# Weighted estimates of higher order commutators generated by BMO-functions and the fractional integral operator on Morrey spaces

## Abstract

The purpose of this paper is to investigate the weighted estimates of commutators generated by BMO-functions and the fractional integral operator on Morrey spaces. The main result generalizes the Sawano, Sugano, and Tanaka result to a weighted setting.

## Introduction

The aim of this paper is to investigate the weighted inequalities of commutators generated by BMO-functions and the fractional integral operator on Morrey spaces. The main results particularly is related to  and . The authors introduced the condition of weights in . Under a certain condition of the weights, we investigate the weighted estimates of commutators generated by BMO-functions and the fractional integral operator on Morrey spaces. The results recover the inequality in .

For $$1< p<\infty$$, we define $$p':=\frac{p}{p-1}$$. In this paper, a symbol C is a positive constant. Whenever we evaluate the operator, the constant C may be change from one constant to another. Let $$|E|$$ denote the Lebesgue measure of E. Let $$\mathcal{D}(\mathbb{R}^{n})$$ be the collection of all dyadic cubes on $$\mathbb{R}^{n}$$. All cubes are assumed to have their sides parallel to the coordinate axes. For a cube $$Q\subset\mathbb{R}^{n}$$, we use $$l(Q)$$ to denote the side-length $$l(Q)$$ and cQ to denote the cube with the same center as Q but with side-length $$cl(Q)$$. The integral average of a measurable function f over Q is written

$$m_{Q}(f)= \fint_{Q} f(x) \,dx= \frac{1}{|Q|} \int_{Q} f(x) \,dx.$$

By a ‘weight’ we will mean a non-negative function w that is positive measure a.e. on $$\mathbb{R}^{n}$$. Given a weight w and a measurable set E, let

$$w(E):= \int_{E} w(x)\,dx.$$

First we define the Morrey spaces.

### Definition 1

Let $$1< p\leq p_{0}<\infty$$. We define the Morrey space $$\mathcal{M}_{p}^{p_{0}}(\mathbb{R}^{n})$$ by

$$\mathcal{M}_{p}^{p_{0}}\bigl(\mathbb{R}^{n}\bigr) := \bigl\{ f\in L^{p}_{\mathrm{loc}}\bigl(\mathbb{R}^{n}\bigr); \Vert f \Vert _{\mathcal{M}_{p}^{p_{0}}}< \infty \bigr\} ,$$

where for all measurable functions f, we define

$$\Vert f \Vert _{\mathcal{M}_{p}^{p_{0}}}:= \sup_{Q\in\mathcal{D}(\mathbb{R}^{n})} |Q|^{\frac{1}{p_{0}}} \biggl( \fint_{Q} \bigl\vert f(x)\bigr\vert ^{p} \,dx \biggr)^{\frac{1}{p}}.$$

### Remark 1

1. (a)

The ordinary Morrey norm is equivalent to the Morrey norm in this paper (see ):

$$\sup_{\substack{Q\subset\mathbb{R}^{n},\\Q\text{: cubes}}} |Q|^{\frac {1}{p_{0}}} \biggl( \fint_{Q} \bigl\vert f(x)\bigr\vert ^{p} \,dx \biggr)^{\frac{1}{p}}\cong \Vert f\Vert _{\mathcal{M}_{p}^{p_{0}}(\mathbb{R}^{n})}.$$
2. (b)

Hölder’s inequality gives us the following inequality: If $$1< p\leq q \leq p_{0}<\infty$$, then we have

$$\Vert f \Vert _{\mathcal{M}_{p}^{p_{0}}} \leq \Vert f \Vert _{\mathcal{M}_{q}^{p_{0}}}.$$

We define the BMO space (see [3, 4]) as follows.

### Definition 2

For an $$L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$$-function b, define

$$\Vert b \Vert _{\operatorname{BMO}}:=\sup_{Q\subset\mathbb{R}^{n}} \fint_{Q} \bigl\vert b(x)-m_{Q}(b)\bigr\vert \,dx,$$

where the supremum is taken over all cubes $$Q\subset\mathbb{R}^{n}$$. Define

$$\operatorname{BMO}\bigl(\mathbb{R}^{n}\bigr):= \bigl\{ b\in L^{1}_{\mathrm {loc}}\bigl(\mathbb{R}^{n}\bigr) : \Vert b \Vert _{\operatorname{BMO}}< \infty \bigr\} .$$

We define the fractional maximal and integral operators.

### Definition 3

1. (1)

Let $$0\leq\alpha< n$$,

$$M_{\alpha}f(x):=\sup_{Q\ni x} l(Q)^{\alpha} \fint_{Q} \bigl\vert f(y)\bigr\vert \,dy,$$

where the supremum is taken over all cubes $$Q\subset\mathbb{R}^{n}$$ such that $$x\in Q$$.

2. (2)

Let $$0<\alpha<n$$,

$$I_{\alpha}f(x):= \int_{\mathbb{R}^{n}} \frac{f(y)}{|x-y|^{n-\alpha}} \,dy.$$

The point-wise inequality holds:

$$M_{\alpha}f(x)\leq CI_{\alpha}f(x),$$

for all positive measurable function f.

It is well known that the following inequality holds (see ). The celebrated result is called the Adams inequality.

### Theorem A

Let $$0<\alpha<n$$, $$1< p\leq p_{0}<\infty$$ and $$1< q\leq q_{0}<\infty$$. Assume that

$$\frac{1}{q_{0}}=\frac{1}{p_{0}}-\frac{\alpha}{n}\quad \textit{and}\quad \frac{q}{q_{0}}=\frac{p}{p_{0}}.$$

Then we have

$$\Vert I_{\alpha}f \Vert _{\mathcal{M}_{q}^{q_{0}}}\leq C\Vert f\Vert _{\mathcal{M}_{p}^{p_{0}}},$$

for all $$f\in\mathcal{M}_{p}^{p_{0}}(\mathbb{R}^{n})$$.

Let $$m\in\mathbb{Z}_{+}$$. The m-fold commutator $$[b, I_{\alpha}]^{(m)}$$ is given by the following definition.

### Definition 4

Let $$0<\alpha<n$$ and $$b\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$$. Then we define

$$[ b, I_{\alpha} ]^{(m)} f(x) := \int_{\mathbb{R}^{n}} \frac{ ( b(x)-b(y) )^{m}}{|x-y|^{n-\alpha}} f(y) \,dy,$$

as long as the integral in the right-hand side makes sense.

### Remark 2

The following inequality holds:

$$\bigl\vert [ b, I_{\alpha} ]^{(m)} f(x) \bigr\vert \leq \int_{\mathbb{R}^{n}} \frac{\vert b(x)-b(y) \vert ^{m}}{|x-y|^{n-\alpha}} \bigl\vert f(y)\bigr\vert \,dy.$$
(1)

As shall be verified in the proof of Theorem 1, we virtually consider the operator

$$x\mapsto \int_{\mathbb{R}^{n}} \frac{\vert b(x)-b(y) \vert ^{m}}{|x-y|^{n-\alpha}} f(y) \,dy$$

and hence we may assume that the integral defining $$[b,I_{\alpha}]^{(m)}f(x)$$ converges for a.e. $$x\in\mathbb{R}^{n}$$.

Di-Fazio and Ragusa  obtained the next theorem.

### Theorem B

Let $$0<\alpha<n$$, $$1< p\leq p_{0}<\infty$$ and $$1< q\leq q_{0}<\infty$$. Assume that

$$\frac{1}{q_{0}}=\frac{1}{p_{0}}-\frac{\alpha}{n}\quad \textit{and}\quad \frac{q}{q_{0}}=\frac{p}{p_{0}}.$$

If $$b\in\operatorname{BMO}(\mathbb{R}^{n})$$, then we have

$$\bigl\Vert [b,I_{\alpha}]^{(1)}f \bigr\Vert _{\mathcal{M}_{q}^{q_{0}}} \leq C\Vert f\Vert _{\mathcal{M}_{p}^{p_{0}}}.$$

Conversely if $$n-\alpha$$ is an even integer and

$$\bigl\Vert [b,I_{\alpha}]^{(1)} f \bigr\Vert _{\mathcal{M}_{q}^{q_{0}}} \leq C \Vert f\Vert _{\mathcal{M}_{p}^{p_{0}}},$$

then $$b\in\operatorname{BMO}(\mathbb{R}^{n})$$.

Komori and Mizuhara  removed the restriction ‘$$n-\alpha$$ is an even integer’.

### Theorem C

Let $$0<\alpha<n$$, $$1< p\leq p_{0}<\infty$$ and $$0< q\leq q_{0}<\infty$$. Assume that

$$\frac{1}{q_{0}}=\frac{1}{p_{0}}-\frac{\alpha}{n} \quad \textit{and}\quad \frac{q}{q_{0}}=\frac{p}{p_{0}}.$$

If $$b\in\operatorname{BMO}(\mathbb{R}^{n})$$, then we have

$$\bigl\Vert [b,I_{\alpha}]^{(1)}f \bigr\Vert _{\mathcal{M}_{q}^{q_{0}}} \leq C\Vert f\Vert _{\mathcal{M}_{p}^{p_{0}}}.$$

Conversely if

$$\bigl\Vert [b,I_{\alpha}]^{(1)} f \bigr\Vert _{\mathcal{M}_{q}^{q_{0}}} \leq C \Vert f\Vert _{\mathcal{M}_{p}^{p_{0}}},$$

then $$b\in\operatorname{BMO}(\mathbb{R}^{n})$$.

Sawano et al.  proved the following inequality.

### Theorem D

Let $$0<\alpha<n$$, $$1< p\leq p_{0}<\infty$$, $$1< q\leq q_{0}<\infty$$ and $$1< r\leq r_{0}<\infty$$. Assume that

$$q< r,\qquad \frac{1}{p_{0}}>\frac{\alpha}{n}\geq\frac{1}{r_{0}},\qquad \frac{1}{q_{0}}=\frac{1}{p_{0}}+\frac{1}{r_{0}}-\frac{\alpha}{n}\quad \textit{and}\quad \frac{q}{q_{0}}=\frac{p}{p_{0}}.$$

Suppose that $$v\in\mathcal{M}_{r}^{r_{0}}(\mathbb{R}^{n})$$. Then, for $$b\in\operatorname{BMO}(\mathbb{R}^{n})$$, we have

$$\bigl\Vert \bigl( [ b, I_{\alpha} ]^{(m)} f \bigr) v \bigr\Vert _{\mathcal{M}_{q}^{q_{0}}} \leq C\Vert b \Vert _{\operatorname{BMO}}^{m} \Vert v\Vert _{\mathcal{M}_{r}^{r_{0}}} \Vert f \Vert _{\mathcal{M}_{p}^{p_{0}}}.$$

In the case of $$m=0$$, we refer to [1, 8, 9]. In this paper, we generalize Theorem D to a weighted setting. On the other hand, in , the following theorem is proved.

### Theorem E

Let $$0<\alpha<n$$, $$1< p\leq p_{0}<\infty$$ and $$1< q\leq q_{0}< r_{0}<\infty$$. Assume that

$$\frac{1}{p_{0}}>\frac{\alpha}{n}\geq\frac{1}{r_{0}},\qquad \frac{1}{q_{0}}=\frac{1}{p_{0}}+\frac{1}{r_{0}}-\frac{\alpha}{n},\qquad \frac{q}{q_{0}}=\frac{p}{p_{0}}$$

and $$1< a<\frac{r_{0}}{q_{0}}$$. Suppose that the weights v and w satisfy the following condition:

\begin{aligned}{} [ v,w ]_{aq_{0}, r_{0},aq,p/a} :=& \sup_{Q\subset Q'} \biggl( \frac{|Q|}{|Q'|} \biggr)^{\frac{1}{aq_{0}}} \bigl\vert Q' \bigr\vert ^{\frac{1}{r_{0}}} \biggl( \fint_{Q} v(x)^{aq} \,dx \biggr)^{\frac{1}{aq}} \biggl( \fint_{Q'} w(x)^{-(p/a)'} \,dx \biggr)^{\frac{1}{(p/a)'}} \\ < & \infty. \end{aligned}
(2)

Then we have

$$\bigl\Vert (I_{\alpha} f ) v \bigr\Vert _{\mathcal{M}_{q}^{q_{0}}} \leq C [ v,w ]_{aq_{0}, r_{0},aq,p/a} \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}}.$$

In this paper, we investigate the boundedness of higher order commutators generated by BMO-functions and the fractional integral operator on Morrey spaces corresponding to Theorem E.

