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.

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 â€˜weightâ€™ 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)

HÃ¶lderâ€™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 TheoremÂ 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 â€˜$$n-\alpha$$ is an even integerâ€™.

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 TheoremÂ 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 TheoremÂ E.

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

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

CorollaryÂ 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

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

CorollaryÂ 3 and TheoremÂ 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 TheoremÂ 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 TheoremÂ 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 [10â€“12] and [4]).

Definition 5

Let $$0<\alpha<n$$, a measurable function Î© on $$\mathbb{R}^{n}\backslash\{ 0\}$$ and a measurable functionÂ 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 TheoremÂ 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 TheoremÂ 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]. TheoremÂ 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 [14â€“16]. 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 TheoremÂ  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

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

â€ƒâ–¡

Remark 5

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

4 Proof of Theorem 1

Proof of TheoremÂ 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 SectionsÂ 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 LemmaÂ 2. By HÃ¶lderâ€™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 LemmaÂ 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 LemmaÂ 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 HÃ¶lderâ€™s inequality for $$\theta_{2}>1$$ as in LemmaÂ 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 LemmaÂ 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 LemmaÂ 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 LemmaÂ 3. By HÃ¶lderâ€™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 HÃ¶lderâ€™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 HÃ¶lderâ€™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 LemmaÂ 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 HÃ¶lderâ€™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 HÃ¶lderâ€™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 TheoremÂ 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 LemmaÂ 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 HÃ¶lderâ€™s inequality for $$\theta_{3}>1$$ as in LemmaÂ 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 LemmaÂ 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 HÃ¶lderâ€™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 HÃ¶lderâ€™s inequality for $$\frac{p}{a_{*}\theta_{1}}>1$$ as in LemmaÂ 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 LemmaÂ 3, we have $$\theta_{1} ( \frac{p}{\theta_{1}a_{*}} )'\leq ( \frac {p}{a} )'$$. By HÃ¶lderâ€™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 TheoremÂ 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 HÃ¶lderâ€™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 HÃ¶lderâ€™s inequality for $$\theta_{4}>1$$ as in LemmaÂ 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 $$k=1,2,\ldots$$â€‰, 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 DefinitionÂ 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 HÃ¶lderâ€™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 LemmaÂ 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 HÃ¶lderâ€™s inequality for $$\theta_{5}\in(1,p)$$ in LemmaÂ 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 HÃ¶lderâ€™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 LemmaÂ 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 HÃ¶lderâ€™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. â€ƒâ–¡

References

1. Iida, T, Sato, E, Sawano, Y, Tanaka, H: Weighted norm inequalities for multilinear fractional operators on Morrey spaces. Stud. Math. 205, 139-170 (2011)

2. Sawano, Y, Sugano, S, Tanaka, H: A bilinear estimate for commutators of fractional integral operators. In: Potential Theory and Its Related Fields. RIMS KÃ´kyÃ»roku Bessatsu, vol.Â B43, pp.Â 155-170. Res. Inst. Math. Sci., Kyoto (2013)

3. Grafakos, L: Modern Fourier Analysis, 2nd edn. Graduate Texts in Math., vol.Â 250. Springer, New York (2008)

4. Lu, S, Ding, Y, Yan, D: Singular Integrals and Related Topics. World Scientific, Singapore (2007)

5. Adams, D: A note on Riesz potentials. Duke Math. J. 42, 765-778 (1975)

6. Di-Fazio, G, Ragusa, MA: Commutators and Morrey spaces. Boll. UMI 7, 323-332 (1991)

7. Komori, Y, Mizuhara, T: Notes on commutators and Morrey spaces. Hokkaido Math. J. 32, 345-353 (2003)

8. Iida, T, Sawano, Y, Tanaka, H: Atomic decomposition for Morrey spaces. Z. Anal. Anwend. 33, 149-170 (2014)

9. Sawano, Y, Sugano, S, Tanaka, H: Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces. Trans. Am. Math. Soc. 363, 6481-6503 (2011)

10. Ding, Y, Lu, S: Weighted norm inequalities for fractional integral operators with rough kernel. Can. J. Math. 50, 29-39 (1998)

11. Ding, Y, Lu, S: Higher order commutators for a class of rough operators. Ark. Mat. 37, 33-44 (1999)

12. Iida, T: Weighted inequalities on Morrey spaces for linear and multilinear fractional integrals with homogeneous kernels. Taiwan. J. Math. 18, 147-185 (2014)

13. Duoandikoetxea, J: Fourier Analysis. Grad. Studies in Math., vol.Â 29. Am. Math. Soc., Providence (2001)

14. PÃ©rez, C: Two weighted inequalities for potential and fractional type maximal operators. Indiana Univ. Math. J. 43, 663-683 (1994)

15. PÃ©rez, C: On sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between $$L^{p}$$ spaces with different weights. Proc. Lond. Math. Soc. (3) 71, 135-157 (1995)

16. PÃ©rez, C: Sharp $$L^{p}$$-weighted Sobolev inequalities. Ann. Inst. Fourier (Grenoble) 45, 809-824 (1995)

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.

Author information

Authors

Corresponding author

Correspondence to Takeshi Iida.

Competing interests

The author declares that they have no competing interests.

Rights and permissions

Reprints and permissions

Iida, T. Weighted estimates of higher order commutators generated by BMO-functions and the fractional integral operator on Morrey spaces. J Inequal Appl 2016, 4 (2016). https://doi.org/10.1186/s13660-015-0953-4