{m})u^{(1)}(x;{ \xi }) \cdots u^{(m)}(x;{\xi })\,dy_{1}\cdots dy_{m} \biggr) \end{aligned}$$

(19)

for $x\notin \bigcap ^{m}_{i=1}\operatorname{supp} u^{(i)}_{I}$, $B\subset \Omega$, and the kernel function $K(y_{1},\ldots ,y_{m})$ satisfies

$$\begin{aligned} & \bigl\vert K(x,y_{1},\ldots , y_{m})-K \bigl(x',y_{1},\ldots , y_{m}\bigr) \bigr\vert \\ &\quad \le \frac{A \vert x-x' \vert }{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn-1}} \end{aligned}$$

(20)

for $|x-x'|\le \frac{1}{2}\max_{1\le j\le m}|x-y_{j}|$.

Next, we give the Poincaré-type inequality for the multilinear Calderón–Zygmund operator ${\widetilde{\mathcal{L}}}$ on a differential form in the local domain.

Theorem 2

Let $\widetilde{\mathcal{L}}$ be a multilinear Calderón–Zygmund operator defined by (19), $B\subset \Omega$, the kernel function K satisfies condition (2) in Definition 1, and $u^{(1)},\ldots ,u^{(m)} \in L^{p}_{loc}(\Omega ,\Lambda ^{l})$ satisfy A-harmonic equation (1), then for $\theta ={1+\frac{2}{n}-\frac{m-1}{p}}$, we have

$$\begin{aligned} & \bigl\Vert \widetilde{\mathcal{L}}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr) (x)-\bigl( \widetilde{\mathcal{L}}\bigl(u^{(1)}, \ldots ,u^{(m)}\bigr) (x)\bigr)_{B} \bigr\Vert _{p,B} \\ &\quad \le C \vert B \vert ^{\theta } \bigl\Vert u^{(1)} \bigr\Vert _{p,\sigma B}\cdots \bigl\Vert u^{(m-1)} \bigr\Vert _{p, \sigma B} \bigl\Vert u^{(m)} \bigr\Vert _{p, B}. \end{aligned}$$

Proof

Applying the decomposition theorem for a differential form and Lemma 1, we get

$$\begin{aligned} & \bigl\Vert \widetilde{\mathcal{L}} \bigl(u^{(1)},\ldots ,u^{(m)}\bigr) (x)-\bigl( \widetilde{ \mathcal{L}}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr) (x) \bigr)_{B} \bigr\Vert _{p,B} \\ &\quad = \bigl\Vert T\bigl(\mathrm{d}\widetilde{\mathcal{L}} \bigl(u^{(1)},\ldots ,u^{(m)}\bigr) (x)\bigr)+ \mathrm{d}\bigl(T \widetilde{\mathcal{L}}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr) (x) \bigr) -\bigl( \widetilde{\mathcal{L}}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr) (x)\bigr)_{B} \bigr\Vert _{p,B} \\ &\quad = \bigl\Vert T\bigl(\mathrm{d}\bigl(\widetilde{\mathcal{L}} \bigl(u^{(1)},\ldots ,u^{(m)}\bigr) (x)\bigr)\bigr) \bigr\Vert _{p,B} \\ &\quad \leq C \vert B \vert \operatorname{diam}(B) \bigl\Vert \mathrm{d} \bigl(\widetilde{\mathcal{L}}\bigl(u^{(1)}, \ldots ,u^{(m)} \bigr) (x)\bigr) \bigr\Vert _{p,B}. \end{aligned}$$

(21)

For convenience, we write

$$ \widetilde{\mathcal{L}}_{I}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr) (x)= \int _{B^{m}}K(x,y_{1}, \ldots ,y_{m})u^{(1)}_{I}u^{(2)}_{I} \cdots u^{(m)}_{I}\,\mathrm{d}y_{1} \cdots \mathrm{d}y_{m}. $$