## Main results and their corollaries

In this paper, we obtain two main theorems.

### Theorem 1

Let $$0<\alpha<n$$, $$1< p\leq p_{0}<\infty$$ and $$1< q\leq q_{0}< r_{0}<\infty$$. Assume that

$$\frac{1}{p_{0}}>\frac{\alpha}{n}\geq\frac{1}{r_{0}},\qquad \frac{1}{q_{0}}=\frac{1}{p_{0}}+\frac{1}{r_{0}}-\frac{\alpha}{n},\qquad \frac{q}{q_{0}}=\frac{p}{p_{0}}$$

and $$1< a<\frac{r_{0}}{q_{0}}$$. Suppose that the weights v and w satisfy the condition (2). Then, for $$b\in\operatorname{BMO}(\mathbb{R}^{n})$$, we have

$$\bigl\Vert \bigl( [ b, I_{\alpha} ]^{(m)} f \bigr) v \bigr\Vert _{\mathcal{M}_{q}^{q_{0}}} \leq C\Vert b \Vert _{\operatorname{BMO}}^{m} [ v,w ]_{aq_{0}, r_{0},aq,p/a} \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}}.$$

### Remark 3

The condition of Theorem 1 corresponds with the condition of Theorem E. This implies that Theorem 1 gives us the same type of corollaries as in Theorem E.

Taking $$w(x)=M_{\frac{aq}{r_{0}}n} ( v^{aq} )(x)^{\frac{1}{aq}}$$, we have the following corollary.

### Corollary 1

Let $$0<\alpha<n$$, $$1< p\leq p_{0}<\infty$$ and $$1< q\leq q_{0}< r_{0}<\infty$$. Assume that

$$\frac{1}{p_{0}}>\frac{\alpha}{n}\geq\frac{1}{r_{0}},\qquad \frac{1}{q_{0}}=\frac{1}{p_{0}}+\frac{1}{r_{0}}-\frac{\alpha}{n},\qquad \frac{q}{q_{0}}=\frac{p}{p_{0}}$$

and $$1< a<\frac{r_{0}}{q_{0}}$$. Let v be a weight. Suppose that $$b\in\operatorname{BMO}(\mathbb{R}^{n})$$, then we have

$$\bigl\Vert \bigl( [ b, I_{\alpha} ]^{(m)} f \bigr) v \bigr\Vert _{\mathcal{M}_{q}^{q_{0}}} \leq C\Vert b \Vert _{\operatorname{BMO}}^{m} \bigl\Vert f M_{\frac{aq}{r_{0}}n}\bigl(v^{aq}\bigr)^{\frac{1}{aq}} \bigr\Vert _{\mathcal{M}_{p}^{p_{0}}}.$$

Taking $$w(x)\equiv1$$, we obtain the following corollary.

### Corollary 2

Let $$0<\alpha<n$$, $$1< p\leq p_{0}<\infty$$ and $$1< q\leq q_{0}< r_{0}<\infty$$. Assume that

$$\frac{1}{p_{0}}>\frac{\alpha}{n}\geq\frac{1}{r_{0}},\qquad \frac{1}{q_{0}}=\frac{1}{p_{0}}+\frac{1}{r_{0}}-\frac{\alpha}{n},\qquad \frac{q}{q_{0}}=\frac{p}{p_{0}}$$

and $$1< a<\frac{r_{0}}{q_{0}}$$. Suppose that $$v\in\mathcal{M}_{aq}^{r_{0}}(\mathbb{R}^{n})$$. Then, for $$b\in\operatorname{BMO}(\mathbb{R}^{n})$$, we have

$$\bigl\Vert \bigl( [ b, I_{\alpha} ]^{(m)} f \bigr) v \bigr\Vert _{\mathcal{M}_{q}^{q_{0}}} \leq C\Vert b \Vert _{\operatorname{BMO}}^{m} \Vert v\Vert _{\mathcal{M}_{aq}^{r_{0}}} \Vert f \Vert _{\mathcal{M}_{p}^{p_{0}}}.$$

On the other hand, letting $$r_{0}\to\infty$$, we obtain the weighted Adams type inequality for the m-fold commutator $$[b, I_{\alpha}]^{(m)}$$.

### Corollary 3

Let $$0<\alpha<n$$, $$1< p\leq p_{0}<\infty$$ and $$1< q\leq q_{0}<\infty$$. Assume that

$$\frac{1}{q_{0}}=\frac{1}{p_{0}}-\frac{\alpha}{n},\qquad \frac{q}{q_{0}}= \frac {p}{p_{0}}$$

and $$1< a<\frac{r_{0}}{q_{0}}$$. Suppose that the weights v and w satisfy the following condition:

$$[ v,w ]_{aq_{0},aq,p/a} := \sup_{Q\subset Q'} \biggl( \frac{|Q|}{|Q'|} \biggr)^{\frac{1}{aq_{0}}} \biggl( \fint_{Q} v(x)^{aq} \,dx \biggr)^{\frac{1}{aq}} \biggl( \fint_{Q'} w(x)^{-(p/a)'} \,dx \biggr)^{\frac{1}{(p/a)'}}< \infty.$$
(3)

Then, for $$b\in\operatorname{BMO}(\mathbb{R}^{n})$$, we have

$$\bigl\Vert \bigl( [ b, I_{\alpha} ]^{(m)} f \bigr) v \bigr\Vert _{\mathcal{M}_{q}^{q_{0}}} \leq C\Vert b \Vert _{\operatorname{BMO}}^{m} [ v,w ]_{aq_{0},aq,p/a} \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}}.$$

Corollary 3 gives us the following inequality in letting $$p=p_{0}$$, $$q=q_{0}$$ and $$v=w$$.

### Corollary 4

Let $$0<\alpha<n$$, $$1< p<\frac{n}{\alpha}$$ and $$1< q<\infty$$. Assume that

$$\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}.$$

Suppose that $$w\in A_{p,q}(\mathbb{R}^{n})$$, i.e.

$$[ w ]_{A_{p,q}(\mathbb{R}^{n})} := \sup_{Q\subset\mathbb{R}^{n}} \biggl( \fint_{Q} w(x)^{q} \,dx \biggr)^{\frac{1}{q}} \biggl( \fint_{Q} w(x)^{-p'} \,dx \biggr)^{\frac{1}{p'}}< \infty.$$
(4)

Then, for $$b\in\operatorname{BMO}(\mathbb{R}^{n})$$, we have

$$\bigl\Vert \bigl( [ b, I_{\alpha} ]^{(m)} f \bigr) \bigr\Vert _{L^{q}(w^{q})} \leq C\Vert b \Vert _{\operatorname{BMO}}^{m} [ w ]_{A_{p,q}(\mathbb{R}^{n})} \Vert f \Vert _{L^{p}(w^{p})}\quad (m=0,1,2,\ldots).$$

Corollary 3 and Theorem C give us the following corollary.

### Corollary 5

Let $$0<\alpha<n$$, $$1< p\leq p_{0}<\infty$$ and $$1< q\leq q_{0}<\infty$$. Assume that

$$\frac{1}{q_{0}}=\frac{1}{p_{0}}-\frac{\alpha}{n},\qquad \frac{q}{q_{0}}= \frac {p}{p_{0}}$$

and $$a>1$$. Suppose that the weights v and w satisfy the condition (3). If

$$\bigl\Vert [b,I_{\alpha}]^{(1)} f\bigr\Vert _{\mathcal{M}_{q}^{q_{0}}} \leq C\Vert f\Vert _{\mathcal{M}_{p}^{p_{0}}}$$

holds, then we have for $$b\in\operatorname{BMO}(\mathbb{R}^{n})$$,

$$\bigl\Vert \bigl( [ b, I_{\alpha} ]^{(m)} f \bigr) v \bigr\Vert _{\mathcal{M}_{q}^{q_{0}}} \leq C\Vert b \Vert _{\operatorname{BMO}}^{m} [ v,w ]_{aq_{0},aq,p/a} \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}}.$$

According to Theorem 1.8 in , we can pass our result to the operator given by

$$[ \vec{b}, I_{\alpha} ] f(x) := \int_{\mathbb{R}^{n}} \frac{f(y)}{|x-y|^{n-\alpha}} \prod _{j=1}^{m} \bigl(b_{j}(x)-b_{j}(y) \bigr) \,dy,$$

where $$\vec{b}=(b_{1},\ldots, b_{m})$$. By a similar argument to , as a consequence of Theorem 1 in this paper, we can obtain the following estimate.

### Corollary 6

Let $$0<\alpha<n$$, $$1< p\leq p_{0}<\infty$$ and $$1< q\leq q_{0}< r_{0}<\infty$$. Assume that

$$\frac{1}{p_{0}}>\frac{\alpha}{n}\geq\frac{1}{r_{0}},\qquad \frac{1}{q_{0}}=\frac{1}{p_{0}}+\frac{1}{r_{0}}-\frac{\alpha}{n},\qquad \frac{q}{q_{0}}=\frac{p}{p_{0}}$$

and $$1< a<\frac{r_{0}}{q_{0}}$$. Suppose that the weights v and w satisfy the condition (2). Then, for $$\vec{b}=(b_{1},\ldots, b_{m})\in\operatorname{BMO}(\mathbb {R}^{n})\times\cdots\times\operatorname{BMO}(\mathbb{R}^{n})$$, we have

$$\bigl\Vert \bigl( [ \vec{b}, I_{\alpha} ] f \bigr) v \bigr\Vert _{\mathcal{M}_{q}^{q_{0}}} \leq C \Biggl( \prod_{j=1}^{m} \Vert b_{j} \Vert _{\operatorname {BMO}} \Biggr) [ v,w ]_{aq_{0}, r_{0},aq,p/a} \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}}.$$

### Fractional integral operators having rough kernel

We define the following operators (see  and ).

### Definition 5

Let $$0<\alpha<n$$, a measurable function Ω on $$\mathbb{R}^{n}\backslash\{ 0\}$$ and a measurable function b. Then we define

$$I_{\Omega,\alpha}f(x):= \int_{\mathbb{R}^{n}} \frac{\Omega(x-y) f(y)}{|x-y|^{n-\alpha}}\,dy$$

and

$$[ b,I_{\Omega,\alpha} ]^{(m)} f(x):= \int_{\mathbb{R}^{n}} \frac{\Omega(x-y) (b(x)-b(y))^{m} f(y)}{|x-y|^{n-\alpha}}\,dy.$$

### Remark 4

The following inequality holds:

$$\bigl\vert [ b, I_{\Omega,\alpha} ]^{(m)} f(x) \bigr\vert \leq \int_{\mathbb{R}^{n}} \frac{\vert \Omega(x-y)\vert \vert b(x)-b(y) \vert ^{m}}{|x-y|^{n-\alpha}} \bigl\vert f(y)\bigr\vert \,dy.$$
(5)

As shall be verified in the proof of Theorem 2, we consider the operator

$$x\mapsto \int_{\mathbb{R}^{n}} \frac{\vert \Omega(x-y)\vert \vert b(x)-b(y) \vert ^{m}}{|x-y|^{n-\alpha}} f(y) \,dy$$

and hence we may assume that the integral defining $$[b,I_{\Omega,\alpha }]^{(m)}f(x)$$ converges for a.e. $$x\in\mathbb{R}^{n}$$.

By a similar argument to the proof of Theorem 1, we have the following estimate.

