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

Journal of Inequalities and Applications20162016:4

https://doi.org/10.1186/s13660-015-0953-4

  • Received: 16 September 2015
  • Accepted: 21 December 2015
  • Published:

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.

Keywords

  • commutator
  • fractional integral operator
  • Morrey space
  • BMO space
  • weight

MSC

  • 26A33
  • 42B25
  • 42B35

1 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 [1] and [2]. The authors introduced the condition of weights in [1]. 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 [2].

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 [1]):
    $$\sup_{\substack{Q\subset\mathbb{R}^{n},\\Q\text{: cubes}}} |Q|^{\frac {1}{p_{0}}} \biggl( \fint_{Q} \bigl\vert f(x)\bigr\vert ^{p} \,dx \biggr)^{\frac{1}{p}}\cong \Vert f\Vert _{\mathcal{M}_{p}^{p_{0}}(\mathbb{R}^{n})}. $$
     
  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 [5]). 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 [6] 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 [7] 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. [2] 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 [1], 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.

2 Main results and their corollaries

In this paper, we obtain two main theorems.

2.1 One of the main results

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 [2], 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 [2], 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
$$\bigl\Vert \bigl( [ \vec{b}, I_{\alpha} ] f \bigr) v \bigr\Vert _{\mathcal{M}_{q}^{q_{0}}} \leq C \Biggl( \prod_{j=1}^{m} \Vert b_{j} \Vert _{\operatorname {BMO}} \Biggr) [ v,w ]_{aq_{0}, r_{0},aq,p/a} \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}}. $$

2.2 Fractional integral operators having rough kernel

We define the following operators (see [1012] and [4]).

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

3 Some lemmas

In this section, we prepare some lemmas for proving main results. We recall the following inequalities (see [3, 13] and [4]).

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 [1416]. 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.

4 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.

4.1 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
$$\Vert \mathit{II}\cdot v\Vert _{\mathcal{M}_{q}^{q_{0}}} \leq C [v,w]_{aq_{0},r_{0},aq,p/a} \Vert fw \Vert _{\mathcal{M}_{p}^{p_{0}}}. $$

4.2 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.  □

Declarations

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.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of General Education, National Institute of Technology, Fukushima College, Fukushima 970-8034, Japan

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