Then,

$$\begin{aligned} &\mathrm{d}\bigl(\widetilde{\mathcal{L}} \bigl(u^{(1)},\ldots ,u^{(m)}\bigr) (x)\bigr) \\ &\quad =\mathrm{d} \biggl(\sum_{I} \bigl( \bigl( \widetilde{\mathcal{L}}_{I}\bigl(u^{(1)}, \ldots ,u^{(m)}\bigr) (x) \bigr)\,\mathrm{d}x_{I} \bigr) \biggr) \\ &\quad =\sum^{n}_{k=1}\sum _{I} \frac{\partial \widetilde{\mathcal{L}}_{I}(u^{(1)},\ldots ,u^{(m)})(x)}{\partial x_{k}} \,\mathrm{d}x_{k}\wedge \mathrm{d}x_{I} \\ &\quad =\sum^{n}_{k=1}\sum _{I}\lim_{\eta \rightarrow 0} \frac{\widetilde{\mathcal{L}}_{I}(u^{(1)},\ldots ,u^{(m)})(x+\eta e_{k})-\widetilde{\mathcal{L}}_{I}(u^{(1)},\ldots ,u^{(m)})(x)}{\eta } \,\mathrm{d}x_{k}\wedge \mathrm{d}x_{I} \\ &\quad =\sum^{n}_{k=1}\sum _{I}\lim_{\eta \rightarrow 0} \int _{B^{m}} \biggl( \frac{K(x+\eta e_{k},y_{1},\ldots ,y_{m})-K(x,y_{1},\ldots ,y_{m})}{\eta } \\ &\qquad {}\times u^{(1)}_{I} \cdots u^{(m)}_{I} \,\mathrm{d}y_{1}\cdots \mathrm{d}y_{m} \biggr) \mathrm{d}x_{k}\wedge \mathrm{d}x_{I}. \end{aligned}$$

(22)

In terms of the conditions of kernel function K, we know

$$\begin{aligned} &K(x+\eta e_{k},y_{1},\ldots ,y_{m})-K(x,y_{1},\ldots ,y_{m}) \\ &\quad \leq \frac{A \vert x+\eta e_{k}-x \vert }{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn-1}} \\ &\quad = \frac{A\eta }{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn-1}}. \end{aligned}$$

(23)

It follows from (23) and (22) that

$$\begin{aligned} &\mathrm{d}\bigl(\widetilde{\mathcal{L}}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr) (x)\bigr) \\ &\quad \leq \sum^{n}_{k=1}\sum _{I}\lim_{\eta \rightarrow 0} \frac{\int _{B^{m}} (\frac{A\eta }{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn-1}} )u^{(1)}_{I}\cdots u^{(m)}_{I}\,\mathrm{d}y_{1}\cdots \mathrm{d}y_{m}}{\eta } \,\mathrm{d}x_{k}\wedge \mathrm{d}x_{I} \\ &\quad =\sum^{n}_{k=1}\sum _{I}\lim_{\eta \rightarrow 0} \int _{B^{m}} \frac{Au^{(1)}_{I}\cdots u^{(m)}_{I}\,\mathrm{d}y_{1}\cdots \mathrm{d}y_{m}}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn-1}} \,\mathrm{d}x_{k} \wedge \mathrm{d}x_{I} \\ &\quad =\sum^{n}_{k=1}\sum _{I} \int _{B^{m}} \frac{Au^{(1)}_{I}\cdots u^{(m)}_{I}\,\mathrm{d}y_{1}\cdots \mathrm{d}y_{m}}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn-1}} \,\mathrm{d}x_{k} \wedge \mathrm{d}x_{I}. \end{aligned}$$