### Theorem 2

Let $$1< s\leq\infty$$, $$0<\alpha<n$$, $$1\leq s'< p\leq p_{0}<\infty$$, $$1< q\leq q_{0}<\infty$$ and $$1< r\leq r_{0}<\infty$$. Assume that

$$\frac{1}{p_{0}}>\frac{\alpha}{n}\geq\frac{1}{r_{0}},\qquad \frac{1}{q_{0}}=\frac{1}{p_{0}}+\frac{1}{r_{0}}-\frac{\alpha}{n},\qquad \frac{q}{q_{0}}=\frac{p}{p_{0}}$$

and $$1< a<\frac{r_{0}}{q_{0}}$$. Suppose that the weights v and w satisfy $$[ v^{s'},w^{s'} ]_{\frac{aq_{0}}{s'},\frac{r_{0}}{s'},\frac {aq}{s'},\frac{p}{s'a}}^{\frac{1}{s'}}<\infty$$. Moreover, suppose that $$\Omega\in L^{s}(\mathbb{S}^{n-1})$$ is homogeneous of order 0: For any $$\lambda>0$$, $$\Omega(\lambda x)=\Omega(x)$$. Then, for $$b\in\operatorname{BMO}(\mathbb{R}^{n})$$, we have

$$\bigl\Vert \bigl( [ b, I_{\Omega, \alpha} ]^{(m)} f \bigr) v \bigr\Vert _{\mathcal{M}_{q}^{q_{0}}} \leq C\Vert b \Vert _{\operatorname{BMO}}^{m} \Vert \Omega \Vert _{L^{s}(\mathbb{S}^{n-1})} \bigl[ v^{s'},w^{s'} \bigr]_{\frac{aq_{0}}{s'},\frac{r_{0}}{s'},\frac {aq}{s'},\frac{p}{s'a}}^{\frac{1}{s'}} \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}}.$$

Since $$[b,I_{\Omega,\alpha} ]^{(0)}=I_{\Omega,\alpha}$$, we refer to . Theorem 2 recovers the following result (see [4, 11]).

### Corollary 7

Let $$1< s\leq\infty$$, $$0<\alpha<n$$, $$1\leq s'< p<\frac{n}{\alpha}$$ and $$1< q<\infty$$. Assume that

$$\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$$

and $$w^{s'}\in A_{\frac{p}{s'},\frac{q}{s'}}(\mathbb{R}^{n})$$. Suppose that $$\Omega\in L^{s}(\mathbb{S}^{n-1})$$ is homogeneous of order 0: For any $$\lambda>0$$, $$\Omega(\lambda x)=\Omega(x)$$. Then we have, for $$b\in\operatorname{BMO}(\mathbb{R}^{n})$$,

$$\bigl\Vert \bigl( [ b, I_{\Omega, \alpha} ]^{(m)} f \bigr) \bigr\Vert _{L^{q}(w^{q})} \leq C \bigl[ w^{s'} \bigr]_{ A_{\frac{p}{s'},\frac{q}{s'}}(\mathbb{R}^{n}) }^{\frac{1}{s'}} \Vert b \Vert _{\operatorname{BMO}}^{m} \Vert \Omega \Vert _{L^{s}(\mathbb{S}^{n-1})} \Vert f \Vert _{L^{p}(w^{p})}.$$

## Some lemmas

In this section, we prepare some lemmas for proving main results. We recall the following inequalities (see [3, 13] and ).

### Lemma 1

(The John-Nirenberg inequality)

Let $$1\leq p<\infty$$ and let Q be a cube. Then there exists a constant $$C>0$$ such that

$$\biggl( \fint_{Q} \bigl\vert b(x)-m_{Q}(b) \bigr\vert ^{p} \,dx \biggr)^{\frac {1}{p}}\leq C \Vert b \Vert _{\operatorname{BMO}},$$

for all $$b\in\operatorname{BMO}(\mathbb{R}^{n})$$.

We invoke the following decomposition which is derived in . We omit the details; see [1, 12] for the proof.

Let $$\mathcal{D}(Q_{0})$$ be the collection of all dyadic subcubes of $$Q_{0}$$, that is, all those cubes obtained by dividing $$Q_{0}$$ into $$2^{n}$$ congruent cubes of half its length, dividing each of those into $$2^{n}$$ congruent cubes. By convention $$Q_{0}$$ itself to $$\mathcal{D}(Q_{0})$$, and so on.

### Lemma 2

Let $$\gamma:=m_{3Q_{0}}(f)$$ and $$A>2\cdot18^{n}$$. For $$k=1,2,\ldots$$ we take

$$D_{k}:=\bigcup \bigl\{ Q\in\mathcal{D}(Q_{0}): m_{3Q}(f) >\gamma A^{k} \bigr\} .$$

For $$\theta_{1}>1$$, let

$$\gamma':= \biggl( \fint_{3Q_{0}} \bigl\vert f(y)\bigr\vert ^{\theta_{1}}\,dy \biggr)^{\frac{1}{\theta_{1}}}$$

and $$A'> ( 2\cdot18^{n} )^{\frac{1}{\theta_{1}}}$$. For $$k=1,2,\ldots$$ we take

$$D_{k}':=\bigcup \biggl\{ Q\in \mathcal{D}(Q_{0}): \biggl( \fint_{3Q} \bigl\vert f(y)\bigr\vert ^{\theta_{1}} \,dy \biggr)^{\frac{1}{\theta_{1}}} >\gamma' A^{\prime k} \biggr\} .$$

Considering the maximality cube, we have

$$D_{k}=\bigcup_{j} Q_{k,j} \quad \textit{and}\quad D_{k}'= \bigcup _{j} Q_{k,j}'.$$

Then we have

$$\gamma A^{k} < m_{3Q_{k,j}}(f) \leq2^{n} \gamma A^{k}\quad \textit{and} \quad \gamma' A^{\prime k} < \biggl( \fint_{3Q_{k,j}'} \bigl\vert f(y)\bigr\vert ^{\theta_{1}} \,dy \biggr)^{\frac{1}{\theta _{1}}} \leq 2^{\frac{n}{\theta_{1}}} \gamma' A^{k}.$$

Let $$E_{k,j}:=Q_{k,j} \backslash D_{k+1}$$ and $$E_{k,j}':=Q_{k,j}'\backslash D_{k+1}'$$. Moreover we obtain

$$\vert Q_{k,j}\vert \leq2\vert E_{k,j}\vert \quad \textit{and}\quad \bigl\vert Q_{k,j}'\bigr\vert \leq2 \bigl\vert E_{k,j}' \bigr\vert .$$

### Lemma 3

Under the condition of Theorem  1, we can choose auxiliary indices $$\theta_{1}$$, $$\theta_{2}$$, $$\theta_{3}$$, $$\theta _{4}$$ and $$\theta_{5}$$ so that the following conditions hold:

1. 1.

$$\theta_{1}$$, $$\theta_{2}$$, $$\theta_{3}$$, $$\theta_{4}$$ and $$\theta_{5}\in(1,p)$$.

2. 2.

$$L>1$$ and $$s\in(q,r)$$ such that $$s\theta_{2}< Lq$$ and $$s'\theta_{2}< q'$$.

3. 3.

For the index $$\theta_{1}\in(1,p)$$, we can choose $$a_{*}>1$$ such that $$a_{*}\theta_{1}< p$$.

Assume in addition that, for these indices,

$$a\geq\max \biggl\{ \theta_{4}, L, \frac{p}{ ( \theta_{5} ( \frac {p}{\theta_{5}} )' )'}, \frac{p}{ ( \theta_{1} ( \frac{p}{\theta_{1}a_{*}} )' )'}, \theta_{3} \biggr\} >1.$$

Then we obtain

$$\max \biggl\{ \theta_{5} \biggl( \frac{p}{\theta_{5}} \biggr)', \theta_{1} \biggl( \frac{p}{\theta_{1}a_{*}} \biggr)' \biggr\} \leq \biggl( \frac{p}{a} \biggr)'.$$

### Proof

We examine the second item; $$s\theta_{2}< Lq$$ and $$s'\theta_{2}< q'$$. For $$0<\varepsilon<1$$, we take $$\delta=\frac{\varepsilon}{q^{2}}<\varepsilon$$. If $$s=q+\varepsilon$$ and $$\theta_{2}=1+\delta$$, then we have the following estimate:

\begin{aligned} s\theta_{2} & = (q+\varepsilon) (1+\delta)= q+q\delta+\varepsilon+ \varepsilon\delta \\ &\leq q+q\max\{\varepsilon, \delta\}+\max\{\varepsilon, \delta\}+ \max\{\varepsilon, \delta\}^{2} \\ &< q+q\max\{\varepsilon, \delta\}+2\max\{\varepsilon, \delta\} \\ &< q+q\max\{ \varepsilon, \delta\}+2q\max\{\varepsilon, \delta\} \\ &=q \bigl( 1+3\max\{ \varepsilon, \delta\} \bigr)=q(1+3\varepsilon)=L q. \end{aligned}

On the other hand, we check $$s'\theta_{2}< q'$$:

$$q'-s'\theta_{2} = \frac{\frac{\varepsilon^{2}}{q^{2}}+ \frac{\varepsilon}{q} ( 1-\varepsilon ) }{(q-1)(q+\varepsilon-1)}>0.$$

Next we check $$\frac{p}{ ( \theta_{5} ( \frac{p}{\theta_{5}} )' )'}>1$$. Since $$\theta_{5}>1$$, we obtain

$$\theta_{5} \biggl( \frac{p}{\theta_{5}} \biggr)'> \biggl( \frac{p}{\theta _{5}} \biggr)'>p'.$$

Therefore we have

$$\biggl( \theta_{5} \biggl( \frac{p}{\theta_{5}} \biggr)' \biggr)' < \bigl( p' \bigr)'=p.$$

This gives us

$$\frac{p}{ ( \theta_{5} ( \frac{p}{\theta_{5}} )' )'}>1.$$

By a similar argument, we obtain

$$\frac{p}{ ( \theta_{1} ( \frac{p}{\theta_{1}a_{*}} )' )'}>1.$$

□

### Remark 5

The index $$\theta_{1}$$ in Lemma 2 corresponds with the index $$\theta_{1}$$ in Lemma 3.

## Proof of Theorem 1

### Proof of Theorem 1

Fix a dyadic cube $$Q_{0}\in\mathcal{D}(\mathbb{R}^{n})$$. Let $$\mathcal{D}_{\nu}$$ be the collection of dyadic cubes. The volume of the elements of $$\mathcal{D}_{\nu}$$ is $$2^{n\nu}$$. For $$x\in Q_{0}$$, we have

\begin{aligned} \bigl\vert [ b, I_{\alpha} ]^{(m)} f(x)\bigr\vert &\leq C \sum_{\nu\in\mathbb{Z}} \sum_{\substack{Q\in\mathcal{D}_{\nu}, \\ |Q|=2^{\nu n}}} 2^{-\nu(n-\alpha)} \chi_{Q} (x) \int_{3Q} \bigl\vert b(x)-b(y)\bigr\vert ^{m} \bigl\vert f(y)\bigr\vert \,dy \\ &= C \sum_{\nu\in\mathbb{Z}} \biggl( \sum _{\substack{Q\in\mathcal{D}_{\nu},\\ Q\subseteq Q_{0}}} + \sum_{\substack{Q\in\mathcal{D}_{\nu},\\ Q\supsetneq Q_{0}}} \biggr) 2^{-\nu(n-\alpha)} \chi_{Q} (x) \int_{3Q} \bigl\vert b(x)-b(y)\bigr\vert ^{m} \bigl\vert f(y)\bigr\vert \,dy \\ &=:C(A+B). \end{aligned}

We evaluate A and B in Sections 4.1 and 4.2, respectively.

### The estimate of A

By $$| b(x)-b(y) |^{m}\leq2^{m-1} ( |b(x)-m_{Q}(b)|^{m} + |m_{Q}(b)-b(y)|^{m} )$$, we obtain

\begin{aligned} A =& \sum_{Q\in\mathcal{D}(Q_{0})} l(Q)^{\alpha-n} \chi_{Q}(x) \int_{3Q} \bigl\vert b(x)-b(y)\bigr\vert ^{m} \bigl\vert f(y)\bigr\vert \,dy \\ \leq& C \sum_{Q\in\mathcal{D}(Q_{0})} l(Q)^{\alpha} \chi_{Q}(x) \bigl\vert b(x)-m_{Q}(b)\bigr\vert ^{m} \fint_{3Q} \bigl\vert f(y)\bigr\vert \,dy \\ &{} + C \sum_{Q\in\mathcal{D}(Q_{0})} l(Q)^{\alpha} \chi_{Q}(x) \fint_{3Q} \bigl\vert m_{Q}(b)-b(y)\bigr\vert ^{m} \bigl\vert f(y)\bigr\vert \,dy. \end{aligned}

We take $$\theta_{1}>1$$ as in Lemma 2. By Hölder’s inequality for $$\theta_{1}>1$$, we have

\begin{aligned} A \leq& C \sum_{Q\in\mathcal{D}(Q_{0})} l(Q)^{\alpha} \chi_{Q}(x) \bigl\vert b(x)-m_{Q}(b)\bigr\vert ^{m} \fint_{3Q} \bigl\vert f(y)\bigr\vert \,dy \\ &{} + C \sum_{Q\in\mathcal{D}(Q_{0})} l(Q)^{\alpha} \chi_{Q}(x) \biggl( \fint_{3Q} \bigl\vert m_{Q}(b)-b(y)\bigr\vert ^{m\theta_{1}'} \,dy \biggr)^{\frac {1}{\theta_{1}'}} \biggl( \fint_{3Q} \bigl\vert f(y)\bigr\vert ^{\theta_{1}} \,dy \biggr)^{\frac{1}{\theta_{1}}}. \end{aligned}

By Lemma 1, we have

\begin{aligned} A \leq& C \sum_{Q\in\mathcal{D}(Q_{0})} l(Q)^{\alpha} \chi_{Q}(x) \bigl\vert b(x)-m_{Q}(b)\bigr\vert ^{m} \fint_{3Q} \bigl\vert f(y)\bigr\vert \,dy \\ &{} + C \Vert b\Vert _{\operatorname{BMO}}^{m} \sum _{Q\in\mathcal{D}(Q_{0})} l(Q)^{\alpha} \chi_{Q}(x) \biggl( \fint_{3Q} \bigl\vert f(y)\bigr\vert ^{\theta_{1}} \,dy \biggr)^{\frac{1}{\theta_{1}}} \\ =& C \bigl( I+\Vert b\Vert _{\operatorname{BMO}}^{m} \mathit{II} \bigr). \end{aligned}

We evaluate I. Let

$$\mathcal{D}_{0}(Q_{0}) := \biggl\{ Q\in \mathcal{D}(Q_{0}); \biggl( \fint_{3Q} \bigl\vert f(y)\bigr\vert \,dy \biggr) \leq \gamma A \biggr\}$$

and

$$\mathcal{D}_{k,j}(Q_{0}) := \biggl\{ Q\in \mathcal{D}(Q_{0}); Q\subset Q_{k,j}, \gamma A^{k}< \biggl( \fint_{3Q} \bigl\vert f(y)\bigr\vert \,dy \biggr) \leq \gamma A^{k+1} \biggr\} ,$$

where $$Q_{k,j}$$ is in Lemma 2. Then we have

$$\mathcal{D}(Q_{0})= \mathcal{D}_{0}(Q_{0}) \cup \biggl( \bigcup_{k,j} \mathcal {D}_{k,j}(Q_{0}) \biggr).$$

By the duality argument, we have

$$\biggl( \int_{Q_{0}} I^{q}\cdot v(x)^{q} \,dx \biggr)^{\frac{1}{q}} =\sup_{\Vert g \Vert _{L^{q'}(Q_{0})}=1} \biggl( \int_{Q_{0}} I \cdot v(x) \bigl\vert g(x)\bigr\vert \,dx \biggr).$$

Let $$g\geq0$$, $$\operatorname{supp}(g)\subset Q_{0}$$, $$\Vert g \Vert _{L^{q'}(Q_{0})}=1$$. Then we have

\begin{aligned} \int_{Q_{0}} I \cdot v(x) \bigl\vert g(x)\bigr\vert \,dx \leq& C \sum_{Q\in\mathcal{D}(Q_{0})} l(Q)^{\alpha} \biggl( \fint_{3Q} \bigl\vert f(y)\bigr\vert \,dy \biggr) \\ &{}\times \int_{Q} \bigl\vert b(x)-m_{Q}(b) \bigr\vert ^{m} v(x) g(x) \,dx \\ =& C \biggl( \sum_{Q\in\mathcal{D}_{0}(Q_{0})}+\sum _{k,j}\sum_{Q\in\mathcal {D}_{k,j}(Q_{0})} \biggr) l(Q)^{\alpha} \biggl( \fint_{3Q} \bigl\vert f(y)\bigr\vert \,dy \biggr) \\ &{} \times \int_{Q} \bigl\vert b(x)-m_{Q}(b) \bigr\vert ^{m} v(x) g(x) \,dx \\ =&I_{0}+\sum_{k,j}I_{k,j}. \end{aligned}

We evaluate $$I_{k,j}$$. If $$Q\in\mathcal{D}_{k,j}(Q_{0})$$, then we have

$$\fint_{3Q} \bigl\vert f(y)\bigr\vert \,dy\leq\gamma A^{k+1}.$$

Hence we obtain

\begin{aligned} \begin{aligned} I_{k,j} & \leq\sum_{Q\in\mathcal{D}_{k,j}(Q_{0})} l(Q)^{\alpha} \gamma A^{k+1} \int_{Q} \bigl\vert b(x)-m_{Q}(b)\bigr\vert ^{m} v(x) g(x) \,dx \\ &\leq A \sum_{Q\in\mathcal{D}_{k,j}(Q_{0})} l(Q_{k,j})^{\alpha} \gamma A^{k} \int_{Q} \bigl\vert b(x)-m_{Q}(b)\bigr\vert ^{m} v(x) g(x) \,dx. \end{aligned} \end{aligned}

Since

$$\gamma A^{k}< \fint_{3Q_{k,j}} \bigl\vert f(y)\bigr\vert \,dy,$$

we obtain

$$I_{k,j}\leq A \sum_{Q\in\mathcal{D}_{k,j}(Q_{0})} l(Q_{k,j})^{\alpha} \fint _{3Q_{k,j}} \bigl\vert f(y)\bigr\vert \,dy \int_{Q} \bigl\vert b(x)-m_{Q}(b)\bigr\vert ^{m} v(x) g(x) \,dx.$$

By Hölder’s inequality for $$\theta_{2}>1$$ as in Lemma 3, we obtain

\begin{aligned} I_{k,j} \leq& A l(Q_{k,j})^{\alpha} m_{3Q_{k,j}} \bigl(\vert f\vert \bigr) \sum_{Q\in\mathcal{D}_{k,j}(Q_{0})} \biggl( \int_{Q} \bigl\vert b(x)-m_{Q}(b)\bigr\vert ^{m}v(x)g(x)\,dx \biggr) \\ \leq& A l(Q_{k,j})^{\alpha} m_{3Q_{k,j}}\bigl(\vert f \vert \bigr) \sum_{Q\in\mathcal{D}_{k,j}(Q_{0})} |Q| \biggl( \fint_{Q} \bigl\vert b(x)-m_{Q}(b)\bigr\vert ^{m\theta_{2}'} \,dx \biggr)^{\frac {1}{\theta_{2}'}} \\ &{}\times\biggl( \fint_{Q} v(x)^{\theta_{2}}g(x)^{\theta_{2}} \,dx \biggr)^{\frac {1}{\theta_{2}}}. \end{aligned}

By Lemma 1, we obtain

\begin{aligned} I_{k,j}&\leq A \Vert b\Vert _{\operatorname{BMO}}^{m} l(Q_{k,j})^{\alpha} m_{3Q_{k,j}}\bigl(\vert f\vert \bigr) \sum_{Q\in\mathcal{D}_{k,j}(Q_{0})} \int_{Q} \biggl( \fint_{Q} \bigl(v(y)g(y)\bigr)^{\theta_{2}} \,dy \biggr)^{\frac {1}{\theta_{2}}} \,dx \\ &\leq A \Vert b\Vert _{\operatorname{BMO}}^{m} l(Q_{k,j})^{\alpha} m_{3Q_{k,j}}\bigl(\vert f\vert \bigr) \sum_{Q\in\mathcal{D}_{k,j}(Q_{0})} \int_{Q} M \bigl[ (v_{k,j} g)^{\theta_{2}} \bigr](x)^{\frac{1}{\theta _{2}}} \,dx, \end{aligned}

where $$v_{k,j}=v\chi_{Q_{k,j}}$$ and the symbol M is the ordinary Hardy-Littlewood maximal operator. By Lemma 2, we have

\begin{aligned} I_{k,j}&\leq A \Vert b\Vert _{\operatorname{BMO}}^{m} |Q_{k,j}| l(Q_{k,j})^{\alpha} m_{3Q_{k,j}}\bigl( \vert f\vert \bigr) \biggl( \fint_{Q_{k,j}} M \bigl[ (v_{k,j} g)^{\theta_{2}} \bigr](x)^{\frac{1}{\theta_{2}}} \,dx \biggr) \\ &\leq2A \Vert b\Vert _{\operatorname{BMO}}^{m} |E_{k,j}| l(Q_{k,j})^{\alpha} m_{3Q_{k,j}}\bigl(\vert f\vert \bigr) \biggl( \fint_{Q_{k,j}} M \bigl[ (v_{k,j} g)^{\theta_{2}} \bigr](x)^{\frac{1}{\theta_{2}}} \,dx \biggr) \\ &= 2A \Vert b\Vert _{\operatorname{BMO}}^{m} \int_{E_{k,j}} l(Q_{k,j})^{\alpha} m_{3Q_{k,j}}\bigl(\vert f\vert \bigr) \biggl( \fint_{Q_{k,j}} M \bigl[ (v_{k,j} g)^{\theta_{2}} \bigr](x)^{\frac{1}{\theta_{2}}} \,dx \biggr) \,dy. \end{aligned}

We take $$s\in(q,r)$$ and $$L>1$$ as in Lemma 3. By Hölder’s inequality for $$s>1$$, we have

$$M \bigl[ (v_{k,j} g)^{\theta_{2}} \bigr](x)^{\frac{1}{\theta_{2}}} \leq M \bigl[ v_{k,j}^{s\theta_{2}} \bigr](x)^{\frac{1}{s\theta_{2}}} M \bigl[ g^{s'\theta_{2}} \bigr](x)^{\frac{1}{s'\theta_{2}}}.$$

By Hölder’s inequality for $$Lq>1$$, we obtain the following inequality:

\begin{aligned}& \biggl( \fint_{Q_{k,j}} M \bigl[ (v_{k,j} g)^{\theta_{2}} \bigr](x)^{\frac{1}{\theta_{2}}} \,dx \biggr) \\& \quad \leq \biggl( \fint_{Q_{k,j}} M \bigl[ v_{k,j}^{s\theta_{2}} \bigr](x)^{\frac {Lq}{s\theta_{2}}} \,dx \biggr)^{\frac{1}{Lq}} \biggl( \fint_{Q_{k,j}} M \bigl[ g^{s'\theta_{2}} \bigr](x)^{\frac {(Lq)'}{s'\theta_{2}}} \,dx \biggr)^{\frac{1}{(Lq)'}}. \end{aligned}

Since $$s\theta_{2}< Lq$$, the boundedness of $$M: L^{\frac{Lq}{s\theta_{2}}}(\mathbb{R}^{n}) \to L^{\frac{Lq}{s\theta _{2}}}(\mathbb{R}^{n})$$ gives us the following inequality:

\begin{aligned}& \biggl( \fint_{Q_{k,j}} M \bigl[ (v_{k,j} g)^{\theta_{2}} \bigr](x)^{\frac{1}{\theta_{2}}} \,dx \biggr) \\& \quad \leq C \biggl( \frac{1}{|Q_{k,j}|} \int_{\mathbb{R}^{n}} v_{k,j}(x)^{Lq} \,dx \biggr)^{\frac{1}{Lq}} \biggl( \fint_{Q_{k,j}} M \bigl[ g^{s'\theta_{2}} \bigr](x)^{\frac {(Lq)'}{s'\theta_{2}}} \,dx \biggr)^{\frac{1}{(Lq)'}}. \end{aligned}

Since $$a\geq L>1$$, by Hölder’s inequality for $$\frac{a}{L}\geq1$$,

\begin{aligned}& \biggl( \fint_{Q_{k,j}} M \bigl[ (v_{k,j} g)^{\theta_{2}} \bigr](x)^{\frac{1}{\theta_{2}}} \,dx \biggr) \\& \quad \leq C \biggl( \fint_{Q_{k,j}} v(x)^{aq} \,dx \biggr)^{\frac{1}{aq}} \biggl( \fint_{Q_{k,j}} M \bigl[ g^{s'\theta_{2}} \bigr](x)^{\frac {(Lq)'}{s'\theta_{2}}} \,dx \biggr)^{\frac{1}{(Lq)'}}. \end{aligned}