Further, we get

$$\begin{aligned} & \bigl\Vert \mathrm{d}\bigl(\widetilde{\mathcal{L}}\bigl(u^{(1)}, \ldots ,u^{(m)}\bigr) (x)\bigr) \bigr\Vert _{p,B} \\ &\quad \leq \Biggl\Vert \sum^{n}_{k=1} \sum_{I} \int _{B^{m}} \frac{Au^{(1)}_{I}\cdots u^{(m)}_{I}\,\mathrm{d}y_{1}\cdots \mathrm{d}y_{m}}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn-1}} \,\mathrm{d}x_{k} \wedge \mathrm{d}x_{I} \Biggr\Vert _{p,B} \\ &\quad \leq \Biggl\Vert \Biggl(\sum^{n}_{k=1} \sum_{I} \biggl\vert \int _{B^{m}} \frac{Au^{(1)}_{I}\cdots u^{(m)}_{I}\,\mathrm{d}y_{1}\cdots \mathrm{d}y_{m}}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn-1}} \biggr\vert ^{2} \Biggr)^{\frac{1}{2}} \Biggr\Vert _{p,B} \\ &\quad \leq \Biggl\Vert C_{1}\sum^{n}_{k=1} \sum_{I} \biggl\vert \int _{B^{m}} \frac{Au^{(1)}_{I}\cdots u^{(m)}_{I}\,\mathrm{d}y_{1}\cdots \mathrm{d}y_{m}}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn-1}} \biggr\vert \Biggr\Vert _{p,B} \\ &\quad \leq \Biggl\Vert C_{1}A\sum^{n}_{k=1} \sum_{I} \biggl\vert \int _{B^{m-1}} \biggl( \int _{B} \frac{u^{(1)}_{I} \,\mathrm{d}y_{1}}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn-1}} \biggr) \\ &\qquad {}\times u^{(2)}_{I}\cdots u^{(m)}_{I} \,\mathrm{d}y_{2} \cdots \mathrm{d}y_{m} \biggr\vert \Biggr\Vert _{p,B} \\ &\quad \leq C_{1}A\sum^{n}_{k=1} \sum_{I} \biggl\Vert \int _{B^{m-1}} \biggl( \int _{B} \frac{u^{(1)}_{I} \,\mathrm{d}y_{1}}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn-1}} \biggr) \\ &\qquad {}\times u^{(2)}_{I}\cdots u^{(m)}_{I} \,\mathrm{d}y_{2} \cdots \mathrm{d}y_{m} \biggr\Vert _{p,B}. \end{aligned}$$

Apply Hölder’s inequality to the innermost layer integral of the right-hand side of the inequality above with Hölder index satisfying $1=\frac{n(m-1)-\varepsilon }{n(m-1)}+\frac{\varepsilon }{n(m-1)}$, $0<\varepsilon <1$. For the convenience of writing, we write $\frac{1}{\zeta }=\frac{n(m-1)-\varepsilon }{n(m-1)}$, $\frac{1}{\eta }= \frac{\varepsilon }{n(m-1)}$, then it follows that

$$\begin{aligned} & \int _{B} \frac{u^{(1)}_{I} \,\mathrm{d}y_{1}}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn-1}} \\ &\quad \leq \int _{B} \biggl\vert \frac{u^{(1)}_{I} }{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn-1}} \biggr\vert \,\mathrm{d}y_{1} \\ &\quad \leq \biggl( \int _{B} \biggl\vert \frac{1}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{{\zeta }mn-{\zeta }}} \biggr\vert \,\mathrm{d}y_{1} \biggr)^{\frac{1}{{\zeta }}} \bigl\Vert u^{(1)}_{I} \bigr\Vert _{{\eta },B} \\ &\quad \leq \biggl( \int _{\mathbb{R}^{n}} \frac{1}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{{\zeta }mn-{\zeta }}}\,\mathrm{d}y_{1} \biggr)^{\frac{1}{{\zeta }}} \bigl\Vert u^{(1)}_{I} \bigr\Vert _{{\eta },B} \\ &\quad = \biggl(\omega _{n} \int ^{\infty }_{0} \frac{r^{n-1}}{(r+ \vert x-y_{2} \vert +\cdots + \vert x-y_{m} \vert )^{{\zeta }mn-{\zeta }}} \,\mathrm{d}r \biggr)^{\frac{1}{{\zeta }}} \bigl\Vert u^{(1)}_{I} \bigr\Vert _{{\eta },B} \\ &\quad \leq \biggl(\omega _{n} \int ^{\infty }_{0} \frac{\mathrm{d}r}{(r+ \vert x-y_{2} \vert +\cdots + \vert x-y_{m} \vert )^{{\zeta }mn-{\zeta }-n+1}} \biggr)^{\frac{1}{{\zeta }}} \bigl\Vert u^{(1)}_{I} \bigr\Vert _{{\eta },B} \\ &\quad \leq \frac{C_{2}}{( \vert x-y_{2} \vert +\cdots + \vert x-y_{m} \vert )^{mn-n+\frac{\varepsilon }{m-1}-1}} \bigl\Vert u^{(1)}_{I} \bigr\Vert _{{\eta },B}. \end{aligned}$$