By Lemma 2, this implies that

$$I_{k,j}\leq 2A\Vert b \Vert _{\operatorname{BMO}}^{m} \int_{E_{k,j}} M_{\alpha ,aq}(f,v) (x)\cdot M \bigl[ M \bigl[ g^{s'\theta_{2}} \bigr]^{\frac{(Lq)'}{s'\theta_{2}}} \bigr](x)^{\frac{1}{(Lq)'}}\,dx,$$

where

$$M_{\alpha,aq}(f,v) (x) :=\sup_{Q\ni x} l(Q)^{\alpha} m_{3Q}(f) \biggl( \fint_{Q} v(x)^{aq} \,dx \biggr)^{\frac{1}{aq}}.$$

A similar argument gives us the following estimate:

$$I_{0}\leq 2A\Vert b \Vert _{\operatorname{BMO}}^{m} \int_{E_{0}} M_{\alpha ,aq}(f,v) (x)\cdot M \bigl[ M \bigl[ g^{s'\theta_{2}} \bigr]^{\frac{(Lq)'}{s'\theta_{2}}} \bigr](x)^{\frac{1}{(Lq)'}}\,dx.$$

By summing up $$I_{0}$$ and $$I_{k,j}$$, we obtain

$$I_{0}+\sum_{k,j}I_{k,j} \leq 2A\Vert b \Vert _{\operatorname{BMO}}^{m} \int_{Q_{0}} M_{\alpha ,aq}(f,v) (x)\cdot M \bigl[ M \bigl[ g^{s'\theta_{2}} \bigr]^{\frac{(Lq)'}{s'\theta_{2}}} \bigr](x)^{\frac{1}{(Lq)'}}\,dx.$$

By Hölder’s inequality for $$q>1$$, we have

\begin{aligned} \begin{aligned} &\int_{Q_{0}} M_{\alpha,aq}(f,v) (x)\cdot M \bigl[ M \bigl[ g^{s'\theta_{2}} \bigr]^{\frac{(Lq)'}{s'\theta_{2}}} \bigr](x)^{\frac{1}{(Lq)'}}\,dx \\ &\quad \leq \biggl( \int_{Q_{0}} M_{\alpha,aq}(f,v) (x)^{q} \,dx \biggr)^{\frac{1}{q}} \biggl( \int_{Q_{0}} M \bigl[ M \bigl[ g^{s'\theta_{2}} \bigr]^{\frac{(Lq)'}{s'\theta_{2}}} \bigr](x)^{\frac{q'}{(Lq)'}}\,dx \biggr)^{\frac{1}{q'}}. \end{aligned} \end{aligned}

Since $$(Lq)'< q'$$, the boundedness of $$M:L^{\frac{q'}{ (Lq)'}}(\mathbb{R}^{n})\to L^{\frac {q'}{(Lq)'}}(\mathbb{R}^{n})$$ gives us the following inequality:

\begin{aligned} \biggl( \int_{Q_{0}} M \bigl[ M \bigl[ g^{s'\theta_{2}} \bigr]^{\frac{(Lq)'}{s'\theta_{2}}} \bigr](x)^{\frac{q'}{(Lq)'}}\,dx \biggr)^{\frac{1}{q'}} &\leq C \biggl( \int_{\mathbb{R}^{n}} M \bigl[ g^{s'\theta_{2}} \bigr](x)^{\frac{(Lq)'}{s'\theta_{2}}\cdot\frac{q'}{(Lq)'}} \,dx \biggr)^{\frac{1}{q'}} \\ &= C \biggl( \int_{\mathbb{R}^{n}} M \bigl[ g^{s'\theta_{2}} \bigr](x)^{\frac{q'}{s'\theta_{2}}} \,dx \biggr)^{\frac{1}{q'}}. \end{aligned}

Since $$s'\theta_{2}< q'$$, the boundedness of $$M:L^{\frac{q'}{s'\theta_{2}}}(\mathbb{R}^{n}) \to L^{\frac{q'}{s'\theta_{2}}}(\mathbb{R}^{n})$$ gives us the following inequality:

\begin{aligned} \biggl( \int_{\mathbb{R}^{n}} M \bigl[ g^{s'\theta_{2}} \bigr](x)^{\frac{q'}{s'\theta_{2}}} \,dx \biggr)^{\frac{1}{q'}} \leq& C \biggl( \int_{Q_{0}} \bigl\vert g(x)\bigr\vert ^{s'\theta_{2}\cdot\frac{q'}{s'\theta_{2}}} \,dx \biggr)^{\frac{1}{q'}} \\ =& C \biggl( \int_{Q_{0}} \bigl\vert g(x)\bigr\vert ^{q'} \,dx \biggr)^{\frac{1}{q'}} =C. \end{aligned}

By Hölder’s inequality for $$\frac{p}{a}>1$$, we obtain

$$M_{\alpha,aq}(f,v) (x) \leq\sup_{Q\ni x} l(Q)^{\alpha} m_{3Q} \bigl( |fw|^{\frac{p}{a}} \bigr)^{\frac{a}{p}} \biggl( \fint_{Q} v(y)^{aq} \,dy \biggr)^{\frac{1}{aq}} \biggl( \fint_{3Q} w(y)^{-(p/a)'}\,dy \biggr)^{\frac{1}{(p/a)'}}.$$

By the condition (2), we obtain

\begin{aligned} M_{\alpha,aq}(f,v) (z)& \leq C[v,w]_{aq_{0},r_{0},aq,p/a} \sup _{Q\ni z} l(Q)^{\alpha-\frac{n}{r_{0}}} m_{3Q} \bigl( |fw|^{\frac {p}{a}} \bigr)^{\frac{a}{p}} \\ & \leq C[v,w]_{aq_{0},r_{0},aq,p/a} M_{ ( \alpha-\frac{n}{r_{0}} )\frac{p}{a}} \bigl( (fw)^{\frac {p}{a}} \bigr) (z)^{\frac{a}{p}}. \end{aligned}

This implies that

$$|Q_{0}|^{\frac{1}{q_{0}}} \biggl( \fint_{Q_{0}} M_{\alpha,aq} ( f,v ) (z)^{q}\, dz \biggr)^{\frac{1}{q}} \leq C[v,w]_{aq_{0},r_{0},aq,p/a} \bigl\Vert M_{ ( \alpha-\frac{n}{r_{0}} )\frac{p}{a}} \bigl( (fw)^{\frac{p}{a}} \bigr) \bigr\Vert _{\mathcal{M}_{\frac{aq}{p}}^{\frac{aq_{0}}{p}}}^{\frac{a}{p}}.$$

Since

$$\frac{1}{q_{0}}\cdot\frac{p}{a} = \frac{1}{p_{0}}\cdot \frac{p}{a} -\frac{ ( \alpha-\frac {n}{r_{0}} )\cdot\frac{p}{a}}{n} \quad \text{and}\quad \frac{\frac{ap_{0}}{p}}{\frac{aq_{0}}{p}}= \frac{a}{\frac{aq}{p}},$$

by Theorem A, we have

\begin{aligned}& |Q_{0}|^{\frac{1}{q_{0}}} \biggl( \fint_{Q_{0}} M_{\alpha,aq} ( f,v ) (z)^{q} \, dz \biggr)^{\frac{1}{q}} \\& \quad \leq C[v,w]_{aq_{0},r_{0},aq,p/a} \bigl\Vert (fw)^{\frac{p}{a}} \bigr\Vert _{\mathcal{M}_{a}^{\frac {ap_{0}}{p}}}^{\frac{a}{p}} \\& \quad =C[v,w]_{aq_{0},r_{0},aq,p/a} \biggl( \sup_{Q} |Q|^{\frac{p}{ap_{0}}} \biggl( \fint_{Q} \bigl\vert f(x)w(x)\bigr\vert ^{\frac{p}{a}\cdot a} \,dx \biggr)^{\frac{1}{a}} \biggr)^{\frac{a}{p}} \\& \quad =C[v,w]_{aq_{0},r_{0},aq,p/a} \sup_{Q} |Q|^{\frac{1}{p_{0}}} \biggl( \fint_{Q} \bigl\vert f(x)\bigr\vert ^{p}w(x)^{p} \,dx \biggr)^{\frac{1}{p}} \\& \quad =C[v,w]_{aq_{0},r_{0},aq,p/a} \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}}. \end{aligned}

We evaluate II. Let

$$\mathcal{D}_{0}'(Q_{0}) := \biggl\{ Q\in \mathcal{D}(Q_{0}); \biggl( \fint_{3Q} \bigl\vert f(y)\bigr\vert ^{\theta_{1}}\,dy \biggr)^{\frac{1}{\theta_{1}}} \leq\gamma' A' \biggr\}$$

and

$$\mathcal{D}_{k,j}'(Q_{0}) := \biggl\{ Q\in \mathcal{D}(Q_{0}); Q\subset Q_{k,j}', \gamma' A^{\prime k}< \biggl( \fint _{3Q} \bigl\vert f(y)\bigr\vert ^{\theta_{1}} \,dy \biggr)^{\frac{1}{\theta_{1}}} \leq\gamma' A^{\prime k+1} \biggr\} ,$$

where $$Q_{k,j}'$$ is found in Lemma 2. Then we have

$$\mathcal{D}(Q_{0})= \mathcal{D}_{0}'(Q_{0}) \cup \biggl( \bigcup_{k,j} \mathcal {D}_{k,j}'(Q_{0}) \biggr).$$

By the duality argument, we have

$$\biggl( \int_{Q_{0}} \mathit{II}^{q}\cdot v(x)^{q} \,dx \biggr)^{\frac{1}{q}} =\sup_{\Vert g \Vert _{L^{q'}(Q_{0})}=1} \biggl( \int_{Q_{0}} \mathit{II} \cdot v(x) \bigl\vert g(x)\bigr\vert \,dx \biggr).$$

Let $$g\geq0$$ be such that $$\operatorname{supp}(g)\subset Q_{0}$$ and $$\Vert g \Vert _{L^{q'}(Q_{0})}=1$$. We have

\begin{aligned} \int_{Q_{0}} \mathit{II} \cdot v(x) g(x) \,dx &\leq \sum _{Q\in\mathcal{D}(Q_{0})} l(Q)^{\alpha} \biggl( \fint_{3Q} \bigl\vert f(y)\bigr\vert ^{\theta_{1}} \,dy \biggr)^{\frac{1}{\theta_{1}}} \Vert v g\Vert _{L^{1}(Q)} \\ &\leq \biggl( \sum_{Q\in\mathcal{D}_{0}'(Q_{0})} +\sum _{k,j} \sum_{Q\in\mathcal{D}_{k,j}'(Q_{0})} \biggr) l(Q)^{\alpha} \biggl( \fint_{3Q} \bigl\vert f(y)\bigr\vert ^{\theta_{1}} \,dy \biggr)^{\frac{1}{\theta_{1}}} \Vert v g\Vert _{L^{1}(Q)} \\ &\leq \biggl( \mathit{II}_{0}+\sum_{k,j} \mathit{II}_{k,j} \biggr). \end{aligned}

We evaluate $$\mathit{II}_{k,j}$$. If $$Q\in\mathcal{D}_{k,j}'(Q_{0})$$, then we have

$$\biggl( \fint_{3Q} \bigl\vert f(y)\bigr\vert ^{\theta_{1}} \,dy \biggr)^{\frac{1}{\theta_{1}}} \leq\gamma' A^{\prime k+1}.$$

Therefore we obtain

\begin{aligned} \mathit{II}_{k,j} & \leq\sum_{Q\in\mathcal{D}_{k,j}'(Q_{0})} l(Q)^{\alpha} \biggl( \fint_{3Q} \bigl\vert f(y)\bigr\vert ^{\theta_{1}} \,dy \biggr)^{\frac {1}{\theta_{1}}} \int_{Q} v(x)g(x) \,dx \\ &\leq\gamma' A^{\prime k+1} \sum _{Q\in\mathcal{D}_{k,j}'(Q_{0})} l(Q)^{\alpha} \int_{Q} v(x)g(x) \,dx. \end{aligned}

Since

$$\gamma' A^{\prime k} \leq \biggl( \fint_{3Q_{k,j}'} \bigl\vert f(y)\bigr\vert ^{\theta_{1}} \,dy \biggr)^{\frac {1}{\theta_{1}}},$$

we obtain

$$\mathit{II}_{k,j} \leq A' \biggl( \fint_{3Q_{k,j}'} \bigl\vert f(y)\bigr\vert ^{\theta_{1}} \,dy \biggr)^{\frac{1}{\theta_{1}}} l\bigl(Q_{k,j}'\bigr)^{\alpha} \biggl( \fint_{Q_{k,j}'} v(x)g(x) \,dx \biggr) \bigl\vert Q_{k,j}'\bigr\vert .$$

By Hölder’s inequality for $$\theta_{3}>1$$ as in Lemma 3, we have

\begin{aligned} \mathit{II}_{k,j} \leq& A' \biggl( \fint_{3Q_{k,j}'} \bigl\vert f(y)\bigr\vert ^{\theta_{1}} \,dy \biggr)^{\frac{1}{\theta_{1}}} l\bigl(Q_{k,j}'\bigr)^{\alpha} \biggl( \fint_{Q_{k,j}'} v(x)^{\theta_{3}q}\,dx \biggr)^{\frac{1}{\theta_{3}q}} \\ &{}\times\biggl( \fint_{Q_{k,j}'} g(x)^{(\theta_{3}q)'} \,dx \biggr)^{\frac {1}{(\theta_{3}q)'}} \bigl\vert Q_{k,j}'\bigr\vert . \end{aligned}

By Lemma 2, we obtain

\begin{aligned} \mathit{II}_{k,j} \leq& 2A' \biggl( \fint_{3Q_{k,j}'} \bigl\vert f(y)\bigr\vert ^{\theta_{1}} \,dy \biggr)^{\frac{1}{\theta_{1}}} l\bigl(Q_{k,j}'\bigr)^{\alpha} \biggl( \fint_{Q_{k,j}'} v(x)^{\theta_{3}q}\,dx \biggr)^{\frac{1}{\theta_{3}q}} \\ &{}\times\biggl( \fint_{Q_{k,j}'} g(x)^{(\theta_{3}q)'} \,dx \biggr)^{\frac {1}{(\theta_{3}q)'}} \bigl\vert E_{k,j}'\bigr\vert \\ =& 2A' \int_{E_{k,j}'} l\bigl(Q_{k,j}' \bigr)^{\alpha} \biggl( \fint_{3Q_{k,j}'} \bigl\vert f(x)\bigr\vert ^{\theta_{1}} \,dx \biggr)^{\frac{1}{\theta_{1}}} \biggl( \fint_{Q_{k,j}'} v(x)^{\theta_{3}q}\,dx \biggr)^{\frac{1}{\theta_{3}q}} \\ &{}\times\biggl( \fint_{Q_{k,j}'} g(x)^{(\theta_{3}q)'} \,dx \biggr)^{\frac {1}{(\theta_{3}q)'}} \,dy \\ \leq&2A' \int_{E_{k,j}'} \tilde{M}_{\alpha, \theta_{1}, \theta_{3}q} \bigl( f^{\theta_{1}},v \bigr) (y) \cdot M \bigl[ g^{(\theta_{3}q)'} \bigr](y)^{\frac{1}{(\theta_{3}q)'}} \,dy, \end{aligned}

where

$$\tilde{M}_{\alpha, \theta_{1}, \theta_{3}q} \bigl( f^{\theta_{1}},v \bigr) (y):= \sup _{Q\ni y}l(Q)^{\alpha} \biggl( \fint_{3Q} \bigl\vert f(x)\bigr\vert ^{\theta_{1}} \,dx \biggr)^{\frac{1}{\theta_{1}}} \biggl( \fint_{Q} v(x)^{\theta_{3}q} \,dx \biggr)^{\frac{1}{\theta_{3}q}}.$$

A similar argument gives us the following estimate:

$$\mathit{II}_{0}\leq2A' \int_{E_{0}'} \tilde{M}_{\alpha, \theta_{1}, \theta_{3}q} \bigl( f^{\theta_{1}},v \bigr) (y) \cdot M \bigl[ g^{(\theta_{3}q)'} \bigr](y)^{\frac{1}{(\theta_{3}q)'}} \,dy.$$

By summing up $$\mathit{II}_{0}$$ and $$\mathit{II}_{k,j}$$, we obtain

$$\mathit{II}_{0}+\sum_{k,j} \mathit{II}_{k,j} \leq 2A' \int_{Q_{0}} \tilde{M}_{\alpha, \theta_{1}, \theta_{3}q} \bigl( f^{\theta_{1}},v \bigr) (y) \cdot M \bigl[ g^{(\theta_{3}q)'} \bigr](y)^{\frac{1}{(\theta_{3}q)'}} \,dy.$$

By Hölder’s inequality for $$q>1$$, we have

\begin{aligned}& \int_{Q_{0}} \tilde{M}_{\alpha, \theta_{1}, \theta_{3}q} \bigl( f^{\theta_{1}},v \bigr) (y) \cdot M \bigl[ g^{(\theta_{3}q)'} \bigr](y)^{\frac{1}{(\theta_{3}q)'}} \,dy \\& \quad \leq \biggl( \int_{Q_{0}} \tilde{M}_{\alpha, \theta_{1}, \theta_{3}q} \bigl( f^{\theta_{1}},v \bigr) (y)^{q} \,dy \biggr)^{\frac{1}{q}}\cdot \biggl( \int_{Q_{0}} M \bigl[ g^{(\theta_{3}q)'} \bigr](y)^{\frac{q'}{(\theta_{3}q)'}} \,dy \biggr)^{\frac{1}{q'}}. \end{aligned}

Since $$(\theta_{3}q)'< q'$$ and $$\operatorname{supp}(g)\subset Q_{0}$$, by the boundedness of $$M:L^{\frac{q'}{(\theta_{3}q)'}}(\mathbb{R}^{n})\to L^{\frac{q'}{(\theta _{3}q)'}}(\mathbb{R}^{n})$$, we have

$$\biggl( \int_{Q_{0}} M \bigl[ g^{(\theta_{3}q)'} \bigr](y)^{\frac{q'}{(\theta_{3}q)'}} \,dy \biggr)^{\frac{1}{q'}} \leq C \biggl( \int_{Q_{0}} g(x)^{(\theta_{3}q)'\cdot\frac{q'}{(\theta _{3}q)'}} \,dx \biggr)^{\frac{1}{q'}} =C.$$

Therefore we have

$$\int_{Q_{0}} \tilde{M}_{\alpha, \theta_{1}, \theta_{3}q} \bigl( f^{\theta_{1}},v \bigr) (y) \cdot M \bigl[ g^{(\theta_{3}q)'} \bigr](y)^{\frac{1}{(\theta_{3}q)'}} \,dy \leq C \biggl( \int_{Q_{0}} \tilde{M}_{\alpha, \theta_{1}, \theta_{3}q} \bigl( f^{\theta_{1}},v \bigr) (y)^{q} \,dy \biggr)^{\frac{1}{q}}.$$

By Hölder’s inequality for $$\frac{p}{a_{*}\theta_{1}}>1$$ as in Lemma 3, we have

\begin{aligned} \tilde{M}_{\alpha, \theta_{1}, \theta_{3}q} \bigl( f^{\theta_{1}},v \bigr) (x) \leq& C\sup _{Q\ni x} l(Q)^{\alpha} m_{3Q} \bigl( \vert fw \vert ^{\frac {p}{a_{*}}} \bigr)^{\frac{a_{*}}{p}} \\ &{} \times \biggl( \fint_{3Q} w(y)^{-\theta_{1} ( \frac{p}{\theta_{1} a_{*}} )'} \,dy \biggr)^{\frac{1}{\theta_{1}}\frac{1}{ ( \frac{p}{\theta_{1} a_{*}} )'}} \biggl( \fint_{Q} v(y)^{\theta_{3}q} \,dy \biggr)^{\frac{1}{\theta_{3}q}}. \end{aligned}

By Lemma 3, we have $$\theta_{1} ( \frac{p}{\theta_{1}a_{*}} )'\leq ( \frac {p}{a} )'$$. By Hölder’s inequality, we have

\begin{aligned} \tilde{M}_{\alpha, \theta_{1}, \theta_{3}q} \bigl( f^{\theta_{1}},v \bigr) (x) \leq& C\sup _{Q\ni x} \bigl( l(Q)^{\alpha\cdot\frac{p}{a_{*}}} m_{3Q} \bigl( \vert fw \vert ^{\frac{p}{a_{*}}} \bigr) \bigr)^{\frac{a_{*}}{p}} \biggl( \frac{|3Q|}{|Q|} \biggr)^{\frac{1}{aq_{0}}} |3Q|^{-\frac{1}{r_{0}}} \\ &{} \times \biggl( \frac{|Q|}{|3Q|} \biggr)^{\frac{1}{aq_{0}}} |3Q|^{\frac{1}{r_{0}}} \biggl( \fint_{Q} v(y)^{\theta_{3}q} \,dy \biggr)^{\frac{1}{\theta_{3}q}} \biggl( \fint_{3Q} w(y)^{-(p/a)'} \,dy \biggr)^{\frac{1}{(p/a)'}}. \end{aligned}

By the condition (2), we obtain

\begin{aligned} \tilde{M}_{\alpha, \theta_{1}, \theta_{3}q} \bigl( f^{\theta_{1}},v \bigr) (x) &\leq C[v,w]_{aq_{0},r_{0},aq,p/a} \sup_{Q\ni x} \bigl( l(Q)^{\alpha\cdot\frac{p}{a_{*}}-\frac {n}{r_{0}}\cdot\frac{p}{a_{*}}} m_{3Q} \bigl( \vert fw\vert ^{\frac{p}{a_{*}}} \bigr) \bigr)^{\frac {a_{*}}{p}} \\ &= C[v,w]_{aq_{0},r_{0},aq,p/a}\cdot M_{ ( \alpha-\frac {n}{r_{0}} )\frac{p}{a_{*}}} \bigl( |fw|^{\frac{p}{a_{*}}} \bigr) (x)^{\frac{a_{*}}{p}}. \end{aligned}

This implies that

$$|Q_{0}|^{\frac{1}{q_{0}}} \biggl( \fint_{Q_{0}} \tilde{M}_{\alpha, \theta_{1}, \theta_{3}q} \bigl( f^{\theta_{1}},v \bigr) (x)^{q} \,dx \biggr)^{\frac{1}{q}} \leq C [v,w]_{aq_{0},r_{0},aq,p/a} \bigl\Vert M_{ ( \alpha-\frac{n}{r_{0}} )\frac{p}{a_{*}}} \bigl( |fw|^{\frac{p}{a_{*}}} \bigr) \bigr\Vert _{\mathcal{M}_{\frac{a_{*}q}{p}}^{\frac{a_{*}q_{0}}{p}}}^{\frac {a_{*}}{p}}.$$

Since

$$\frac{1}{q_{0}}\cdot\frac{p}{a_{*}} = \frac{1}{p_{0}}\cdot \frac{p}{a_{*}} - \frac{ ( \alpha-\frac{n}{r_{0}} )\cdot\frac{p}{a_{*}}}{n} \quad \text{and}\quad \frac{\frac{a_{*}p_{0}}{p}}{\frac{a_{*}q_{0}}{p}}= \frac{a_{*}}{\frac{a_{*}q}{p}},$$

by Theorem A, we have

$$\bigl\Vert M_{ ( \alpha-\frac{n}{r_{0}} )\frac{p}{a_{*}}} \bigl( |fw|^{\frac{p}{a_{*}}} \bigr) \bigr\Vert _{\mathcal{M}_{\frac{a_{*}q}{p}}^{\frac{a_{*}q_{0}}{p}}}^{\frac {a_{*}}{p}} \leq\bigl\Vert \vert fw\vert ^{\frac{p}{a_{*}}} \bigr\Vert _{\mathcal {M}_{a_{*}}^{\frac{a_{*}p_{0}}{p}}}^{\frac{a_{*}}{p}} =\Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}}.$$

Therefore we have

$$\Vert \mathit{II}\cdot v\Vert _{\mathcal{M}_{q}^{q_{0}}} \leq C [v,w]_{aq_{0},r_{0},aq,p/a} \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}}.$$