Similarly, applying Hölder’s inequalities $(m-2)$ times with the indexes satisfying $1=\frac{1}{{\zeta }}+\frac{1}{{\eta }}$, we have

$$\begin{aligned} & \int _{B^{m}} \frac{u^{(1)}_{I}\cdots u^{(m)}_{I}\,\mathrm{d}y_{1}\cdots \mathrm{d}y_{m}}{( \vert x-y_{1} \vert +\cdots + \vert x-y_{m} \vert )^{mn-1}} \\ &\quad \le \int _{B} \frac{ u^{(m)}_{I}\,\mathrm{d}y_{m}}{ \vert x-y_{m} \vert ^{n+\varepsilon -1}} \bigl\Vert u^{(1)}_{I} \bigr\Vert _{{\eta },B}\cdots \bigl\Vert u^{(m-1)}_{I} \bigr\Vert _{{\eta },B}. \end{aligned}$$

Let $\chi _{B}$ be a characteristic function as

$$ \chi _{B}=\textstyle\begin{cases} 1, & x\in B, \\ 0, & x\notin B, \end{cases} $$

then we get

$$\begin{aligned} & \biggl\Vert \int _{B} \frac{ u^{(m)}_{I}\,\mathrm{d}y_{m}}{ \vert x-y_{m} \vert ^{n+\varepsilon -1}} \biggr\Vert _{p,B} \\ &\quad = \biggl\Vert \int _{\mathbb{R}^{n}}u^{(m)}_{I}\chi _{B}(y_{m}) \frac{\chi _{B}(x) }{( \vert x-y_{m} \vert ^{n+\varepsilon -1})^{\frac{1}{p}}} \frac{\chi _{B}(y_{m}) }{( \vert x-y_{m} \vert ^{n+\varepsilon -1})^{\frac{1}{q}}} \,\mathrm{d}y_{m} \biggr\Vert _{p,B}. \end{aligned}$$

(24)

Applying Hölder’s inequalities again with the indexes satisfying $1=\frac{1}{p}+\frac{1}{q}$, we obtain

$$\begin{aligned} & \int _{\mathbb{R}^{n}}u^{(m)}_{I}\chi _{B}(y_{m}) \frac{\chi _{B}(x) }{( \vert x-y_{m} \vert ^{n+\varepsilon -1})^{\frac{1}{p}}} \frac{\chi _{B}(y_{m}) }{( \vert x-y_{m} \vert ^{n+\varepsilon -1})^{\frac{1}{q}}} \,\mathrm{d}y_{m} \\ &\quad \le \biggl( \int _{\mathbb{R}^{n}} \biggl\vert u^{(m)}_{I}\chi _{B}(y_{m}) \frac{\chi _{B}(x) }{( \vert x-y_{m} \vert ^{n+\varepsilon -1})^{\frac{1}{p}}} \biggr\vert ^{p}\,\mathrm{d}y_{m} \biggr)^{\frac{1}{p}} \\ & \qquad {}\times \biggl( \int _{\mathbb{R}^{n}} \biggl\vert \frac{\chi _{B}(y_{m}) }{( \vert x-y_{m} \vert ^{n+\varepsilon -1})^{\frac{1}{q}}} \biggr\vert ^{q}\,\mathrm{d}y_{m} \biggr)^{\frac{1}{q}}. \end{aligned}$$

(25)

The integral in the second bracket above can be simplified by the basic inequality as follows:

$$\begin{aligned} & \int _{\mathbb{R}^{n}} \biggl\vert \frac{\chi _{B}(y_{m}) }{( \vert x-y_{m} \vert ^{n+\varepsilon -1})^{\frac{1}{q}}} \biggr\vert ^{q}\,\mathrm{d}y_{m} \\ &\quad = \int _{\mathbb{R}^{n}} \frac{\chi ^{q}_{B}(z+x) }{ \vert z \vert ^{n+\varepsilon -1}}\,\mathrm{d}z \\ &\quad = \int _{B'}\frac{1 }{ \vert z \vert ^{n+\varepsilon -1}}\,\mathrm{d}z \\ &\quad \le \omega _{n}\bigl(r_{1}+\operatorname{diam}(B) \bigr)^{1-\varepsilon }-r^{1- \varepsilon }_{1}) \\ &\quad \le \omega _{n}\bigl(\bigl(r_{1}+ \operatorname{diam}(B)-r_{1}\bigr)^{1-\varepsilon }\bigr) \\ &\quad =\omega _{n}\bigl(\operatorname{diam}(B)\bigr)^{1-\varepsilon }. \end{aligned}$$

(26)

Here, $B'=B(y_{0}')$ and $B=B(y_{0})$ with $y_{0}'=y_{0}-x$, $r_{1}$ is the distance from 0 to $B'$. And then, by Fubini’s theorem, we have

$$\begin{aligned} & \int _{\mathbb{R}^{n}} \int _{\mathbb{R}^{n}} \bigl\vert u^{(m)}_{I}\chi _{B}(y_{m}) \bigr\vert ^{p} \frac{(\chi _{B}(x))^{p} }{ \vert x-y_{m} \vert ^{n+\varepsilon -1}} \,\mathrm{d}y_{m} \,\mathrm{d}x \\ &\quad = \int _{\mathbb{R}^{n}} \bigl\vert u^{(m)}_{I}\chi _{B}(y_{m}) \bigr\vert ^{p} \int _{ \mathbb{R}^{n}} \frac{(\chi _{B}(x))^{p} }{ \vert x-y_{m} \vert ^{n+\varepsilon -1}}\,\mathrm{d}x \,\mathrm{d}y_{m} \\ &\quad \le \omega _{n}\bigl(\operatorname{diam}(B) \bigr)^{1-\varepsilon } \bigl\Vert u^{(m)}_{I} \bigr\Vert _{p,B}^{p}. \end{aligned}$$

(27)

Combining (24), (25), (26), and (27), we have

$$\begin{aligned} \biggl\Vert \int _{B} \frac{ u^{(m)}_{I}\,\mathrm{d}y_{m}}{ \vert x-y_{m} \vert ^{n+\varepsilon -1}} \biggr\Vert _{p,B} \le C\bigl(\operatorname{diam}(B)\bigr)^{1-\varepsilon } \bigl\Vert u^{(m)}_{I} \bigr\Vert _{p,B}. \end{aligned}$$

(28)

Applying Lemma 2 to $\|u^{(i)}_{I}\|_{{\eta },B}$, $1 \le i\le m-1$, we get

$$\begin{aligned} \bigl\Vert u^{(i)}_{I} \bigr\Vert _{{\eta },B} \le & \vert B \vert ^{\frac{p-{\eta }}{p{\eta }}} \bigl\Vert u^{(i)}_{I} \bigr\Vert _{p,\sigma B} \\ =& \vert B \vert ^{\frac{p\varepsilon -n(m-1)}{pn(m-1)}} \bigl\Vert u^{(i)}_{I} \bigr\Vert _{p,\sigma B} \end{aligned}$$

(29)

for $\sigma >1$ with $\sigma B\subset \Omega$. Combining (21), (28), (29), we obtain

$$\begin{aligned} & \bigl\Vert \widetilde{\mathcal{L}}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr) (x)-\bigl( \widetilde{\mathcal{L}}\bigl(u^{(1)}, \ldots ,u^{(m)}\bigr) (x)\bigr)_{B} \bigr\Vert _{p,B} \\ &\quad \le C \vert B \vert ^{1+\frac{1}{n}+\frac{1}{n}-\frac{\varepsilon }{n}+ \frac{p\varepsilon -n(m-1)}{pn(m-1)}(m-1)} \bigl\Vert u^{(1)}_{I} \bigr\Vert _{p,\sigma B} \cdots \bigl\Vert u^{(m-1)}_{I} \bigr\Vert _{p,\sigma B} \bigl\Vert u^{(m)}_{I} \bigr\Vert _{p, B} \\ &\quad =C \vert B \vert ^{1+\frac{2}{n}-\frac{m-1}{p}} \bigl\Vert u^{(1)}_{I} \bigr\Vert _{p,\sigma B} \cdots \bigl\Vert u^{(m-1)}_{I} \bigr\Vert _{p,\sigma B} \bigl\Vert u^{(m)}_{I} \bigr\Vert _{p, B} \\ &\quad \le C \vert B \vert ^{1+\frac{2}{n}-\frac{m-1}{p}} \bigl\Vert u^{(1)} \bigr\Vert _{p,\sigma B} \cdots \bigl\Vert u^{(m-1)} \bigr\Vert _{p,\sigma B} \bigl\Vert u^{(m)} \bigr\Vert _{p, B}. \end{aligned}$$

(30)

 □

Finally, we give the Poincaré-type inequality for the multilinear Calderón–Zygmund operator ${\widetilde{\mathcal{L}}}$ in a bounded convex domain. We need the following covering lemma.