### The estimate of B

Since $$|b(x)-b(y)|^{m}\leq2^{m-1} ( | b(x)-m_{Q}(b) |^{m}+| m_{Q}(b)-b(y) |^{m} )$$, we have

\begin{aligned}& \int_{3Q} \bigl\vert b(x)-b(y)\bigr\vert ^{m} \bigl\vert f(y)\bigr\vert \,dy \\& \quad \leq C \int_{3Q} \bigl\vert b(x)-m_{Q}(b)\bigr\vert ^{m} \bigl\vert f(y)\bigr\vert \,dy + C \int_{3Q} \bigl\vert m_{Q}(b)-b(y)\bigr\vert ^{m} \bigl\vert f(y)\bigr\vert \,dy. \end{aligned}

Therefore we obtain

\begin{aligned} v(x) B \leq& C v(x) \sum_{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb{R}^{n})}} |Q|^{ ( \frac{\alpha}{n}-1 )} \chi_{Q} (x) \int_{3Q} \bigl\vert b(x)-m_{Q}(b)\bigr\vert ^{m} \bigl\vert f(y)\bigr\vert \,dy \\ &{}+ C v(x) \sum_{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb{R}^{n})}} |Q|^{ ( \frac{\alpha}{n}-1 )} \chi_{Q} (x) \int_{3Q} \bigl\vert m_{Q}(b)-b(y)\bigr\vert ^{m} \bigl\vert f(y)\bigr\vert \,dy \\ =:& C C_{1}[f,v](x)+C C_{2}[f,v](x). \end{aligned}

By Hölder’s inequality and the definition of the Morrey norm we obtain

\begin{aligned} C_{1}[f,v](x) =& v(x)\sum_{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb{R}^{n})}} |Q|^{ ( \frac{\alpha}{n}-1 )} \chi_{Q} (x) \bigl\vert b(x) - m_{Q} (b) \bigr\vert ^{m} \int_{3Q} \bigl\vert f(y)\bigr\vert \,dy \\ \leq& v(x)\sum_{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb{R}^{n})}} |Q|^{ ( \frac{\alpha}{n}-1 )} \chi_{Q} (x) \bigl\vert b(x) - m_{Q} (b) \bigr\vert ^{m} |3Q|^{\frac{1}{p_{0}}} \biggl( \fint_{3Q} \bigl\vert f(y)\bigr\vert ^{p} w(y)^{p}\,dy \biggr)^{\frac{1}{p}} \\ &{} \times|3Q|^{1-\frac{1}{p_{0}}} \biggl( \fint_{3Q} w(y)^{-p'} \,dy \biggr)^{\frac{1}{p'}} \\ \leq& C \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}} v(x) \sum _{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb{R}^{n})}} |3Q|^{\frac{\alpha}{n}-\frac{1}{p_{0}}} \biggl( \fint_{3Q} w(y)^{-p'} \,dy \biggr)^{\frac{1}{p'}} \bigl\vert b(x)-m_{Q}(b) \bigr\vert ^{m}. \end{aligned}

Since $$\frac{1}{q_{0}}=\frac{1}{p_{0}}+\frac{1}{r_{0}}- \frac{\alpha}{n}$$, the integral of $$C_{1}[f,v](x)^{q}$$ on $$Q_{0}$$ is evaluated as follows:

\begin{aligned}& |Q_{0}|^{\frac{1}{q_{0}}} \biggl( \fint_{Q_{0}} C_{1}[f,v](x)^{q} \,dx \biggr)^{\frac{1}{q}} \\& \quad \leq C \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}} \sum _{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb{R}^{n})}} |Q_{0}|^{\frac{1}{q_{0}}} |3Q|^{\frac{1}{r_{0}}-\frac{1}{q_{0}}} \biggl( \fint_{3Q} w(y)^{-p'} \,dy \biggr)^{\frac{1}{p'}} \\& \qquad {}\times \biggl( \fint_{Q_{0}} v(x)^{q} \bigl\vert b(x)-m_{Q}(b) \bigr\vert ^{mq} \,dx \biggr)^{\frac{1}{q}} \\& \quad = C \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}} \sum_{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb {R}^{n})}} \biggl( \frac{|Q_{0}|}{|3Q|} \biggr)^{\frac{1}{q_{0}}} |3Q|^{\frac{1}{r_{0}}} \biggl( \fint_{3Q} w(y)^{-p'} \,dy \biggr)^{\frac{1}{p'}} \\& \qquad {} \times \biggl( \fint_{Q_{0}} v(x)^{q} \bigl\vert b(x)-m_{Q}(b) \bigr\vert ^{mq} \,dx \biggr)^{\frac{1}{q}}. \end{aligned}

By Hölder’s inequality for $$\theta_{4}>1$$ as in Lemma 3, we have

\begin{aligned}& \biggl( \fint_{Q_{0}} v(x)^{q} \bigl\vert b(x)-m_{Q}(b) \bigr\vert ^{mq} \,dx \biggr)^{\frac{1}{q}} \\& \quad \leq \biggl( \fint_{Q_{0}} v(x)^{q\theta_{4}} \,dx \biggr)^{\frac{1}{q\theta_{4}}} \biggl( \fint_{Q_{0}} \bigl\vert b(x)-m_{Q}(b) \bigr\vert ^{mq\theta_{4}'} \,dx \biggr)^{\frac{1}{q\theta_{4}'}}. \end{aligned}
(6)

We evaluate $$\vert b(x)-m_{Q}(b)\vert$$. If $$Q\supsetneqq Q_{0}$$ and $$Q\in\mathcal{D}(\mathbb{R}^{n})$$, then there exists $$k=1,2,\ldots$$ , such that $$Q_{k}:=Q$$, $$Q_{j}\in\mathcal{D}(\mathbb{R}^{n})$$, $$Q_{j}\supsetneqq Q_{j-1}$$ and $$|Q_{j}|=2^{n}|Q_{j-1}|$$ ($$j=1,2,\ldots,k$$). By the triangle inequality, we obtain

\begin{aligned} \bigl\vert b(x)-m_{Q}(b) \bigr\vert & \leq\bigl\vert b(x)-m_{Q_{0}}(b)\bigr\vert + \bigl\vert m_{Q_{0}}(b)-m_{Q}(b) \bigr\vert \\ & = \bigl\vert b(x)-m_{Q_{0}}(b)\bigr\vert + \Biggl\vert \sum _{j=1}^{k} \bigl( m_{Q_{j-1}}(b)-m_{Q_{j}}(b) \bigr) \Biggr\vert \\ & \leq\bigl\vert b(x)-m_{Q_{0}}(b)\bigr\vert + \sum _{j=1}^{k} \bigl\vert m_{Q_{j-1}}(b)-m_{Q_{j}}(b) \bigr\vert . \end{aligned}

Moreover, we have

\begin{aligned} \bigl\vert m_{Q_{j-1}}(b)-m_{Q_{j}} (b) \bigr\vert &=\biggl\vert \fint_{Q_{j-1}} b(y) \,dy-m_{Q_{j}} (b) \biggr\vert \\ &=\biggl\vert \fint_{Q_{j-1}} \bigl( b(y)- m_{Q_{j}}(b) \bigr) \,dy \biggr\vert \\ &\leq \fint_{Q_{j-1}} \bigl\vert b(y)-m_{Q_{j}}(b) \bigr\vert \,dy \\ & \leq \frac{2^{n}}{|Q_{j}|} \int_{Q_{j}} \bigl\vert b(y)-m_{Q_{j}}(b) \bigr\vert \,dy \\ &\leq2^{n} \Vert b\Vert _{\operatorname{BMO}}\quad (j=1,2, \ldots), \end{aligned}

where we invoke Definition 2 for the last line. By the inequality $$(a+b)^{m}\leq2^{m-1}(a^{m}+b^{m})$$:

\begin{aligned} \bigl\vert b(x)-m_{Q}(b) \bigr\vert ^{m} \leq& \bigl( \bigl\vert b(x)-m_{Q_{0}}(b)\bigr\vert + 2^{n}k \Vert b \Vert _{\operatorname{BMO}} \bigr)^{m} \\ \leq& C \bigl( \bigl\vert b(x)-m_{Q_{0}}(b)\bigr\vert ^{m}+2^{mn} k^{m}\Vert b \Vert _{\operatorname{BMO}}^{m} \bigr). \end{aligned}
(7)

By the estimates (6), (7), and Hölder’s inequality for $$(p/a)'>p'$$, we obtain

\begin{aligned}& \sum_{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb{R}^{n})}} \biggl( \frac{|Q_{0}|}{|3Q|} \biggr)^{\frac{1}{q_{0}}} |3Q|^{\frac{1}{r_{0}}} \biggl( \fint_{3Q} w(y)^{-p'} \,dy \biggr)^{\frac{1}{p'}} \biggl( \fint_{Q_{0}} v(x)^{q} \bigl\vert b(x)-m_{Q}(b) \bigr\vert ^{mq} \,dx \biggr)^{\frac{1}{q}} \\& \quad \leq C \sum_{k=1}^{\infty} \sum _{\substack{Q_{k}\in\mathcal{D}(\mathbb{R}^{n}),\\ Q_{k}\supset Q_{0}, |Q_{k}|=2^{kn}|Q_{0}| }} \biggl( \frac{|Q_{0}|}{|3Q_{k}|} \biggr)^{\frac{1}{q_{0}}} |3Q_{k}|^{\frac{1}{r_{0}}} \biggl( \fint_{3Q_{k}} w(y)^{-(p/a)'} \,dy \biggr)^{\frac{1}{(p/a)'}} \\& \qquad {}\times\biggl( \fint_{Q_{0}} v(x)^{q\theta_{4}} \,dx \biggr)^{\frac{1}{q\theta_{4}}} \biggl( \fint_{Q_{0}} \bigl( \bigl\vert b(x)-m_{Q_{0}}(b)\bigr\vert ^{m}+2^{mn} k^{m}\Vert b\Vert _{\operatorname{BMO}}^{m} \bigr)^{q\theta_{4}'} \,dx \biggr)^{\frac{1}{q\theta_{4}'}}. \end{aligned}

By the triangle inequality on $$L^{q\theta_{4}'}(\mathbb{R}^{n})$$, we obtain

\begin{aligned}& \biggl( \fint_{Q_{0}} \bigl( \bigl\vert b(x)-m_{Q_{0}}(b)\bigr\vert ^{m}+2^{mn} k^{m}\Vert b\Vert _{\operatorname{BMO}}^{m} \bigr)^{q\theta_{4}'} \,dx \biggr)^{\frac{1}{q\theta_{4}'}} \\& \quad \leq \biggl( \fint_{Q_{0}} \bigl\vert b(x)-m_{Q_{0}}(b)\bigr\vert ^{mq\theta_{4}' }\,dx \biggr)^{\frac{1}{q\theta_{4}'}} + \biggl( \fint_{Q_{0}} \bigl( 2^{mn} k^{m}\Vert b \Vert _{\operatorname{BMO}}^{m} \bigr)^{q\theta _{4}'} \,dx \biggr)^{\frac{1}{q\theta_{4}'}}. \end{aligned}
(8)

By the estimate (8), we obtain

\begin{aligned}& \sum_{k=1}^{\infty} \sum _{\substack{Q_{k}\in\mathcal{D}(\mathbb{R}^{n}),\\ Q_{k}\supset Q_{0}, |Q_{k}|=2^{kn}|Q_{0}| }} \biggl( \frac{|Q_{0}|}{|3Q_{k}|} \biggr)^{\frac{1}{q_{0}}} |3Q_{k}|^{\frac{1}{r_{0}}} \biggl( \fint_{3Q_{k}} w(y)^{-(p/a)'} \,dy \biggr)^{\frac{1}{(p/a)'}} \biggl( \fint_{Q_{0}} v(x)^{q\theta_{4}} \,dx \biggr)^{\frac{1}{q\theta_{4}}} \\& \qquad {}\times \biggl( \fint_{Q_{0}} \bigl( \bigl\vert b(x)-m_{Q_{0}}(b)\bigr\vert ^{m}+2^{mn} k^{m}\Vert b\Vert _{\operatorname{BMO}}^{m} \bigr)^{q\theta_{4}'} \,dx \biggr)^{\frac{1}{q\theta_{4}'}} \\& \quad \leq \sum_{k=1}^{\infty} \sum _{\substack{Q_{k}\in\mathcal{D}(\mathbb{R}^{n}),\\ Q_{k}\supset Q_{0}, |Q_{k}|=2^{kn}|Q_{0}| }} \biggl( \frac{|Q_{0}|}{|3Q_{k}|} \biggr)^{\frac{1}{q_{0}}} |3Q_{k}|^{\frac{1}{r_{0}}} \\& \qquad {}\times\biggl( \fint_{3Q_{k}} w(y)^{-(p/a)'} \,dy \biggr)^{\frac{1}{(p/a)'}} \biggl( \fint_{Q_{0}} v(x)^{q\theta_{4}} \,dx \biggr)^{\frac{1}{q\theta_{4}}} \\& \qquad {}\times \biggl\{ \biggl( \fint_{Q_{0}} \bigl\vert b(x)-m_{Q_{0}}(b)\bigr\vert ^{mq\theta _{4}' }\,dx \biggr)^{\frac{1}{q\theta_{4}'}} + \biggl( \fint_{Q_{0}} \bigl( 2^{mn} k^{m}\Vert b \Vert _{\operatorname{BMO}}^{m} \bigr)^{q\theta _{4}'} \,dx \biggr)^{\frac{1}{q\theta_{4}'}} \biggr\} . \end{aligned}

By Lemma 1, we have

$$\biggl( \fint_{Q_{0}} \bigl\vert b(x)-m_{Q_{0}}(b)\bigr\vert ^{mq\theta_{4}' }\,dx \biggr)^{\frac{1}{q\theta_{4}'}} \leq C \Vert b \Vert _{\operatorname{BMO}}^{m}.$$
(9)

The estimate (9) gives us the following:

\begin{aligned}& \biggl( \fint_{Q_{0}} \bigl\vert b(x)-m_{Q_{0}}(b)\bigr\vert ^{mq\theta_{4}' }\,dx \biggr)^{\frac{1}{q\theta_{4}'}} + \biggl( \fint_{Q_{0}} \bigl( 2^{mn} k^{m}\Vert b \Vert _{\operatorname{BMO}}^{m} \bigr)^{q\theta _{4}'} \,dx \biggr)^{\frac{1}{q\theta_{4}'}} \\& \quad \leq C \Vert b\Vert _{\operatorname{BMO}}^{m} \bigl( 1 +2^{mn} k^{m} \bigr). \end{aligned}
(10)

As a consequence of (10), we obtain the following inequality:

\begin{aligned}& \sum_{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb{R}^{n})}} \biggl( \frac{|Q_{0}|}{|3Q|} \biggr)^{\frac{1}{q_{0}}} |3Q|^{\frac{1}{r_{0}}} \biggl( \fint_{3Q} w(y)^{-p'} \,dy \biggr)^{\frac{1}{p'}} \biggl( \fint_{Q_{0}} v(x)^{q} \bigl\vert b(x)-m_{Q}(b) \bigr\vert ^{mq} \,dx \biggr)^{\frac{1}{q}} \\& \quad \leq C \Vert b\Vert _{\operatorname{BMO}}^{m} \sum _{k=1}^{\infty} \sum_{\substack{Q_{k}\in\mathcal{D}(\mathbb{R}^{n}),\\ Q_{k}\supset Q_{0}, |Q_{k}|=2^{kn}|Q_{0}| }} \biggl( \frac{|Q_{0}|}{|3Q_{k}|} \biggr)^{\frac{1}{aq_{0}}} |3Q_{k}|^{\frac{1}{r_{0}}} \biggl( \fint_{3Q_{k}} w(y)^{-(p/a)'} \,dy \biggr)^{\frac{1}{(p/a)'}} \\& \qquad {} \times \biggl( \fint_{Q_{0}} v(x)^{q\theta_{4}} \,dx \biggr)^{\frac{1}{q\theta_{4}}}\bigl( 1+ 2^{mn} k^{m} \bigr) \biggl( \frac{|Q_{0}|}{|3Q_{k}|} \biggr)^{\frac{1}{q_{0}} ( 1-\frac{1}{a} )}. \end{aligned}

By the condition (2), we have

\begin{aligned}& \sum_{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb{R}^{n})}} \biggl( \frac{|Q_{0}|}{|3Q|} \biggr)^{\frac{1}{q_{0}}} |3Q|^{\frac{1}{r_{0}}} \biggl( \fint_{3Q} w(y)^{-p'} \,dy \biggr)^{\frac{1}{p'}} \biggl( \fint_{Q_{0}} v(x)^{q} \bigl\vert b(x)-m_{Q}(b) \bigr\vert ^{mq} \,dx \biggr)^{\frac{1}{q}} \\& \quad \leq C \Vert b\Vert _{\operatorname{BMO}}^{m} [v,w]_{aq_{0},r_{0},aq,p/a} \sum_{k=1}^{\infty} \sum _{\substack{Q_{k}\in\mathcal{D}(\mathbb{R}^{n}),\\ Q_{k}\supset Q_{0}, |Q_{k}|=2^{kn}|Q_{0}| }} \bigl( 1+ 2^{mn} k^{m} \bigr) 2^{-\frac{kn}{q_{0}} ( 1-\frac{1}{a} )} \\& \quad = C \Vert b\Vert _{\operatorname{BMO}}^{m} [v,w]_{aq_{0},r_{0},aq,p/a} \sum_{k=1}^{\infty} \bigl( 1+ 2^{mn} k^{m} \bigr) 2^{-\frac{kn}{q_{0}} ( 1-\frac{1}{a} )} \\& \quad \leq C \Vert b\Vert _{\operatorname{BMO}}^{m} [v,w]_{aq_{0},r_{0},aq,p/a} . \end{aligned}

Therefore we obtain

$$\bigl\Vert C_{1}[f,v] \bigr\Vert _{\mathcal{M}_{q}^{q_{0}}}\leq C [v,w]_{aq_{0},r_{0},aq,p/a} \Vert b\Vert _{\operatorname{BMO}}^{m} \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}}.$$
(11)

Next, we evaluate $$C_{2}[f,v](x)$$. By Hölder’s inequality for $$\theta_{5}\in(1,p)$$ in Lemma 3, we have

\begin{aligned} C_{2}[f,v](x) =&v(x) \sum_{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb {R}^{n})}} l(Q)^{\alpha-n} \chi_{Q}(x) \biggl( \int_{3Q} \bigl\vert m_{Q}(b)-b(y) \bigr\vert ^{m} f(y) \,dy \biggr) \\ \leq& v(x) \sum_{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb{R}^{n})}} l(Q)^{\alpha-n} \chi_{Q}(x) \biggl( \int_{3Q} \bigl\vert m_{Q}(b)-b(y) \bigr\vert ^{m\theta_{5}' } \,dy \biggr)^{\frac{1}{\theta_{5}'}} \\ &{}\times\biggl( \int_{3Q} \bigl\vert f(y)\bigr\vert ^{\theta_{5}} \,dy \biggr)^{\frac{1}{\theta_{5}}}. \end{aligned}

By Hölder’s inequality for $$\frac{p}{\theta_{5}}>1$$, we obtain

\begin{aligned} C_{2}[f,v](x) \leq& v(x) \sum_{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb{R}^{n})}} l(Q)^{\alpha-\frac{n}{p_{0}}} \chi_{Q}(x) \biggl( \fint_{3Q} \bigl\vert m_{Q}(b)-b(y) \bigr\vert ^{m\theta_{5}' } \,dy \biggr)^{\frac{1}{\theta_{5}'}} |3Q|^{\frac{1}{p_{0}}} \\ &{} \times \biggl( \fint_{3Q} w(y)^{-\theta_{5} ( \frac{p}{\theta_{5}} )'}\,dy \biggr)^{\frac{1}{\theta_{5} ( \frac{p}{\theta_{5}} )'}} \biggl( \fint_{3Q} \bigl\vert f(y)\bigr\vert ^{p}w(y)^{p} \,dy \biggr)^{\frac{1}{p}}. \end{aligned}

Taking the Morrey norm, we obtain

\begin{aligned} C_{2}[f,v](x) \leq&\Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}} v(x) \sum _{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb{R}^{n})}} l(Q)^{\alpha-\frac{n}{p_{0}}} \chi_{Q}(x) \biggl( \fint_{3Q} \bigl\vert m_{Q}(b)-b(y) \bigr\vert ^{m\theta_{5}' } \,dy \biggr)^{\frac{1}{\theta _{5}'}} \\ &{} \times \biggl( \fint_{3Q} w(y)^{-\theta_{5} ( \frac{p}{\theta _{5}} )'}\,dy \biggr)^{\frac{1}{\theta_{5} ( \frac{p}{\theta _{5}} )'}}. \end{aligned}

Using Lemma 1, we have

$$C_{2}[f,v](x) \leq \Vert b\Vert _{\operatorname{BMO}}^{m} \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}} v(x) \sum_{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb{R}^{n})}} l(Q)^{\alpha-\frac{n}{p_{0}}} \chi_{Q}(x) \biggl( \fint_{3Q} w(y)^{-\theta_{5} ( \frac{p}{\theta_{5}} )'}\,dy \biggr)^{\frac{1}{\theta_{5} ( \frac{p}{\theta_{5}} )'}}.$$

Since we have the assumption that $$a\geqq\frac{p}{ ( \theta_{5} ( \frac{p}{\theta_{5}} )' )'}>1$$, using Hölder’s inequality, we obtain

$$C_{2}[f,v](x) \leq C \Vert b\Vert _{\operatorname{BMO}}^{m} \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}} v(x) \sum_{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb{R}^{n})}} l(Q)^{\alpha-\frac{n}{p_{0}}} \chi_{Q}(x) \biggl( \fint_{3Q} w(y)^{-(p/a)'}\,dy \biggr)^{\frac{1}{(p/a)'}}.$$

The integral of $$C_{2}[f,v](x)^{q}$$ on $$Q_{0}$$ is evaluated as follows:

\begin{aligned}& |Q_{0}|^{\frac{1}{q_{0}}} \biggl( \fint_{Q_{0}} C_{2}[f,v](x)^{q} \,dx \biggr)^{\frac{1}{q}} \\& \quad \leq C \Vert b\Vert _{\operatorname{BMO}}^{m} \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}} \sum_{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb{R}^{n})}} l(Q)^{\alpha-\frac{n}{p_{0}}} |Q_{0}|^{\frac{1}{q_{0}}} \\& \qquad {}\times\biggl( \fint_{Q_{0}} v(x)^{q} \,dx \biggr)^{\frac{1}{q}} \biggl( \fint_{3Q} w(y)^{-(p/a)'} \,dy \biggr)^{\frac{1}{(p/a)'}} \\& \quad \leq C \Vert b\Vert _{\operatorname{BMO}}^{m} \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}} \sum_{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb{R}^{n})}} \biggl( \frac{|Q_{0}|}{|3Q|} \biggr)^{\frac{1}{aq_{0}}} |3Q|^{\frac{1}{r_{0}}} \\& \qquad {}\times\biggl( \fint_{Q_{0}} v(x)^{aq} \,dx \biggr)^{\frac{1}{aq}} \biggl( \fint _{3Q} w(y)^{-(p/a)'} \,dy \biggr)^{\frac{1}{(p/a)'}} \biggl( \frac{|Q_{0}|}{|3Q|} \biggr)^{\frac{1}{q_{0}} ( 1-\frac{1}{a} )}. \end{aligned}

By the condition (2), we have

\begin{aligned}& |Q_{0}|^{\frac{1}{q_{0}}} \biggl( \fint_{Q_{0}} C_{2}[f,v](x)^{q} \,dx \biggr)^{\frac{1}{q}} \\& \quad \leq C [v,w]_{aq_{0},r_{0},aq,p/a} \Vert b\Vert _{\operatorname{BMO}}^{m} \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}} \sum_{\substack{Q\supsetneq Q_{0},\\ Q\in\mathcal{D}(\mathbb{R}^{n})}} \biggl( \frac{|Q_{0}|}{|3Q|} \biggr)^{\frac{1}{q_{0}} ( 1-\frac {1}{a} )} \\& \quad \leq C [v,w]_{aq_{0},r_{0},aq,p/a} \Vert b\Vert _{\operatorname{BMO}}^{m} \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}}. \end{aligned}

We obtain the desired result.  □

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## Acknowledgements

The author wish to thank Professor Y Sawano for a rich lecture of higher order commutators generated by BMO-functions and the fractional integral operator.

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Correspondence to Takeshi Iida. 