Lemma 3

([21])

There exists a cover ${\mathcal{V}} = \{B_{i}\}$ for any bounded subset Ω in an n-dimensional Euclidean space $\mathbb{R}^{n}$ satisfying

$$ \bigcup _{i} B_{i} = \Omega , \sum _{B_{i} \in \mathcal{V}} \chi _{\sqrt{ \frac{5}{4}}B_{i}} \leq N \chi _{\Omega } $$

for a constant $N>1$. And if $B_{i} \cap B_{j} \neq \emptyset$, then there exists a cube Q included in $B_{i} \cap B_{j}$ with $B_{i} \cup B_{j} \subset NR$, and the cube Q does not have to be an element in the set family $\mathcal{V}$.

Theorem 3

Let $\widetilde{\mathcal{L}}$ be a multilinear Calderón–Zygmund operator defined by (19), $\Omega _{1}\subset \subset \Omega$, kernel function K satisfies condition (2) in Definition 1, $u^{(1)},\ldots ,u^{(m)} \in L^{p}_{loc}(\Omega ,\Lambda ^{l})$ satisfy A-harmonic equation (1) and $2p+(p+1-m)n>0$, then

$$\begin{aligned} & \bigl\Vert \widetilde{\mathcal{L}}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr) (x)-\bigl( \widetilde{\mathcal{L}}\bigl(u^{(1)}, \ldots ,u^{(m)}\bigr) (x)\bigr)_{\Omega } \bigr\Vert _{p, \Omega _{1}} \\ &\quad \le C \bigl\Vert u^{(1)} \bigr\Vert _{p,\Omega }\cdots \bigl\Vert u^{(m-1)} \bigr\Vert _{p,\Omega } \bigl\Vert u^{(m)} \bigr\Vert _{p, \Omega }. \end{aligned}$$

Proof

By Lemma 3 and Theorem 2, we have

$$\begin{aligned} & \bigl\Vert T\bigl(\mathrm{d}\widetilde{\mathcal{L}}\bigl(u^{(1)}, \ldots ,u^{(m)}\bigr) (x)\bigr) \bigr\Vert _{p, \Omega _{1}} \\ &\quad \leq \sum_{B\in \mathcal{V}} \bigl\Vert T\bigl( \,\mathrm{d}\widetilde{\mathcal{L}}\bigl(u^{(1)}, \ldots ,u^{(m)} \bigr) (x)\bigr) \bigr\Vert _{p,B} \\ &\quad \le \sum_{B\in \mathcal{V}}\bigl( C_{1} \vert B \vert ^{1+\frac{2}{n}-\frac{m-1}{p}} \bigl\Vert u^{(1)} \bigr\Vert _{p,\sigma B}\cdots \bigl\Vert u^{(m-1)} \bigr\Vert _{p,\sigma B} \bigl\Vert u^{(m)} \bigr\Vert _{p, B} \bigr) \\ &\quad \le C_{2}N \bigl\Vert u^{(1)} \bigr\Vert _{p,\Omega }\cdots \bigl\Vert u^{(m-1)} \bigr\Vert _{p,\Omega } \bigl\Vert u^{(m)} \bigr\Vert _{p, \Omega }, \end{aligned}$$

where N is the constant in Lemma 3. Next, from the decomposition theorem of homotopy operator, we get

$$\begin{aligned} & \bigl\Vert \widetilde{\mathcal{L}}\bigl(u^{(1)},\ldots ,u^{(m)}\bigr) (x)-\bigl( \widetilde{\mathcal{L}}\bigl(u^{(1)}, \ldots ,u^{(m)}\bigr) (x)\bigr)_{\Omega } \bigr\Vert _{p, \Omega _{1}} \\ &\quad \le \bigl\Vert T\bigl(\mathrm{d}\widetilde{\mathcal{L}} \bigl(u^{(1)},\ldots ,u^{(m)}\bigr) (x)\bigr) \bigr\Vert _{p,\Omega _{1}} \\ &\quad \le C_{2}N \bigl\Vert u^{(1)} \bigr\Vert _{p,\Omega }\cdots \bigl\Vert u^{(m-1)} \bigr\Vert _{p,\Omega } \bigl\Vert u^{(m)} \bigr\Vert _{p, \Omega }, \end{aligned}$$

which completes the proof of Theorem 3. □